American Mathematics Competition 8 - 2013

Problem 1

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars.

What is the smallest number of additional cars she must buy in order to be able to arrange her cars in this way?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer :

(A) 1

Problem 2

A sign at the fish market says, " $50 \%$ off, today only: half-pound packages for just $\$ 3$ per package."

What is the regular price for a full pound of fish, in dollars?
(A) 6
(B) 9
(C) 10
(D) 12
(E) 15

Answer:

(D) 12

Problem 3

What is the value of $4 \cdot(-1+2-3+4-5+6-7+\cdots+1000)$ ?
(A) -10
(B) 0
(C) 1
(D) 500
(E) 2000

Answer:

(E) 2000

Problem 4

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money,

each of her seven friends paid an extra $\$ 2.50$ to cover her portion of the total bill. What was the total bill?
(A) $\$ 120
(B) $\$ 128
(C) $\$ 140
(D) $\$ 144
(E) $\$ 160

Answer:

Problem 5

Hammie is in the $6^{\text {th }}$ grade and weighs 106 pounds.

His quadruplet sisters are tiny babies and weigh $5,5,6$, and 8 pounds.

Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
(A) median, by 60
(B) median, by 20
(C) average, by 5
(D) average, by 15
(E) average, by 20

Answer:

(E) average, by 20

Problem 6

The number in each box below is the product of the numbers in the two boxes that touch it in the row above.

For example, $30=6 \times 5$. What is the missing number in the top row?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer:

Peoblem 7

Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass,

Trey counted 6 cars in the first 10 seconds.

It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed.

Which of the following was the most likely number of cars in the train?
(A) 60
(B) 80
(C) 100
(D) 120
(E) 140

Answer:

Problem 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
(A) 1/8
(B) 1/4
(C) 3/8
(D) 1/2
(E) 3/4

Answer:

Problem 9
The Incredible Hulk can double the distance he jumps with each succeeding jump.

If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on,

then on which jump will he first be able to jump more than 1 kilometer?
(A) 9th
(B) 10th
(C) 11th
(D) 12th
(E) 13th

Answer:

Problem 10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
(A) 110
(B) 165
(C) 330
(D) 625
(E) 660

Answer:

Problem 11
Ted's grandfather used his treadmill on 3 days this week.

He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour.

He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday.

If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill.

How many minutes less?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

(D) 4

Problem 12
At the 2013 Winnebago Country Fair a vendor is offering a "fair special" on sandals.

If you buy one pair of sandals at the regular price of $\$ 50$, you get a second pair at a $40 \%$ discount,

and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals.

What percentage of the $\$ 150$ regular price did he save?
(A) 25
(B) 30
(C) 33
(D) 40
(E) 45

Answer:

(B) 30

Problem 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score.

By which of the following might her incorrect sum have differed from the correct one?
(A) 45
(B) 46
(C) 47
(D) 48
(E) 49

Answer:

(A) 45

Problem 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand.

Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
(A) 1/4
(B) 1/3
(C) 3/8
(D) 1/2
(E) 2/3

Answer:

Problem 15
If $3^{p}+3^{4}=90,2^{r}+44=76$, and $5^{3}+6^{s}=1421$, what is the product of $p, r$, and $s$ ?
(A) 27
(B) 40
(C) 50
(D) 70
(E) 90

Answer:

(B) 40

Problem 16
A number of students from Fibonacci Middle School are taking part in a community service project.

The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $5: 3$, and the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$-graders is $8: 5$.

What is the smallest number of students that could be participating in the project?
(A) 16
(B) 40
(C) 55
(D) 79
(E) 89

Answer:

(E) 89

Problem 17
The sum of six consecutive positive integers is 2013 . What is the largest of these six integers?
(A) 335
(B) 338
(C) 340
(D) 345
(E) 350

Answer:

(B) 338

Problem 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high.

The floor and the four walls are all one foot thick. How many blocks does the fort contain?
(A) 204
(B) 280
(C) 320
(D) 340
(E) 600

Answer:

(B) 280

Problem 19
Bridget, Cassie, and Hannah are discussing the results of their last math test.

Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show their tests to anyone.

Cassie says, "I didn't get the lowest score in our class," and Bridget adds, "I didn't get the highest score.

" What is the ranking of the three girls from highest to lowest?
(A) Hannah, Cassie, Bridget
(B) Hannah, Bridget, Cassie
(C) Cassie, Bridget, Hannah
(D) Cassie, Hannah, Bridget
(E) Bridget, Cassie, Hannah

Answer:

(D) Cassie, Hannah, Bridget

Problem 20
A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter.

What is the area of the semicircle?
(A) $\frac{\pi}{2}$
(B) $\frac{2 \pi}{3}$
(C) $\pi$
(D) $\frac{4 \pi}{3}$
(E) $\frac{5 \pi}{3}$

Answer:

Problem 21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park.

Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park.

On school days she bikes on streets to the southwest corner of City Park,

then takes a diagonal path through the park to the northeast corner of City Park, and then bikes on streets to school.

If her route is as short as possible, how many different routes can she take?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 18

Answer:

(E) 18

Problem 22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks high.

How many toothpicks are used altogether?
(A) 1920
(B) 1952
(C) 1980
(D) 2013
(E) 3932

Answer:

(E) 3932

Problem 23
Angle $A B C$ of $\triangle A B C$ is a right angle.

The sides of $\triangle A B C$ are the diameters of semicircles as shown.

The area of the semicircle on $\overline{A B}$ equals $8 \pi$,

and the arc of the semicircle on $\overline{A C}$ has length $8.5 \pi$.

What is the radius of the semicircle on $\overline{B C}$ ?
(A) 7
(B) 7.5
(C) 8
(D) 8.5
(E) 9

Answer:

(B) 7.5

Problem 24
Squares $A B C D, E F G H$, and $G H I J$ are equal in area. Points $C$ and $D$ are the midpoints of sides $I H$ and $H E$, respectively. What is the ratio of the area of the shaded pentagon $A J I C B$ to the sum of the areas of the three squares?

(A) $\frac{1}{4}$
(B) $\frac{7}{24}$
(C) $\frac{1}{3}$
(D) $\frac{3}{8}$
(E) $\frac{5}{12}$

Answer:

Problem 25
A ball with diameter 4 inches starts at point $A$ to roll along the track shown.

The track is comprised of 3 semicircular arcs whose radii are $R_{1}=100$ inches,

$R_{2}=60$ inches, and $R_{3}=80$ inches, respectively.

The ball always remains in contact with the track and does not slip.

