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Ramanujan Contest (NMTC Inter 2018 - XI and XII Grades) - Stage I- Problems and Solution

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Part A

Problem 1

In the addition shown, each of the letters $\mathrm{T}, \mathrm{H}, \mathrm{I}, \mathrm{S}$ represents a non zero digit. What is $\mathrm{T}+\mathrm{H}+\mathrm{I}+\mathrm{S}$ ?

(A) 34
(B) 32
(C) 24
(D) 22

Problem 2

We have four sets $S_1, S_2, S_3, S_4$ each containing a number of parallel lines. The set $S_1$ contains $i+1$ parallel lines $i=1,2,3,4$. A line in $S_i$ is not parallel to lines in $S_j$ when $i \neq j$. In how many points do these lines intersect?

(A) 54
(B) 63
(C) 71
(D) 95

Problem 3

An old tanker is $100 \mathrm{~km}$ due north of a cruise liner. The tanker sails Southeast at a speed of 20 kilometers per hour and the liner sail Northwest at a speed of 10 kilometres per hour. What is the shortest distance between the two boats during the subsequent motion?

(A) $50 \sqrt{2} \mathrm{~km}$
(B) $60 \mathrm{~km}$
(C) $80 \mathrm{~km}$
(D) $100 \mathrm{~km}$

Problem 4

Volume A equals one fourth of the sum of the volumes B and C, while volume B equals one sixth of the sum of the volumes $A$ and $C$. The ratio of volume $C$ to the sum of volumes of $A$ and $B$ is

(A) $2: 3$
(B) $9: 10$
(C) $7: 12$
(D) $12: 23$

Problem 5

In the ninety-nine shop every item costs some whole number of rupees plus 99 paise. Rhea spent sixty five rupees and seventy six paise in buying some items from the shop. How many items did she buy?

(A) 23
(B) 24
(C) 65
(D) 66

Problem 6

The diagram shows a rectangle $A B C D$ where $A B: A D=1: 2$. Point $E$ on $A C$ is such that $D E$ is perpendicular to $A C$. What is the ratio of the area of the triangle DCE to the rectangle ABCD?

(A) $1: 4 \sqrt{2}$
(B) $1: 6$
(C) $1: 8$
(D) $1: 10$

Problem 7

The numbers $2,3,12,14,15,20,21$ may be divided into two sets so that the product of the numbers in each set is the same. What is this product?

(A) 420
(B) 1260
(C) 2520
(D) 6720

Problem 8

$A B C D$ is a trapezium with $A D=D C=C D=10$ units and $A B=22$ units. Semi circles are drawn as shown in the figure. The area of the region bounded by these semi circles in square units is

(A) $128+48 \pi$
(B) $128+24 \pi$
(C) $116+48 \pi$
(D) $116+24 \pi$

Problem 9

Consider the number of ways in which five girls and five boys sit in ten seats that are equally spaced around a circle. The proportion of the seating arrangements in which no two girls sit at the ends of a diameter is

(A) $\frac{1}{2}$
(B) $\frac{8}{63}$
(C) $\frac{55}{63}$
(D) None of the above

Problem 10

Let $A=1^{-4}+2^{-4}+3^{-4}+\ldots \ldots \ldots \ldots$, the sum of reciprocals of fourth powers of integers and $\mathrm{B}=1^{-4}+3^{-4}+5^{-4}+\ldots \ldots \ldots \ldots$, the sum of reciprocals of fourth powers of odd positive integers. The value of $\mathrm{A} / \mathrm{B}$ as a fraction is

(A) $\frac{16}{15}$
(B) $\frac{32}{31}$
(C) $\frac{64}{63}$
(D) $\frac{128}{127}$

Problem 11

The number $5^{\left(6^7\right)}$ is written on the board (in base 10). Gia takes two of the digits at a time, erases them but appends the sum of those digits at the end. She repeats this till she ends up with one digit on the board. What is the digit that remains on the board?

(A) 1
(B) 5
(C) 6
(D) 7

Problem 12

Seven points are marked on the circumference of a circle and all pairs of points are joined by straight lines. No three of these lines have a common point and any two intersect at a point inside the circle. Into how many regions is the interior of the circle divided by these lines?

(A) 64
(B) 63
(C) 57
(D) 56

Problem 13

The diagram below shows a regular hexagon with side length 1 , insceibed in a square. Two of the vertices lie on the diagonal of the square and the remaining vertices lie on its sides. What is the area of the square?

