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Ramanujan Contest (NMTC Inter 2019 - XI and XII Grades) - Stage I - Problems and Solution

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Part A

Problem 1

Ram and Shyam play table tennis with Ram's chance of winning a game being $\frac{3}{5}$ and Shyam's $\frac{2}{5}$. The winner gets 1 point and loser 0 points. The match terminates when one player has 2 points more than the other. The probability of Ram winning the game at exactly the end of $6^{th}$ game, not before, is
(A) $\frac{364}{15625}$
(B) $\frac{1296}{15625}$
(C) $\frac{432}{3125}$
(D) $\frac{2592}{15625}$

Problem 2

Thirty volunteers are distributed to three poling booths. Each booth must have at least one and all must have different number of volunteers allotted. Then the number of ways of allocating volunteers is :
(A) 406
(B) 496
(C) 378
(D) None of these

Problem 3

The number of values of a for which the function $f(x)=\cos 2 x+2 a(1+\cos x)$ has a minimum value $\frac{1}{2}$ is :
(A) 0
(B) 1
(C) 2
(D) 3

Problem 4

Let $f(x)=\frac{x}{\sqrt{x^2-1}}$. If $f^2(x)=f(f(x)), f^3(x)=f\left(f^2(x)\right), \ldots \ldots, f^{n+1}(x)=f\left(f^n(x)\right)$, then $f^{2019}(\sqrt{2})$ is :
(A) 1
(B) 0
(C) $\sqrt{2}$
(D) not define

Problem 5

The area of the curve enclosed by $|x-2 \sqrt{2}|+|y-\sqrt{5}|=2$ is :
(A) 16
(B) 12
(C) 8
(D) 4

Problem 6

Let $a$ be an irrational number. How many lines through the point $(a, 2a)$ contain at least two points with both coordinates rational ?
(A) Infinitely many
(B) At least two but finitely many
(C) Only one
(D) None

Problem 7

Suppose $A, A_2, \ldots \ldots, A_{33}$ be 33 sets each containing 6 elements and $B_1, B_2, \ldots . ., B_n$ be $n$ sets each with 8 elements. if \[\bigcup_{i=1}^{33} A_{i} = \bigcup_{i=1}^{n} B_{i}=S\] and if each element of $S$ occurs exactly 9 times in $A_1, \ldots$ $A_2, A_{33}$ and exactly 4 times in $B_1, \ldots B_2, B_n$, then $n$ is :
(A) 22
(B) 33
(C) 12
(D) 11

Problem 8

Let $a, b$ and $c$ be real numbers such that $2 a^2-b c-9 a+10=0$ and $4 b^2+c^2+b c-7 a-8=0$. Then the set of real values that a can take is given by
(A) $[1,4.2]$
(B) $(-\infty, 1) \cup(4.2, \infty)$
(C) $(1,4.2)$
(D) $[1,4.2)$

Problem 9

Let $g(x)=\left[\frac{1}{\operatorname{cosec}(x)}\right]$, then the range of $g(x)$ is $(\mathbb{Z}$ is the set of integers)
(A) $\mathbb{Z}$
(B) $\mathbb{Z}$-{0}
(C) {0}
(D) {0,1,-1}

Problem 10

The ordered pair of numbers $(x, y)$ satisfy both the equations $x+y=3$ and $x^5+y^5+162=0$. Then

(A) There are 5 pairs of real solutions
(B) there are four pairs of real solutions
(C) The are two pairs of real and two pairs of non-real solutions
(D) All four pairs are non-real solutions

Problem 11

In a rectangle $A B C D$, point $E$ lies on $B C$ such that $\frac{B E}{E C}=2$ and point $F$ lies on $C D$ such that $\frac{C F}{F D}=$ 2. Lines $A E$ and $A C$ intersect $B F$ at $X$ and $Y$ respectively. If $F Y: Y X: X B=a: b: c$, are relatively prime positive integers, then the minimum value of $a+b+c$ is :

(A) 4
(B) 8
(C) 12
(D) 16

Problem 12

Rita takes a train home at $4: 00$, arriving at the station at 6:00 Every day, driving the same rate, rate, her husband meets her at the station at 6:00. On day she takes the train an hour early and arrives at 5:00. Her husband leaves home to meet her at the usual time, so Rita begins to walk home. he meets her on the way and hey reach home 20 minutes earlier than usual. The number of minutes Rita was walking before she met her husband on the way is :
(A) 20
(B) 40
(C) 50
(D) 60

Problem 13

A regular polygon has 100 sides each of length. A another regular polygon has 200 sided each of length 2 . When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
(A) 2
(B) 4
(C) 8
(D) 16

Problem 14

The function $f$ satisfies $f(f(x))=f(x+2)-3$ for all integers $x$. If $f(1)=4 ; f(4)=3$, then $f(5)$ equals
(A) 3
(B) 6
(C) 9
(D) 12

Problem 15

If $x$ and $y$ are positive real numbers such that $x+y=1$, then maximum value of $x y^4+x^4 y$ is
(A) $\frac{1}{16}$
(B) $\frac{1}{12}$
(C) $\frac{1}{8}$
(D) $\frac{1}{4}$

Part B

Problem 16

Consider all 4 element subsets of the set $A={1,2,3, \ldots 8}$. Each of these subsets has a greatest element. The arithmetic mean of the greatest elements of these 4 element subsets is $\rule{1cm}{0.15mm}$

Problem 17

The number of times the digit occurs in the result of $1+11+111+\ldots . .+111 \ldots . .111$ (100digits) is $\rule{1cm}{0.15mm}$.