What is the distance in inches the center of the ball travels over the course from $A$ to $B$ ?
(A) $238 \pi$
(B) $240 \pi$
(C) $260 \pi$
(D) $280 \pi$
(E) $500 \pi$

Answer:

(A) $238 \pi$

American Mathematics Competition 8 - 2008

Problem 1

Susan had $\$ 50$ to spend at the carnival. She spent $\$ 12$ on food and twice as much on rides. How many dollars did she have left to spend?
(A) 12
(B) 14
(C) 26
(D) 38
(E) 50

Answer : B

Problem 2

The ten-letter code BEST OF LUCK represents the ten digits $0-9$, in order. What 4 -digit number is represented by the code word CLUE?
(A) 8671
(B) 8672
(C) 9781
(D) 9782
(E) 9872

Answer : A

Problem 3

If February is a month that contains Friday the $13^{\text {th }}$, what day of the week is February 1?
(A) Sunday
(B) Monday
(C) Wednesday
(D) Thursday
(E) Saturday

Answer : A

Problem 4

In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1 , and the three trapezoids are congruent. What is the area of one of the trapezoids?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer : C

Problem 5

Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661. What was his average speed in miles per hour?
(A) 15
(B) 16
(C) 18
(D) 20
(E) 22

Answer : E

Problem 6
In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
(A) $3: 10$
(B) $3: 8$
(C) $3: 7$
(D) $3: 5$
(E) $1: 1$

Answer : D

Problem 7

If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$ ?
(A) 27
(B) 29
(C) 45
(D) 105
(E) 127

Answer : E

Problem 8
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
(A) 60
(B) 70
(C) 75
(D) 80
(E) 85

Answer : D

Problem 9
In 2005 Tycoon Tammy invested Dollar $100$ for two years. During the the first year her investment suffered a $15 Dollar $ loss, but during the second year the remaining investment showed a $20 $ Dollar gain. Over the two-year period, what was the change in Tammy's investment?
(A) 5 Dollar loss
(B) 2 Dollar loss
(C) 1 Dollar gain
(D) 2 Dollar gain
(E) 5 Dollar gain

Answer: D

Problem 10
The average age of the 6 people in Room A is 40 . The average age of the 4 people in Room B is 25. If the two groups are combined, what is the average age of all the people?
(A) 32.5
(B) 33
(C) 33.5
(D) 34
(E) 35

Answer : D

11. Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
(A) 7
(B) 13
(C) 19
(D) 39
(E) 46

Answer : A

Problem 12
A ball is dropped from a height of 3 meters. On its first bounce it rises to a height of 2 meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of 0.5 meters?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer : C

Problem 13
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122,125 and 127 pounds. What is the combined weight in pounds of the three boxes?
(A) 160
(B) 170
(C) 187
(D) 195
(E) 354

Answer : C

Problem 14
Three A's, three B's, and three C's are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer : C

Problem 15

In Theresa's first 8 basketball games, she scored $7,4,3,6,8,3,1$ and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
(A) 35
(B) 40
(C) 48
(D) 56
(E) 72

Answer : B

Problem 16

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?

(A) $1: 6$
(B) $7: 36$
(C) $1: 5$
(D) $7: 30$
(E) $6: 25$

Answer : D

Problem 17

Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
(A) 76
(B) 120
(C) 128
(D) 132
(E) 136

Answer : D

Problem 18

Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?

(A) $10 \pi+20$
(B) $10 \pi+30$
(C) $10 \pi+40$
(D) $20 \pi+20$
(E) $20 \pi+40$

Answer : E

Problem 19

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?

(A) $\frac{1}{4}$
(B) $\frac{2}{7}$
(C) $\frac{4}{11}$
(D) $\frac{1}{2}$
(E) $\frac{4}{7}$

Answer : B

Problem 20

The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
(A) 12
(B) 17
(C) 24
(D) 27
(E) 36

Answer : B

Problem 21

Jerry cuts a wedge from a $6-\mathrm{cm}$ cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?

(A) 48
(B) 75
(C) 151
(D) 192
(E) 603

Answer : C

Problem 22

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3 n$ three-digit whole numbers?
(A) 12
(B) 21
(C) 27
(D) 33
(E) 34

Answer : A

Problem 23
In square $A B C E, A F=2 F E$ and $C D=2 D E$. What is the ratio of the area of $\triangle B F D$ to the area of square $A B C E$ ?

(A) $\frac{1}{6}$
(B) $\frac{2}{9}$
(C) $\frac{5}{18}$
(D) $\frac{1}{3}$
(E) $\frac{7}{20}$

Answer : C

Problem 24
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
(A) $\frac{1}{10}$
(B) $\frac{1}{6}$
(C) $\frac{11}{60}$
(D) $\frac{1}{5}$
(E) $\frac{7}{30}$

Answer : C

Problem 25
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?

(A) 42
(B) 44
(C) 45
(D) 46
(E) 48

Answer : A

AMERICAN MATHEMATICS COMPETITION 8 - 2014

Problem 1

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$ ?
(A) -10
(B) -6
(C) 0
(D) 6
(E) 10

Answer

(A) -10

Problem 2

Paul owes Paula 35 cents and has a pocket full of 5 -cent coins, 10 -cent coins, and 25 -cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer

(E) 5

Problem 3

Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
(A) 240
(B) 250
(C) 260
(D) 270
(E) 280

Answer

(B) 250 pages.

Problem 4

The sum of two prime numbers is 85 . What is the product of these two prime numbers?
(A) 85
(B) 91
(C) 115
(D) 133
(E) 166

Answer

(E) 166.

Problem 5

Margie's car can go 32 miles on a gallon of gas, and gas currently costs $\$ 4$ per gallon. How many miles can Margie drive on $\$ 20$ worth of gas?
(A) 64
(B) 128
(C) 160
(D) 320
(E) 640

Answer

Problem 6

Six rectangles each with a common base width of 2 have lengths of $1,4,9,16,25$, and 36 . What is the sum of the areas of the six rectangles?
(A) 91
(B) 93
(C) 162
(D) 182
(E) 202

Answer

(D) 182.

Problem 7

There are four more girls than boys in Ms. Raub's class of 28 students. What is the ratio of number of girls to the number of boys in her class?
(A) $3: 4$
(B) $4: 3$
(C) $3: 2$
(D) $7: 4$
(E) $2: 1$

Answer

(B) $4: 3$

Problem 8

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\$ \underline{1 A 2}$. What is the missing digit $A$ of this 3 -digit number?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer

(D) 3 $\sim$ fn106068.

Problem 9

Answer

(D) 140.

Problem 10

The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8 . In what year was Samantha born?
(A) 1979
(B) 1980
(C) 1981
(D) 1982
(E) 1983

Answer

(A) $1979 \sim$ SweetMango77
corrections made by DrDominic.

Problem 11

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
(A) 4
(B) 5
(C) 6
(D) 8
(E) 10

Answer

(A) 4 is the correct answer.

Problem 12

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?

Answer

(B) Is the correct answer.

Problem 13

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

$\textbf{(A) }$ $n$ and $m$ are even $\qquad\textbf{(B) }$ $n$ and $m$ are odd $\qquad\textbf{(C) }$ $n+m$ is even $\qquad\textbf{(D) }$ $n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

Answer

(D) is odd.

Problem 14

Rectangle

and right triangle

have the same area. They are joined to form a trapezoid, as shown. What is

Answer

(B) Is the correct answer.

Problem 15

The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$ ?

(A) 75

(B) 80

(D) 120

(E) 150

Answer

Problem 16

The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against nonconference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

(A) 60

(B) 88

(D) 144

(E) 160

Answer

(B) 88.

Problem 17

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?

(A) 4

(B) 6

(D) 10

(E) 12

Answer

(B) 6.

Problem 18

Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?

$(\mathbf{A})$ all 4 are boys $(\mathbf{B})$ all 4 are girls $(\mathbf{C})_{2}$ are girls and 2 are boys $(\mathbf{D})_{3}$ are of one gender and 1 is of the other gender $\boldsymbol{(} \mathbf{E} \boldsymbol{)}$ all of these outcomes are equally likely

Answer

(D) Is the correct answer.

Problem 19

A cube with 3 -inch edges is to be constructed from 27 smaller cubes with 1 -inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3 -inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

(A) $\frac{5}{54}$

(B) $\frac{1}{9}$

(D) $\frac{2}{9}$

(E) $\frac{1}{3}$

Answer

(A) $\frac{5}{54}$

Problem 20

Rectangle ABCD has sides $\mathrm{CD}=3$ and $\mathrm{DA}=5$. A circle of radius 1 is centered at A , a circle of radius 2 is centered at B , and a circle of radius 3 is centered at C . Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

(A) 3.5

(B) 4.0

(D) 5.0

(E) 5.5

Answer

(B) 4.0

Problem 21

The 7-digit numbers $\underline{74 A 52 B 1}$ and $\underline{326 A B 4 C}$ are each multiples of 3 . Which of the following could be the value of $C$ ?