(A) $\frac{7}{2}$
(B) 4
(C) $2+\sqrt{3}$
(D) $3+\sqrt{2}$

Problem 14

$\mathrm{AB}$ is a diameter of a semicircle of centre $\mathrm{O}$. C is the midpoint of the arc $\mathrm{AB}$. $\mathrm{AC}$ and the tangent at $B$ to the semicircle meet at P. D is the midpoint of $B P$. If $A C D O$ is a parallelogram and $\angle P A D=\theta$, then $\sin \theta$ is

(A) $\frac{1}{\sqrt{5}}$
(B) $\frac{1}{\sqrt{10}}$
(C) $\frac{2}{\sqrt{10}}$
(D) $\frac{3}{\sqrt{10}}$

Problem 15

The real valued function $f(x)$ satisfies the equation $2 f(1-x)+1=x f(x)$ for all $x$. Then $\left(x^2-x+4\right)$ $f(x)$ equals

(A) $x-1$
(B) $x$
(C) $x+1$
(D) $x-3$

Part B

Problem 16

The number of ways in which 26 identical chocolates be distributed between Amy, Bob, Cathy and Daniel so that each receives at least one chocolate and Amy receives more chocolates than Bob is $\rule{2cm}{0.15mm}$

Problem 17

A set $\mathrm{S}$ contains 11 numbers. The average of the numbers in $\mathrm{S}$ is 302 . The average of the six smallest numbers of $S$ is 100 and the average of the six largest of the numbers is 300 . What is the median of the numbers in $\mathrm{S}$ $\rule{2cm}{0.15mm}$

Problem 18

The sum of the angles $1,2,3,4,5,6,7,8$ in degrees shows in the following figure is $\rule{2cm}{0.15mm}$

Problem 19

The number of positive integers less than 2018 that are divisible by 6 but are not divisible by at least one of the numbers 4 or 9 is $\rule{2cm}{0.15mm}$

Problem 20

\[x(x+1)(x+2) \ldots \ldots(x+23)=\sum_{n=1}^{24} a_n x^n\] the number of coefficients $a_n$ that are multiples of 3 is $\rule{2cm}{0.15mm}$

Problem 21

A square is cut into 37 squares of which 36 have area 1 square $\mathrm{cms}$. The length of the side of the original square is $\rule{2cm}{0.15mm}$

Problem 22

There are 4 coins in a row and all are showing heads to start with. The coins can be flipped with the following rules :
(a) The fourth coin (from the left) can be flipped any time
(b) An intermediate coin can be changed to tail only if its immediate neighbor on the right is heads and all other coins (if any) to its right are tails.
(c) Only one coin can be flipped in one step.

The minimum number of steps required to bring all coins to show tails is $\rule{2cm}{0.15mm}$

Problem 23

A poet met a tortoise sitting under a tree. When the tortoise was the poet's age, the poet was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If all the ages are in whole number of years, and the sum of their ages is now 264 , the age of the tree in years is $\rule{2cm}{0.15mm}$

Problem 24

The sum of all real value of $x$ satisfying $\left(x+\frac{1}{x}-17\right)^2=x+\frac{1}{x}+17$ is $\rule{2cm}{0.15mm}$

Problem 25

On the inside of a square with side length 6 , construct four congruent isosceles triangles each with base 6 and height 5 , and each having one side coinciding with a different side of the square. The area of the octagonal region common to the interiors of all four triangles is $\rule{2cm}{0.15mm}$

Problem 26

In a triangle with integer side lengths, one side is thrice the other. The third side is $15 \mathrm{~cm}$. The greatest possible perimeter of the triangle is (in $\mathrm{cm}$ ) $\rule{2cm}{0.15mm}$

Problem 27

A cube has edge length $x$ (an integer). three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of $x$ is $\rule{2cm}{0.15mm}$

Problem 28

If $f(x)=a x^4-b x^2+x+5$ and $f(3)=8$, the value of $f(-3)$ is $\rule{2cm}{0.15mm}$

Problem 29

Archana has to choose a three-digit code for her bike lock. The digits can be chosen from 1 to 9 . To help her remember them, she decides to choose three different digits in increasing order, for example 278 . The number of such codes she can choose is $\rule{2cm}{0.15mm}$

Problem 30

Let $\mathrm{S}$ be a set of five different positive integers, the largest of which is $\mathrm{n}$. It is impossible to construct a quadrilateral with non-zero area, whose side-lengths are all distinct elements of $\mathrm{S}$. The smallest possible value of $n$ is $\rule{2cm}{0.15mm}$

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