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Bhaskara Contest (NMTC Junior 2019 - IX and X Grades) - Stage I- Problems and Solution

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Part A

Problem 1

The number of 6 digit numbers of the form "ABCABC", which are divisible by 13 , where $A, B$ and $C$ are distinct digits, $A$ and $C$ being even digits is
(A) 200
(B) 250
(C) 160
(D) 128

Problem 2

In $\triangle \mathrm{ABC}$, the medians through $\mathrm{B}$ and $\mathrm{C}$ are perpendicular. Then $\mathrm{b}^2+\mathrm{c}^2$ is equal to
(A) $2 a^2$
(B) $3 a^2$
(C) $4 a^2$
(D) $5 a^2$

Problem 3

In a quadrilateral $A B C D, A B=A D=10, B D=12, C B=C D=13$. Then

(A) $A B C D$ is a cyclic quadrilateral
(B) $A B C D$ has an in-circle
(C) $A B C D$ has both circum-circle and in-circle
(D) It has neither a circum-circle nor an in-circle

Problem 4

Given three cubes with integer side lengths, if the sum of the surface areas of the three cubes is 498 $cm^2$, then the sum of the volumes of the cubes in all possible solutions is
(A) 731
(B) 495
(C) 1226
(D) None of these

Problem 5

In a rhombus of side length 5 , the length of one of the diagonals is at least 6 , and the length of the other diagonal is at most 6 . What is the maximum value of the sum of the diagonals ?
(A) $10 \sqrt{2}$
(B) $14$
(C) $5 \sqrt{6}$
(D) $12$

Problem 6

In the sequence $1,4,8,10,16,21,25,30$ and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is
(A) only one
(B) exactly two
(C) exactly three
(D) exactly four

Problem 7

Let $\mathrm{A}={1,2,3, \ldots \ldots \ldots . ., 17}$. For every nonempty subset $\mathrm{B}$ of $\mathrm{A}$ find the product of the reciprocals of the members of $\mathrm{B}$. The sum of all such product is
(A) $\frac{153}{17 !}$
(B) $\frac{153}{\operatorname{lcm}(1,2, \ldots ., 17)}$
(C) $18$
(D) $17$

Problem 8

The remainder of $f(x)=x^{100}+x^{50}+x^{10}+x^2-6$ when divided by $x^2-1$ is
(A) $x+1$
(B) $-2$
(C) $0$
(D) $2$

Problem 9

The number of acute angled triangles whose vertices are chosen from the vertices of a rectangular box is
(A) 6
(B) 8
(C) 12
(D) 24

Problem 10

In the subtraction below, what is the sum of the digits in the result?

$111 \ldots 111 (\text{100 digits}) -222 \ldots222 (\text{50 digits})$

(A) 375
(B) 420
(C) 429
(D) 450

Problem 11

If $m$ and $n$ are positive integers such that $\frac{m+n}{m^2+m n+n^2}=\frac{4}{49}$, then $m+n$ is equal to
(A) 4
(B) 8
(C) 12
(D) 16

Problem 12

Given a sheet of 16 stamps as shown, the number of ways of choosing three connected stamps (two adjacent stamps must have an edge in common) is

(A) 40
(B) 41
(C) 42
(D) 44

Problem 13

In an election 320 votes were cast for five candidates. The winner's margins over the other four candidates were $9,13,18$ and 25 . The lowest number of votes received by a candidate was
(A) 49
(B) 50
(C) 51
(D) 52

Problem 14

A competition has 25 questions and is marked as follows

(A) Five marks are awarded for each correct answer to questions 1 to 15
(B) Six marks are awarded for each correct answer to questions 16 to 25
(C) Each incorrect answer to questions 16 to 20 loses 1 mark
(D) Each incorrect answer to questions 21 to 25 loses 2 marks

Problem 15

A, M, T, I are positive integers such that $A+M+T+I=10$. The maximum possible value of A $\times$ M $\times$ T $\times$ I+A $\times$ M $\times$ T+A $\times$ M $\times$ I+A $\times$ T $\times$ I+M $\times$ T $\times$ I+A $\times$ M+A $\times$ T+A $\times$ I+M $\times$ T+M $\times$ I +T $\times$ 1

(A) 109
(B) 121
(C) 133
(D) 144

Part B

Problem 16

The three digit number $\mathrm{XYZ}$ when divided by 8 , gives as quotient the two digit number $\mathrm{ZX}$ and remainder $\mathrm{Y}$. The number $\mathrm{XYZ}$ is$\rule{1cm}{0.15mm}$

Problem 17

The digit sum of any number is the sum of its digits. $\mathrm{N}$ is a 3 digit number. When the digit sum of $\mathrm{N}$ is subtracted from $\mathrm{N}$, we obtain the square of the digit sum of $\mathrm{N}$. The number $\mathrm{N}$ is $\rule{1cm}{0.15mm}$.

Problem 18

A $4 \times 4$ anti-magic square is an arrangement of the numbers 1 to 16 in a square so that the totals of each of the four rows, four columns and the two diagonals are ten consecutive numbers in some order. The diagram shows an incomplete anti magic square. When it is completed, the number in the position of ${ }^*$ is $\rule{1cm}{0.15mm} $.

Problem 19

An escalator moves up at a constant rate. John walks up the escalator at the rate of one step per second and reaches the top in twenty seconds. The next day John's rate was two steps per second, and he reached the top in sixteen seconds. The number of steps in the escalator is $\rule{1cm}{0.15mm}$

Problem 20

In a stack of coins, each row has exactly one coin less than the row below. If we have nine coins, two such towers are possible. Of these, the tower on the left is the tallest. If you have 2015 coins, the height of the tallest towers is $\rule{1cm}{0.15mm}$.

Problem 21

Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles A and $B$ are congruent. Circle $C$ has radius 1 unit and passes through the centre of circle $D$. Then the radius of circle B is units $\rule{1cm}{0.15mm}$.

Problem 22

The number of different integers $x$ that satisfy the equation $\left(x^2-5 x+5\right)^{\left(x^2-11 x+30\right)}=1$ is $\rule{1cm}{0.15mm}$.

Problem 23

In a single move a King $\mathrm{K}$ is allowed to move to any of the squares touching the square it is on, including diagonals, as indicated in the figure. The number of different paths using exactly seven moves to go from $A$ to $B$ is $\rule{1cm}{0.15mm}$.

Problem 24

In $\triangle A B C$ shows below, $A B=A C, F$ is a point on $A B$ and $E$ a point on $A C$ such that $A F=E F, H$ is a point in the interior of $\triangle A B C, D$ is a point on $B C$ and $G$ is a point on $A B$ such that $E H=C H=D H=G H=D G$ $=\mathrm{BG}$. Also, $\angle \mathrm{CHE}=\angle \mathrm{HGF}$. The measure of $\angle \mathrm{BAC}$ in degree is $\rule{1cm}{0.15mm}$.

Problem 25

Let $x$ and $y$ be real numbers satisfying $x^4 y^5+y^4 x^5=810$ and $x^3 y^6+y^3 x^6=945$. Then the value of $2 x^3+$ $x^3 y^3+2 y^3$ is $\rule{1cm}{0.15mm}$.

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