Problem 18

In a $38 \times 32$ rectangle $A B C D$, points $P, Q, R, S$ are taken on the sides $A B, B C, C D, D A$ respectively such that the lengths $A P, B Q, C R$ and $D S$ are integers and $P Q R S$ is rectangle. The largest possible area of $P Q R S$ is $\rule{1cm}{0.15mm} $.

Problem 19

6 blue, 7 green and 10 white balls are arranged in row such that every blue ball is between and green and a white ball. Moreover, a white ball and a green ball must not be next to each other. The number of such arrangements is $\rule{1cm}{0.15mm}$

Problem 20

Let us call a sum of integers a cool sum if the first and last terms are 1 and each term differs from its neighbours by at most. For example, $1+2+2+3+3+2+1$ and $1+2+3+4+3+2+1$ are cool sums. The minimum number of terms required to write 2019 as a cool sum is $\rule{1cm}{0.15mm}$.

Problem 21

$O$ is a point inside an equilateral triangle $A B C$. The perpendicular distance $O P, O Q, O R$ to the sides of the triangle are in the ratio $O P: O Q: O R=1: 2: 3$. If $\frac{\text { Area of quadrilateral OPBR }}{\text { Area of triangleABC }}=\frac{a}{b}$, where $\mathrm{a}, \mathrm{b}$ are co-prime positive integers, then $\mathrm{a}+\mathrm{b}$ equals $\rule{1cm}{0.15mm}$.

Problem 22

In $\triangle \mathrm{ABC}, \mathrm{AB}=6, \mathrm{BC}=7$ and $\mathrm{CA}=8$. Point $\mathrm{D}$ lies on $\mathrm{BC}$ and $\mathrm{AD}$ bisects $\angle \mathrm{BAC}$. Point $\mathrm{E}$ lies on $A C$ and $B E$ bisects $\angle A B C$. If the bisectors intersect at $F$, then the ratio $A F: F D=$ $\rule{1cm}{0.15mm}$.

Problem 23

Let $a, b, c$ be real numbers such that the polynomial $f(x)=x^3+a x^2+x+10$ has three distinct roots and each root of $f(x)$ is also a root of the polynomial $h(x)=x^4+x^3+b x^2+13 x+c$. The $h(1)=$ $\rule{1cm}{0.15mm}$.

Problem 24

In quadrilateral $A B C D, A B=10, B C=33, C D=10$ and $D A=15$. If $B D$ is an integer then $B D=$ $\rule{1cm}{0.15mm}$.

Problem 25

For each positive integer $n$ let $f(n)=n^4-3 n^2+9$. Then the sum of all $f(n)$ which are prime is $\rule{1cm}{0.15mm}$.

Problem 26

13 boys are sitting in a row in a theatre. After the intermission, they return and are seated such that either they occupy the same seat or the adjacent seat in such a way that it differs from the original arrangement. The number of ways this is possible is $\rule{1cm}{0.15mm}$.

Problem 27

$\mathrm{A}_1 \mathrm{~A}_2 \mathrm{~A}_3 \ldots . . \mathrm{A}_{15}$ is a 15 sided regular polygon. The number of distinct equilateral triangles in the plane of the polygon, with exactly two of their vertices from the set $\left{A_1, A_2, A_3 \ldots \ldots A_{15}\right}$ is $\rule{1cm}{0.15mm}$.

Problem 28

The polynomial $P(x)=x^3+a x^2+b x+c$ has the property that the mean of its roots, the product of its roots, and the sum of its coefficients are all equal. If the $y$ intercept of the graph $y=P(x)$ is 2 then $b=$ $\rule{1cm}{0.15mm}$.

Problem 29

$A B C D$ is a quadrilateral in the first quadrant where $A=(3,9), B=(1,1), C=(5,3)$ and $D=(p, q)$. The quadrilateral formed by joining the midpoints of $A B, B C, C D$ and $D A$ is a square. Then $p+q=$ $\rule{1cm}{0.15mm}$.

Problem 30

The product of four positive integers $a, b, c$ and $d$ is 9 ! The number $a, b, c, d$ satisfy $a b+a+b=$ $1224, b c+b+c=549$ and $c d+c+d=351$. The $a+b+c+d=\ldots \ldots$ $\rule{1cm}{0.15mm}$.

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