(A) 1

(B) 2

(D) 5

(E) 8

Answer

(A) 1.

Problem 22

A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?

(A) 1

(B) 3

(D) 7

(E) 9

Answer

(E) 9.

Problem 23

Three members of the Euclid Middle School girls' softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all 2 -digit primes.

Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today's date.

What number does Caitlin wear?

(A) 11

(B) 13

(D) 19

(E) 23

Answer

(A) 11.

Problem 24

One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

(A) 2.5

(B) 3.0

(D) 4.0

(E) 4.5

Answer

Problem 25

A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?

Note: 1 mile= 5280 feet

(A) $\frac{\pi}{11}$

(B) $\frac{\pi}{10}$

(D) $\frac{2 \pi}{5}$

(E) $\frac{2 \pi}{3}$

Answer

(B) $\frac{\pi}{10}$

AMERICAN MATHEMATICS COMPETITION 8 - 2023

PROBLEM 1 :

What is the value of $(8 \times 4+2)-(8+4 \times 2)$ ?
(A) 0
(B) 6
(C) 10
(D) 18
(E) 24

ANSWER :

(D) 18

PROBLEM 2 :

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?

ANSWER :

(E)

PROBLEM 3 :

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation

$$
(\text { wind chill })=(\text { air temperature })-0.7 \times(\text { wind speed }),
$$

where temperature is measured in degrees Fahrenheit ( ${ }^{\circ} \mathrm{F}$ ) and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ} \mathrm{F}$ and the wind speed is 18 mph . Which of the following is closest to the approximate wind chill?
(A) 18
(B) 23
(C) 28
(D) 32
(E) 35

ANSWER :

(B) 23

PROBLEM 4 :

The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7 . How many of these four numbers are prime?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

(D) 3

PROBLEM 5 :

A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
(A) 1250
(B) 1500
(C) 1750
(D) 1800
(E) 2000

ANSWER :

(B) 1500

PROBLEM 6 :

The digits $2,0,2$, and 3 are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

(A) 0
(B) 8
(C) 9
(D) 16
(E) 18

ANSWER :

PROBLEM 7 :

A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15,3)$ and $(16,5)$. A line is drawn through points $A(0,0)$ and $B(3,1)$. Another line is drawn through points $C(0,10)$ and $D(2,9)$. How many points on the rectangle lie on at least one of the two lines?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

(B) 1

PROBLEM 8 :

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo's win-loss record?

(A) 000101
(B) 001001
(C) 010000
(D) 010101
(E) 011000

SOLUTION :

(A) 000101

PROBLEM 9 :

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between 4 and 7 meters?

(A) 6
(B) 8
(C) 10
(D) 12
(E) 14

ANSWER :

(B) 8

PROBLEM 10 :

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?
(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{5}{12}$

ANSWER :

(D) $\frac{1}{3}$

PROBLEM 11 :

NASA's Perseverance Rover was launched on July 30 , 2020. After traveling $292,526,838$ miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
(A) 6,000
(B) 12,000
(C) 60,000
(D) 120,000
(E) 600,000

ANSWER :

PROBLEM 12 :

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

(A) $\frac{1}{4}$
(B) $\frac{11}{36}$
(C) $\frac{1}{3}$
(D) $\frac{19}{36}$
(E) $\frac{5}{9}$

ANSWER :

(B) $\frac{11}{36}$

PROBLEM 13 :

Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the start and finish lines. The 3 rd water station is located 2 miles after the 1 st repair station. How long is the race in miles?

(A) 8
(B) 16
(C) 24
(D) 48
(E) 96

ANSWER :

(D) 48

PROBLEM 14 :

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5 -cent, 10 -cent, and 25 -cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$ 7.10$ in postage? (Note: The amount $\$ 7.10$ corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
(A) 45
(B) 46
(C) 51
(D) 54
(E) 55

ANSWER :

(E) 55

PROBLEM 15 :

Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? Here's a hint… if you aren't correct, think about using conversions, maybe that's why you're wrong! -RyanZ4552

(A) 4
(B) 4.2
(C) 4.5
(D) 4.8
(E) 5

ANSWER :

(B) 4.2

PROBLEM 16 :

The letters $\mathrm{P}, \mathrm{Q}$, and R are entered into a $20 \times 20$ table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?

(A) 132 Ps, $134 \mathrm{Qs}, 134 \mathrm{Rs}$
(B) $133 \mathrm{Ps}, 133 \mathrm{Qs}, 134 \mathrm{Rs}$
(C) $133 \mathrm{Ps}, 134 \mathrm{Qs}, 133 \mathrm{Rs}$
(D) $134 \mathrm{Ps}, 132 \mathrm{Qs}, 134 \mathrm{Rs}$
(E) $134 \mathrm{Ps}, 133 \mathrm{Qs}, 133 \mathrm{Rs}$

ANSWER :

PROBLEM 17 :

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$ ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER :

(A) 1

PROBLEM 18 :

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5 pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located 2023 pads to the right of her starting position?
(A) 405
(B) 407
(C) 409
(D) 411
(E) 413

ANSWER :

(D) 411

PROBLEM 19 :

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

(A) $1: 3$
(B) $3: 8$
(C) $5: 12$
(D) $7: 16$
(E) $4: 9$

ANSWER :

PROBLEM 20 :

Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
(A) 56
(B) 57
(C) 58
(D) 60
(E) 61

ANSWER :

(D) 60

PROBLEM 21 :

Alina writes the numbers $1,2, \ldots, 9$ on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

PROBLEM 22 :

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is 4000 . What is the first term?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 10

ANSWER :

(D) 5

PROBLEM 23 :

Each square in a $3 \times 3$ grid is randomly filled with one of the 4 gray and white tiles shown below on the right.

What is the probability that the tiling will contain a large gray diamond in one of the smaller $2 \times 2$ grids? Below is an example of such tiling.

(A) $\frac{1}{1024}$
(B) $\frac{1}{256}$
(C) $\frac{1}{64}$
(D) $\frac{1}{16}$
(E) $\frac{1}{4}$

ANSWER :

PROBLEM 24 :

Isosceles $\triangle A B C$ has equal side lengths $A B$ and $B C$. In the figure below, segments are drawn parallel to $\overline{A C}$ so that the shaded portions of $\triangle A B C$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle A B C$ ? (Diagram not drawn to scale.)

(A) 14.6
(B) 14.8
(C) 15
(D) 15.2
(E) 15.4

ANSWER :

(A) 14.6

PROBLEM 25 :

Fifteen integers $a_1, a_2, a_3, \ldots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that

$$
1 \leq a_1 \leq 10,13 \leq a_2 \leq 20, \text { and } 241 \leq a_{15} \leq 250 .
$$

What is the sum of digits of $a_{14}$ ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

SOLUTION :

(A) 8

AMERICAN MATHEMATICS COMPETITION 8 - 2024

PROBLEM 1 :

What is the unit digit of:

$$
222,222-22,222-2,222-222-22-2 ?
$$

(A) 0
(B) 2
(C) 4
(D) 8
(E) 10

ANSWER :

(B) 2

PROBLEM 2 :

What is the value of this expression in decimal form?

$$
\frac{44}{11}+\frac{110}{44}+\frac{44}{1100}
$$

(A) 6.4
(B) 6.504
(C) 6.54
(D) 6.9
(E) 6.94

ANSWER :

PROBLEM 3 :

Four squares of side length $4,7,9$, and 10 are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?

(A) 42
(B) 45
(C) 49
(D) 50
(E) 52

ANSWER :

(E) 52

PROBLEM 4 :

When Yunji added all the integers from 1 to 9 , she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER :

(E) 9

PROBLEM 5 :

Aaliyah rolls two standard 6 -sided dice. She notices that the product of the two numbers rolled is a multiple of 6 . Which of the following integers cannot be the sum of the two numbers?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER :

(B) 6

PROBLEM 6 :

Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P, Q, R$, and $S$. What is the sorted order of the four paths from shortest to longest?

(A) $P, Q, R, S$
(B) $P, R, S, Q$
(C) $Q, S, P, R$
(D) $R, P, S, Q$
(E) $R, S, P, Q$

ANSWER :

(D) $R, P, S, Q$

PROBLEM 7 :

A $3 \times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2 \times 2,1 \times 4$, and $1 \times 1$, shown below. What is the minimum possible number of $1 \times 1$ tiles used?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER :

(E) 5

PROBLEM 8 :

On Monday, Taye has $\$ 2$. Every day, he either gains $\$ 3$ or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

ANSWER :

(D) 6

PROBLEM 9 :

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles, and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

ANSWER :

(E) 28

PROBLEM 10 :

In January 1980 the Mauna Loa Observatory recorded carbon dioxide (CO2) levels of 338 ppm (parts per million). Over the years the average $C O 2$ reading has increased by about 1.515 ppm each year. What is the expected $C O 2$ level in ppm in January 2030 ? Round your answer to the nearest integer.
(A) 399
(B) 414
(C) 420
(D) 444
(E) 459

ANSWER :

(B) 414

PROBLEM 11 :

The coordinates of $\triangle A B C$ are $A(5,7), B(11,7)$, and $C(3, y)$, with $y>7$. The area of $\triangle A B C$ is 12 . What is the value of $y$ ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

ANSWER :

(D) 11

PROBLEM 12 :

Rohan keeps 90 guppies in 4 fish tanks.

There is 1 more guppy in the 2 nd tank than in the 1 st tank.
There are 2 more guppies in the 3rd tank than in the 2nd tank.
There are 3 more guppies in the 4th tank than in the 3rd tank.

How many guppies are in the 4th tank?
(A) 20
(B) 21
(C) 23
(D) 24
(E) 26

ANSWER :

(E) 26

PROBLEM 13 :

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz Bunny start on the ground, make a sequence of 6 hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

(A) 4
(B) 5
(C) 6
(D) 8
(E) 12

ANSWER :

(B) 5

PROBLEM 14 :

The one-way routes connecting towns $A, M, C, X, Y$, and $Z$ are shown in the figure below(not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?

(A) 28
(B) 29
(C) 30
(D) 31
(E) 32

ANSWER :

(A) 28

PROBLEM 15 :

Let the letters $F, L, Y, B, U, G$ represent distinct digits. Suppose $\underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}$ is the greatest number that satisfies the equation

$$
8 \cdot \underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}=\underline{B} \underline{U} \underline{G} \underline{B} \underline{U} \underline{G} .
$$

What is the value of $\underline{F} \underline{L} \underline{Y}+\underline{B} \underline{U} \underline{G}$ ?
(A) 1089
(B) 1098
(C) 1107
(D) 1116
(E) 1125

ANSWER :

PROBLEM 16 :

Minh enters the numbers 1 through 81 into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3 ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

ANSWER :

(D) 11

PROBLEM 17 :

A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a 3 $x 3$ grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?

(A) 20
(B) 24
(C) 27
(D) 28
(E) 32

ANSWER :

(E) 32

PROBLEM 18 :

Three concentric circles centered at $O$ have radii of 1, 2, and 3 . Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angles $B O C$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle B O C$ in degrees?

(A) 108
(B) 120
(C) 135
(D) 144
(E) 150

ANSWER :

(A) 108

PROBLEM 19 :

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?

(A) 0
(B) $\frac{1}{5}$
(C) $\frac{4}{15}$
(D) $\frac{1}{3}$
(E) $\frac{2}{5}$

ANSWER :

PROBLEM 20 :

Any three vertices of the cube $P Q R S T U V W$, shown in the figure below, can be connected to form a triangle. (For example, vertices $P, Q$, and $R$ can be connected to form isosceles $\triangle P Q R$.) How many of these triangles are equilateral and contain $P$ as a vertex?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 6

ANSWER :

(D) 3

PROBLEM 21 :

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3: 1$. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is $4: 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
(A) 10
(B) 12
(C) 16
(D) 20
(E) 24

ANSWER :

(E) 24

PROBLEM 22 :

A roll of tape is 4 inches in diameter and is wrapped around a ring that is 2 inches in diameter. A cross section of the tape is shown in the figure below. The tape is 0.015 inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 100 inches.

(A) 300
(B) 600
(C) 1200
(D) 1500
(E) 1800

ANSWER :

(B) 600

PROBLEM 23 :

Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the 4 cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time?

(A) 6000
(B) 6500
(C) 7000
(D) 7500
(E) 8000

ANSWER :

PROBLEM 24 :

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is 8 feet high while the other peak is 12 feet high. Each peak forms a $90^{\circ}$ angle, and the straight sides form a $45^{\circ}$ angle with the ground. The artwork has an area of 183 square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$ ?

(A) 4
(B) 5
(C) $4 \sqrt{2}$
(D) 6
(E) $5 \sqrt{2}$

ANSWER :

(B) 5

PROBLEM 25 :

A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?

(A) $\frac{8}{15}$
(B) $\frac{32}{55}$
(C) $\frac{20}{33}$
(D) $\frac{34}{55}$
(E) $\frac{8}{11}$

ANSWER :

8 Cheenta students cracked the Regional Math Olympiad 2025

In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies.

Srishti Verma
Hritam Das
Nupur Layek
Prabhav Bhatia
Ayan kalra
Adhiraj Singh Anand
Neel Avinash
Jayaditya Gupta

The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are below 20 years of age.
These contests help find students with strong mathematical talent. INMO is the higher stage after RMO. It recognises the best problem solvers in the country.

In RMO the top performers may get chances for further training.They may also get opportunities to represent India in international competitions like the International Mathematical Olympiad (IMO).

Achieving success in the Regional Mathematics Olympiad (RMO) and aiming for the Indian National Mathematics Olympiad (INMO) is a remarkable feat, requiring dedication, strategic preparation, and a strong foundation in mathematical problem-solving.

In this post, we share RMO success stories of Ayan kalra & Adhiraj Singh Anand. They performed excellently in RMO. Through their experiences, we learn about their journey, study methods, and useful tips for students who are preparing for RMO and INMO.

IOQM 2025 Questions, Answer Key, Solutions

Answer Key

Answer 1 40	Answer 2 17	Answer 3 18	Answer 4 5	Answer 5 36
Answer 6 18	Answer 7 576	Answer 8 44	Answer 9 28	Answer 10 15
Answer 11 80	Answer 12 38	Answer 13 13	Answer 14 11	Answer 15 75
Answer 16 8	Answer 17 8	Answer 18 1	Answer 19 72	Answer 20 42
Answer 21 80	Answer 22 7	Answer 23 19	Answer 24 66	Answer 25 9
Answer 26 6	Answer 27 37	Answer 28 12	Answer 29 33	Answer 30 97

Problem 1

If $60 \%$ of a number $x$ is 40 , then what is $x \%$ of 60 ?

Problem 2

Find the number of positive integers $n$ less than or equal to 100 , which are divisible by 3 but are not divisible by 2.

Problem 3

The area of an integer-sided rectangle is 20 . What is the minimum possible value of its perimeter?

Problem 4

How many isosceles integer-sided triangles are there with perimeter 23?

Problem 5

How many 3 -digit numbers $a b c$ in base 10 are there with $a \neq 0$ and $c=a+b$ ?

Problem 6

The height and the base radius of a closed right circular cylinder are positive integers and its total surface area is numerically equal to its volume. If its volume is $k \pi$ where $k$ is a positive integer, what is the smallest possible value of $k$ ?

Problem 7

A quadrilateral has four vertices $A, B, C, D$. We want to colour each vertex in one of the four colours red, blue, green or yellow, so that every side of the quadrilateral and the diagonal $A C$ have end points of different colours. In how many ways can we do this?

Problem 8

The sum of two real numbers is a positive integer $n$ and the sum of their squares is $n+1012$. Find the maximum possible value of $n$.

Problem 9

Four sides and a diagonal of a quadrilateral are of lengths $10, 20, 28, 50, 75$, not necessarily in that order. Which amongst them is the only possible length of the diagonal?

Problem 10

The age of a person (in years) in 2025 is a perfect square. His age (in years) was also a perfect square in 2012. His age (in years) will be a perfect cube $m$ years after 2025. Determine the smallest value of $m .=15$

Problem 11

There are six coupons numbered 1 to 6 and six envelopes, also numbered 1 to 6 . The first two coupons are placed together in any one envelope. Similarly, the third and the fourth are placed together in a different envelope, and the last two are placed together in yet another different envelope. How many ways can this be done if no coupon is placed in the envelope having the same number as the coupon?

Problem 12

Consider five-digit positive integers of the form $\overline{a b c a b}$ that are divisible by the two digit number $a b$ but not divisible by 13 . What is the largest possible sum of the digits of such a number?

Problem 13

A function $f$ is defined on the set of integers such that for any two integers $m$ and $n$,

$$
f(m n+1)=f(m) f(n)-f(n)-m+2
$$

holds and $f(0)=1$. Determine the largest positive integer $N$ such that $\sum_{k=1}^N f(k)<100$ .

Problem 14

Consider a fraction $\frac{a}{b} \neq \frac{3}{4}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$ and $b \leq 15$. If this fraction is chosen closest to $\frac{3}{4}$ amongst all such fractions, then what is the value of $a+b$ ?

Problem 15

Three sides of a quadrilateral are $a=4 \sqrt{3}, b=9$ and $c=\sqrt{3}$. The sides $a$ and $b$ enclose an angle of $30^{\circ}$, and the sides $b$ and $c$ enclose an angle of $90^{\circ}$. If the acute angle between the diagonals is $x^{\circ}$, what is the value of $x$ ?

Problem 16

$f(x)$ and $g(x)$ be two polynomials of degree 2 such that

$$
\frac{f(-2)}{g(-2)}=\frac{f(3)}{g(3)}=4
$$

If $g(5)=2, f(7)=12, g(7)=-6$, what is the value of $f(5)$ ?

Problem 17

The triangle $A B C, \angle B=90^{\circ}, A B=1$ and $B C=2$. On the side $B C$ there are two points $D$ and $E$ such that $E$ lies between $C$ and $D$ and $D E F G$ is a square, where $F$ lies on $A C$ and $G$ lies on the circle through $B$ with centre $A$. If the area of $D E F G$ is $\frac{m}{n}$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 18

$M T A I$ is a parallelogram of area $\frac{40}{41}$ square units such that $M I=1 / M T$. If $d$ is the least possible length of the diagonal $M A$, and $d^2=\frac{a}{b}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$, find $|a-b|$.

Problem 19

Let $N$ be the number of nine-digit integers that can be obtained by permuting the digits of 223334444 and which have at least one 3 to the right of the right-most occurrence of 4 . What is the remainder when $N$ is divided by $100$?

Problem 20

Let $f$ be the function defined by

$$
f(n)=\text { remainder when } n^n \text { is divided by } 7,
$$

for all positive integers $n$. Find the smallest positive integer $T$ such that $f(n+T)=f(n)$ for all positive integers $n$.

Problem 21

Let $P(x)=x^{2025}, Q(x)=x^4+x^3+2 x^2+x+1$. Let $R(x)$ be the polynomial remainder when the polynomial $P(x)$ is divided by the polynomial $Q(x)$. Find $R(3)$.

Problem 22

Let $A B C D$ be a rectangle and let $M, N$ be points lying on sides $A B$ and $B C$, respectively. Assume that $M C= C D$ and $M D=M N$, and that points $C, D, M, N$ lic on a circle. If $(A B / B C)^2=m / n$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 23

Let $A B C D$ be a rectangle and let $M, N$ be points lying on sides $A B$ and $B C$, respectively. Assume that $M C= C D$ and $M D=M N$, and that points $C, D, M, N$ lie on a circle. If $(A B / B C)^2=m / n$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 24

There are $m$ blue marbles and $n$ red marbles on a table. Armaan and Babita play a game by taking turns. In each turn the player has to pick a marble of the colour of his/her choice. Armaan starts first, and the player who picks the last red marble wins. For how many choices of $(m, n)$ with $1 \leq m, n \leq 11$ can Armaan force a win?

Problem 25

For some real numbers $m, n$ and a positive integer $a$, the list $(a+1) n^2, m^2, a(n+1)^2$ consists of three consecutive integers written in increasing order. What is the largest possible value of $m^2$ ?

Problem 26

Let $S$ be a circle of radius 10 with centre $O$. Suppose $S_1$ and $S_2$ are two circles which touch $S$ internally and intersect each other at two distinct points $A$ and $B$. If $\angle O A B=90^{\circ}$ what is the sum of the radii of $S_1$ and $S_2$ ?

Solution

Problem 27

A regular polygon with $n \geq 5$ vertices is said to be colourful if it is possible to colour the vertices using at most 6 colours such that each vertex is coloured with exactly one colour, and such that any 5 consecutive vertices have different colours. Find the largest number $n$ for which a regular polygon with $n$ vertices is not colourful.

Solution

Problem 28

Find the number of ordered triples $(a, b, c)$ of positive integers such that $1 \leq a, b, c \leq 50$ which satisfy the relation

$$
\frac{\operatorname{lcm}(a, c)+\operatorname{lcm}(b, c)}{a+b}=\frac{26 c}{27}
$$

Here, by $\operatorname{lcm}(x, y)$ we mean the LCM, that is, least common multiple of $x$ and $y$.

Problem 29

Consider a sequence of real numbers of finite length. Consecutive four term averages of this sequence are strictly increasing, but consecutive seven term averages are strictly decreasing. What is the maximum possible length of such a sequence?

Problem 30

Assume $a$ is a positive integer which is not a perfect square. Let $x, y$ be non-negative integers such that $\sqrt{x-\sqrt{x+a}}=\sqrt{a}-y$. What is the largest possible value of $a$ such that $a<100 ?$

NMTC - Screening Test – Ramanujan Contest 2025

PART – A

Problem 1

If four different positive integers $m, n, p, q$ satisfy the equation
$7-m)(7-n)(7-p)(7-q)=4$

then the sum $m+n+p+q$ is equal to

A. 10
B. 24
C. 28
D. 36

Problem 2

A three member sequence $a, b, c$ is said to be a up-down sequence if $ac$. For example $1,3,2$ is a up-down sequence. The sequence 1342 contains three up-down sequences: $(1,3,2),(1,4,2)$ and $(3,4,2)$. How many up-down sequences are contained in the sequence 132597684?

A. 32
B. 34
C. 36
D. 38

Problem 3

For a positive integer $n$, let $P(n)$ denote the product of the digits of $n$ when $n$ is written in base 10. For example, $P(123)=6$ and $P(788)=448$. If $N$ is the smallest positive integer such that $P(N)>1000$, and $N$ is written as $100 x+y$ where $x, y$ are integers with $0 \leq x, y<100$, then $x+y$ equals

A. 112
B. 114
C. 116
D. 118

Problem 4

The sum of 2025 consecutive odd integers is $2025^{2025}$. The largest of these off numbers is

A. $2025^{2024}+2024$
B. $2025^{2024}-2024$
C. $2025^{2023}+2024$
D. $2025^{2023}-2024$

Problem 5

$A B C$ is an equilateral triangle with side length 6. $P, Q, R$ are points on the sides $A B, B C, C A$ respectively such that $A P=B Q=C R=1$. The ratio of the area of the triangle $A B C$ to the area of the triangle $P Q R$ is

A. $36: 25$
B. $12: 5$
C. $6: 5$
D. $12: 7$

Problem 6

How many three-digit positive integers are there if the digits are the side lengths of some isosceles or equilateral triangle?

A. 45
B. 81
C. 165
D. 216

Problem 7

All the positive integers whose sum of digits is 7 are written in the increasing order. The first few are $7,16,25,34,43, \ldots$. What is the 125 th number in this list?

A. 7000
B. 10006
C. 10024
D. 10042

Problem 8

The bisectors of the angles $A, B, C$ of the triangle $A B C$ meet the circum circle of the triangle again at the points $D, E, F$ respectively. What is the value of
$\frac{A D \cos \frac{A}{2}+B E \cos \frac{B}{2}+C F \cos \frac{C}{2}}{\sin A+\sin B+\sin C}$

if the circum radius of $A B C$ is 1 ?

A. 2
B. 4
C. 6
D. 8

Problem 9

For a real number $x$, let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$. For example, $[1.7]=1$ and $[\sqrt{2}]=1$. Let $N=\left\lfloor\frac{10^{93}}{10^{31}+3}\right\rfloor$. Find the remainder when $N$ is divided by 100.

A. 1
B. 8
C. 22
D. 31

Problem 10

A point $(x, y)$ in the plane is called a lattice point if both its coordinates $x, y$ are integers. The number of lattice points that lie on the circle with center at $(199,0)$ and radius 199 is

A. 4
B. 8
C. 12
D. 16

Problem 11

The sum of all real numbers $p$ such that the equation

$5 x^3-5(p+1) x^2+(71 p-1) x-(66 p-1)=0$

has all its three roots positive integers.

A. 70
B. 74
C. 76
D. 88

Problem 12

If $1-x+x^2-x^3+\cdots+x^{20}$ is rewritten in the form

$a_0+a_1(x-4)+a_2(x-4)^2+\cdots+a_{20}(x-4)^{20}$, where $a_0, a_1, \ldots, a_{20}$

are all real numbers, the value of $a_0+a_1+a_2+\cdots+a_{20}$ is

A. $\frac{5^{21}+1}{6}$
B. $\frac{5^{21}-1}{6}$
C. $\frac{5^{20}+1}{6}$
D. $\frac{5^{20}-1}{6}$

Problem 13

For a positive integer $n$, a distinct 3-partition of $n$ is a triple $ (a, b, c) $ of positive integers such that $a<b<c$ and $a+b+c=n$. For example, $(1,2,4)$ is a distinct 3 -partition of 7 . The number of distinct 3-partitions of 15 is

A. 10
B. 12
C. 13
D. 15

Problem 14

If $m$ and $n$ are positive integers such that $30 m n-6 m-5 n=2019$, what is the value of $30 m n-5 m-6 n ?$

A. 1900
B. 2020
C. 1939
D. Can not be found from the given information

Problem 15

A class of 100 students takes a six question exam. For the first question, a student receives 1 point for answering correctly, -1 point for answering incorrectly or not answering at all. For the second question, the student receives 2 points for answering correctly and -2 points for answering incorrectly or not answering at all and so on. What is the minimum number of students having the same scores?

A. 6
B. 5
C. 0
D. Can not be found from the given information

Part B

Problem 16

The value of

$\frac{1}{2}+\frac{1^2+2^2}{6}+\frac{1^2+2^2+3^2}{12}+\frac{1^2+2^2+3^2+4^2}{20}+\cdots+\frac{1^2+2^2+\cdots+60^2}{3660}$

is ________ .

Problem 17

The largest prime divisor of $3^{21}+1$ is _________

Problem 18

A circular garden divided into 10 equal sectors needs to be planted with flower plants that yield flowers of 3 different colors, in such a way that no two adjacent sectors will have flowers of the same color. The number of ways in which this can be done is _________

Problem 19

We call an integer special if it is positive and we do not need to use the digit 0 to write it down in base 10. For example, 2126 is special whereas 2025 is not. The first 10 special numbers are $1,2,3,4,5,6,7,8,9,11$. The 2025th special number is _________ .

Problem 20

Let $a, b, c$ be non zero real numbers such that $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$. The value of $\frac{5}{a^2+b^2+c^2}$ is _________ .

Problem 21

The equation $x^3-\frac{1}{x}=4$ has two real roots $\alpha, \beta$. The value of $(\alpha+\beta)^2$ is _________

Problem 22

If $x, y, z$ are positive integers satisfying the system of equations

$\begin{aligned} x y+y z+z x & =2024 \ x y z+x+y+z & =2025\end{aligned}$

find $\max (x, y, z)$ . ________

Problem 23

If $p, q, r$ are primes such that $p q+q r+r p=p q r-2025$, find $p+q+r .$. __________

Problem 24

A cyclic quadrilateral has side lengths $3,5,5,8$ in this order. If $R$ is its circumradius, find $3 R^2$. __________

Problem 25

Consider the sequence of numbers $24,2534,253534,25353534, \ldots$. Let $N$ be the first number in the sequence that is divisible by 99 . Find the number of digits in the base 10 representation of $N$. _____________

Problem 26

An isosceles triangle has integer sides and has perimeter 16. Find the largest possible area of the triangle. ____________

Problem 27

Suppose that $a, b, c$ are positive real numbers such that $a^2+b^2=c^2$ and $a b=c$. Find the value of

$\frac{(a+b+c)(a-b+c)(a+b-c)(a-b-c)}{c^2}$ ______________

Problem 28

In a right angled triangle with integer sides, the radius of the inscribed circle is 12. Compute the largest possible length of the hypotenuse. _______________

Problem 29

Points $C$ and $D$ lie on opposite sides of the line $A B$. Let $M$ and $N$ be the centroids of the triangles $A B C$ and $A B D$ respectively. If $A B=25, B C=24, A C=7, A D=20$ and $B D=15$, find $M N$. __________

Problem 30

Let $a_0=1$ and for $n \geq 1$, define $a_n=3 a_{n-1}+1$. Find the remainder when $a_{11}$ is divided by 97. ___________

NMTC - Screening Test – KAPREKAR Contest - 2025

Part 1

Problem 1

$A B$ is a straight road of length 400 metres. From $A$, Samrud runs at a speed of $6 \mathrm{~m} / \mathrm{s}$ towards $B$ and at the same time Saket starts from $B$ and runs towards $A$ at a speed of $5 \mathrm{~m} / \mathrm{s}$. After reaching their destinations, they return with the same speeds. They repeat it again and again. How many times do they meet each other in 15 minutes?

A) 25
B) 23
C) 24
D) 20

Problem 2

In the adjoining figure, the measure of the angle $x$ is

A) $84^{\circ}$
B) $44^{\circ}$
C) $64^{\circ}$
D) $54^{\circ}$

Problem 3

The value of $x$ which satisfies $\frac{1}{x+a}+\frac{1}{x+b}=\frac{1}{x+a+b}+\frac{1}{x}$ is

A) $\frac{a+b}{2}$
B) $\frac{a-b}{2}$
C) $\frac{b-a}{2}$
D) $\frac{-(a+b)}{2}$

Problem 4

Two sides of an isosceles triangle are 23 cm and 17 cm respectively. The perimeter of the triangle (in cm ) is

A) 63
В) 57
C) 63 or 57
D) 40

Problem 5

$A B C D E$ is a pentagon with $\angle B=90^{\circ}$ and $\angle E=150^{\circ}$.
If $\angle C+\angle D=180^{\circ}$ and $\angle A+\angle D=180^{\circ}$, then the external angle $\angle D$ is

A) $120^{\circ}$
B) $110^{\circ}$
C) $105^{\circ}$
D) $115^{\circ}$

Problem 6

The unit's digit of the product $3^{2025} \times 7^{2024}$ is

A) 1
B) 2
C) 3
D) 6

Problem 7

The smallest positive integer $n$ for which $18900 \times n$ is a perfect cube is

A) 1
B) 2
C) 3
D) 6

Problem 8

Two numbers $a$ and $b$ are respectively $20 \%$ and $50 \%$ more of a third number $c$. The percentage of $a$ to $b$ is

A) 120 %
В) 80 %
C) 75 %
D) 110 %

Problem 9

If $a+b=2, \frac{1}{a}+\frac{1}{b}=18$, then $a^3+b^3$ lies between

A) 7 and 8
B) 6 and 7
C) 8 and 9
D) 5 and 6

Problem 10

If $\sqrt{12+\sqrt[3]{x}}=\frac{7}{2}$ and $x=\frac{p}{q^{\prime}}, p, \mathrm{q}$ are natural numbers with G.C.D. $(p, q)=1$, then $p+q$ is

A) 65
В) 56
C) 45
D) 54

Problem 11

The smallest number of 4-digits leaving a remainder 1 when divided by 2 or

A) 5 as its unit digit
B) Only one zero as one of the digits
C) Exactly two zeroes as its digits
D) 7 as its unit digit

Problem 12

If $a: b=2: 3, b: c=4: 5$ and $a+c=736$, then the value of $b$ is

A) 392
B) 378
C) 384
D) 386

Problem 13

In the given figure,

\[
\begin{aligned}
& \angle B=110^{\circ} ; \quad \angle C=80^{\circ} ; \
& \angle F=120^{\circ} ; \quad \angle A D C=30^{\circ} \
& 2 \angle D G F=\angle D E F .
\end{aligned}
\]

The measure of $\angle B H F$ is

A) $115^{\circ}$
B) $135^{\circ}$
C) $100^{\circ}$
D) $130^{\circ}$

Problem 14

If $\frac{1}{b+c}+\frac{1}{c+a}=\frac{2}{a+b}$, then the value of $\frac{a^2+b^2}{c^2}$ is

A) 2
B) 1
C) 1 / 2
D) 3

Problem 15

If 3 men or 4 women can do a job in 43 days, the number of days the same job is done by 7 men and 5 women is

A) 12
B) 10
C) 11
D) 13

Part B

Problem 16

The expression $49(a+b)^2-46(a-b)^2$ is factorized into $(l a+m b)(n a+p b)$, then the numerical value of $(l+m+n+p)$ is _________________

Problem 17

The integer part of the solution of the equation in $x$, $\frac{1}{3}(x-3)-\frac{1}{4}(x-8)=\frac{1}{5}(x-5)$ is ______________

Problem 18

In the adjoining figure, $A B C$ is a triangle in which $\angle B A C=100^{\circ}$, $\angle A C B=30^{\circ}$. An equilateral triangle, a square and a regular hexagon are drawn as shown in the figure. The measure (in degrees) of $(x+y+z)$ is ____________

Problem 19

The mean of 5 numbers is 105 . The first number is $\frac{2}{5}$ times the sum of the other 4 numbers. The first number is ____________

Problem 20

$P Q R S$ is a square. The sides $P Q$ and $R S$ are increased by 30 % each and the sides $Q R$ and $P S$ are increased by 20 % each. The area of the quadrilateral thus obtained exceeds the area of the square by ___________ %.

Problem 21

If $x^2+(2+\sqrt{3}) x-1=0$ and $x^2+\frac{1}{x^2}=a+b \sqrt{c}$, then $(a+b+c)$ is _____________

Problem 22

In the given figure, $A B C D$ is a rectangle.

The measure of angle $x$ is _________________ degrees.

Problem 23

The sum of all positive integers $m, n$ which satisfy $m^2+2 m n+n=44$ is __________________

Problem 24

Given $a=2025, b=2024$, the numerical value of $\left(a+b-\frac{4 a b}{a+b}\right) \div\left(\frac{a}{a+b}-\frac{b}{b-a}+\frac{2 a b}{b^2-a^2}\right)$ is _________________

Problem 25

In the sequence $0,7,26,63,124, \ldots \ldots \ldots$ the $6^{\text {th }}$ term is _____________

Problem 26

\[
\text { If } A=\sqrt{281+\sqrt{53+\sqrt{112+\sqrt{81}}}}, B=\sqrt{92+\sqrt{55+\sqrt{75+\sqrt{36}}}}
\]

then $A-B$ is _______________________

Problem 27

The average of the numbers $a, b, c, d$ is $(b+4)$. The average of pairs $(a, b),(b, c)$ and $(c, a)$ are respectively 16,26 and 25 . Then the average of $d$ and 67 is ___________________

Problem 28

$A B C$ is a quadrant of a circle of radius 10 cm . Two semicircles are drawn as in the figure.

The area of the shaded portion is $k \pi$, where $k$ is a positive integer.

The value of $k$ is __________________

Problem 29

In the figure, $A B C$ and $P Q R$ are two triangles such that $\angle \mathrm{A}: \angle \mathrm{B}: \angle \mathrm{C}=5: 6: 7$ and $\angle P R Q=\angle B$. $P S$ makes an angle $\frac{\angle P}{3}$ with $P Q$ and $R S$ makes an angle $\frac{\angle S R T}{5}$ with $R Q$. Then the measure of $\angle S$ is ______________________

Problem 30

In a two-digit positive integer, the units digit is one less than the tens digit. The product of one less than the units digit and one more than the tens digit is 40. The number of such two-digit integers is _______________

BHASKARA Contest - NMTC - Screening Test – 2025

Problem 1

The greatest 4 -digit number such that when divided by 16,24 and 36 leaves 4 as remainder in each case is
А) 9994
B) 9940
C) 9094
D) 9904

Problem 2

$A B C D$ is a rectangle whose length $A B$ is 20 units and breadth is 10 units. Also, given $A P=8$ units. The area of the shaded region is $\frac{p}{q}$ sq unit, where $p, q$ are natural numbers with no common factors other than 1 . The value of $p+q$ is
A) 167
В) 147
C) 157
D) 137

Problem 3

The solution of $\frac{\sqrt[7]{12+x}}{x}+\frac{\sqrt[7]{12+x}}{12}=\frac{64}{3}(\sqrt[7]{x})$ is of the form $\frac{a}{b}$ where $a, b$ are natural numbers with $\operatorname{GCD}(a, b)=1$; then $(b-a)$ is equal to
A) 115
B) 114
C) 113
D) 125

Problem 4

The value of $(52+6 \sqrt{43})^{3 / 2}-(52-6 \sqrt{43})^{3 / 2}$ is
A) 858
В) 918
C) 758
D) 828

Problem 5

In the adjoining figure $\angle D C E=10^{\circ}$, $\angle C E D=98^{\circ}, \angle B D F=28^{\circ}$
Then the measure of angle $x$ is
A) $72^{\circ}$
B) $76^{\circ}$
C) $44^{\circ}$
D) $82^{\circ}$

Problem 6

$A B C$ is a right triangle in which $\angle \mathrm{B}=90^{\circ}$. The inradius of the triangle is $r$ and the circumradius of the triangle is R . If $\mathrm{R}: r=5: 2$, then the value of $\cot ^2 \frac{A}{2}+\cot ^2 \frac{C}{2}$ is
A) $\frac{25}{4}$
B) 17
C) 13
D) 14

Problem 7

If $(\alpha, \beta)$ and $(\gamma, \beta)$ are the roots of the simultaneous equations:

\[
|x-1|+|y-5|=1 ; \quad y=5+|x-1|
\]

then the value of $\alpha+\beta+\gamma$ is
A) $\frac{15}{2}$
B) $\frac{17}{2}$
C) $\frac{14}{3}$
D) $\frac{19}{2}$

Problem 8

Three persons Ram, Ali and Peter were to be hired to paint a house. Ram and Ali can paint the whole house in 30 days, Ali and Peter in 40 days while Peter and Ram can do it in 60 days. If all of them were hired together, in how many days can they all three complete $50 \%$ of the work?
A) $24 \frac{1}{3}$
B) $25 \frac{1}{2}$
C) $26 \frac{1}{3}$
D) $26 \frac{2}{3}$

Problem 9

$\frac{\sqrt{a+3 b}+\sqrt{a-3 b}}{\sqrt{a+3 b}-\sqrt{a-3 b}}=x$, then the value of $\frac{3 b x^2+3 b}{a x}$ is
A) 1
B) 2
C) 3
D) 4

Problem 10

The number of integral solutions of the inequation $\left|\frac{2}{x-13}\right|>\frac{8}{9}$ is
A) 1
B) 2
C) 3
D) 4

Problem 11

In the adjoining figure, $P$ is the centre of the first circle, which touches the other circle in C . PCD is along the diameter of the second circle. $\angle \mathrm{PBA}=20^{\circ}$ and $\angle \mathrm{PCA}=30^{\circ}$.

The tangents at B and D meet at E . The measure of the angle $x$ is
A) $75^{\circ}$
B) $80^{\circ}$
C) $70^{\circ}$
D) $85^{\circ}$

Problem 12

If $\alpha, \beta$ are the values of $x$ satisfying the equation $3 \sqrt{\log _2 x}-\log _2 8 x+1=0$, where $\alpha<\beta$, then the value of $\left(\frac{\beta}{\alpha}\right)$ is
A) 2
B) 4
C) 6
D) 8

Problem 13

When a natural number is divided by 11 , the remainder is 4 . When the square of this number is divided by 11 , the remainder is
A) 4
B) 5
C) 7
D) 9

Problem 14

The unit's digit of a 2-digit number is twice the ten's digit. When the number is multiplied by the sum of the digits the result is 144 . For another 2-digit number, the ten's digit is twice the unit's digit and the product of the number with the sum of its digits is 567 . Then the sum of the two 2 -digit numbers is
A) 68
В) 86
C) 98
D) 87

Problem 15

$A B C D E$ is a pentagon. $\angle A E D=126^{\circ}, \angle B A E=\angle C D E$ and $\angle A B C$ is $4^{\circ}$ less than $\angle B A E$ and $\angle B C D$ is $6^{\circ}$ less than $\angle C D E . P R, Q R$ the bisectors of $\angle B P C, \angle E Q D$ respectively, meet at $R$. Points $\mathrm{P}, \mathrm{C}, \mathrm{D}, \mathrm{Q}$ are collinear. Then measure of $\angle P R Q$ is
A) $151^{\circ}$
B) $137^{\circ}$
C) $141^{\circ}$
D) $143^{\circ}$

Problem 16

$a, b, c$ are real numbers such that $b-c=8$ and $b c+a^2+16=0$.
The numerical value of $a^{2025}+b^{2025}+c^{2025}$ is $\rule{2cm}{0.2mm}$.

Problem 17

Given $f(x)=\frac{2025 x}{x+1}$ where $x \neq-1$. Then the value of $x$ for which $f(f(x))=(2025)^2$ is $\rule{2cm}{0.2mm}$.

Problem 18

The sum of all the roots of the equation $\sqrt[3]{16-x^3}=4-x$ is $\rule{2cm}{0.2mm}$.

Problem 19

In the adjoining figure, two
Quadrants are touching at $B$.
$C E$ is joined by a straight line, whose mid-point is $F$.

The measure of $\angle C E D$ is $\rule{2cm}{0.2mm}$.

Problem 20

The value of $k$ for which the equation $x^3-6 x^2+11 x+(6-k)=0$ has exactly three positive integer solutions is $\rule{2cm}{0.2mm}$.

Problem 21

The number of 3-digit numbers of the form $a b 5$ (where $a, b$ are digits) which are divisible by 9 is $\rule{2cm}{0.2mm}$.

Problem 22

If $a=\sqrt{(2025)^3-(2023)^3}$, the value of $\sqrt{\frac{a^2-2}{6}}$ is $\rule{2cm}{0.2mm}$.

Problem 23

In a math Olympiad examination, $12 \%$ of the students who appeared from a class did not solve any problem; $32 \%$ solved with some mistakes. The remaining 14 students solved the paper fully and correctly. The number of students in the class is $\rule{2cm}{0.2mm}$.

Problem 24

When $a=2025$, the numerical value of
$\left|2 a^3-3 a^2-2 a+1\right|-\left|2 a^3-3 a^2-3 a-2025\right|$ is $\rule{2cm}{0.2mm}$.

Problem 25

A circular hoop and a rectangular frame are standing on the level ground as shown. The diagonal $A B$ is extended to meet the circular hoop at the highest point $C$. If $A B=18 \mathrm{~cm}, B C=32 \mathrm{~cm}$, the radius of the hoop (in cm ) is $\rule{2cm}{0.2mm}$.

Problem 26

' $n$ ' is a natural number. The number of ' $n$ ' for which $\frac{16\left(n^2-n-1\right)^2}{2 n-1}$ is a natural number is $\rule{2cm}{0.2mm}$.

Problem 27

The number of solutions $(x, y)$ of the simultaneous equations $\log _4 x-\log _2 y=0, \quad x^2=8+2 y^2$ is $\rule{2cm}{0.2mm}$.

Problem 28

In the adjoining figure,
$P A, P B$ are tangents.
$A R$ is parallel to $P B$

$P Q=6 ; Q R=18 .$

Length $S B= \rule{2cm}{0.2mm}$.

Problem 29

A large watermelon weighs 20 kg with $98 \%$ of its weight being water. It is left outside in the sunshine for some time. Some water evaporated and the water content in the watermelon is now $95 \%$ of its weight in water. The reduced weight in kg is $\rule{2cm}{0.2mm}$.

Problem 30

In a geometric progression, the fourth term exceeds the third term by 24 and the sum of the second and third term is 6 . Then, the sum of the second, third and fourth terms is $\rule{2cm}{0.2mm}$.