PART - I

Problem 1

If \(2^{x-1}+2^{x-2}+2^{x-3}=\frac{1}{16}\), find \(2^x\)

(a) \(\frac{1}{14}\)
(b) \(\frac{2}{3}\)
(c) \(\sqrt[14]{2}\)
(d) \(\sqrt[3]{4}\)

Answer: A

Problem 2

If the number of sides of a regular polygon is decreased from 10 to 8, by how much does the measure of each of its interior angles decrease?

(a) \(30^{\circ}\)
(b) \(18^{\circ}\)
(c) \(15^{\circ}\)
(d) \(9^{\circ}\)

Answer: D

Problem 3

Sylvester has 5 black socks, 7 white socks, 4 brown socks, where each sock can be worn on either foot. If he takes socks randomly and without replacement, how many socks would be needed to guarantee that he has at least one pair of socks of each color?

(a) 13
(b) 14
(c) 15
(d) 16

Answer: B

Problem 4

Three dice are simultaneously rolled. What is the probability that the resulting numbers can be arranged to form an arithmetic sequence?

(a) \(\frac{1}{18}\)
(b) \(\frac{11}{36}\)
(c) \(\frac{7}{36}\)
(d) \(\frac{1}{6}\)

Answer: C

Problem 5

Sean and the bases of three buildings A, B, and C are all on level ground. Sean measures the angles of elevation of the tops of buildings A and B to be \(62^{\circ}\) and \(57^{\circ}\), respectively. Meanwhile, on top of building C, CJ spots Sean and determines that the angle of depression of Sean from his location is \(31^{\circ}\). If the distance from Sean to the bases of all three buildings is the same, arrange buildings A, B, and C in order of increasing heights.

(a) C, B, A
(b) B, C, A
(c) A, C, B
(d) A, B, C

Answer: A

Problem 6

A function \(f: \mathbb{R} \rightarrow \mathbb{R}\) satisfies \(f(x y)=f(x) / y^2\) for all positive real numbers x and y. Given that \(f(25)=48\), what is \(f(100)\) ?

(a) 1
(b) 2
(c) 3
(d) 4

Answer: C

Problem 7

A trapezoid has parallel sides of lengths 10 and 15; its two other sides have lengths 3 and 4. Find its area.

(a) 24
(b) 30
(c) 36
(d) 42

Answer: B

Problem 8

Find the radius of the circle tangent to the line \(3 x+2 y+4=0\) at \((-2,1)\) and whose center is on the line \(x-8 y+36=0\).

(a) \(2 \sqrt{13}\)
(b) \(2 \sqrt{10}\)
(c) \(3 \sqrt{5}\)
(d) \(5 \sqrt{2}\)

Answer: A

Problem 9

A circle is inscribed in a rhombus which has a diagonal of length 90 and area 5400. What is the circumference of the circle?

(a) \(36 \pi\)
(b) \(48 \pi\)
(c) \(72 \pi\)
(d) \(90 \pi\)

Answer: C

Problem 10

Suppose that n identical promo coupons are to be distributed to a group of people, with no assurance that everyone will get a coupon. If there are 165 more ways to distribute these to four people than there are ways to distribute these to three people, what is n ?

(a) 12
(b) 11
(c) 10
(d) 9

Answer: D

Problem 11

Let x and y be positive real numbers such that

\((\log x 64+\log \) \({y^2}\) \(16=\frac{5}{3})\) and \((\log x 64+\log \) \({x^2}\) \(16= 1\)

What is the value of \(\log _2(x y)\) ?

(a) 16
(b) 3
(c) \(\frac{1}{3}\)
(d) \(\frac{1}{48}\)

Answer: A

Problem 12

The figure below shows a parallelogram ABCD with CD=18. Point F lies inside ABCD and lines AB and DF meet at E. If AE=12 and the areas of triangles FEB and FCD are 30 and 162 , respectively, find the area of triangle BFC.

(a) 162
(b) 156
(c) 150
(d) 144

Answer: D

Problem 13

A semiprime is a natural number that is the product of two primes, not necessarily distinct. How many subsets of the set \({2,4,6, \ldots, 18,20}\) contain at least one semiprime?

(a) 768
(b) 896
(c) 960
(d) 992

Answer: C

Problem 14

The number whose base- b representation is \(91_b\) is divisible by the number whose base- b representation is \(19_b\). How many possible values of b are there?

(a) 2
(b) 3
(c) 4
(d) 5

Answer: B

Problem 15

The number of ordered pairs (a, b) of relatively prime positive integers such that \(ab=36 !\) is

(a) 128
(b) 1024
(c) 2048
(d) 4096

Answer: C

PART - II

Problem 16

Which of the following cannot be the difference between a positive integer and the sum of its digits?

(a) 603
(b) 684
(c) 765
(d) 846

Answer: B

Problem 17

Evaluate the sum

\((\sum_{n=0}^{2019} \cos \left(\frac{n^2 \pi}{3}\right)).\)

(a) 0
(b) 1
(c) -1
(d) \(\frac{1}{2}\)

Answer: A

Problem 18

There is an unlimited supply of red \(4 \times 1\) tiles and blue \(7 \times 1\) tiles. In how many ways can an \(80 \times 1\) path be covered using nonoverlapping tiles from this supply?

(a) 2381
(b) 3382
(c) 5384
(d) 6765

Answer: C

Problem 19

For a real number \(t,\lfloor t\rfloor\) is the greatest integer less than or equal to t. How many natural numbers n are there such that \(\left\lfloor\frac{n^3}{9}\right\rfloor\) is prime?

(a) 3
(b) 9
(c) 27
(d) infinitely many

Answer: A

Problem 20

A quadrilateral with sides of lengths 7,15,15, and d is inscribed in a semicircle with diameter d, as shown in the figure below.

Find the value of d.
(a) 18
(b) 22
(c) 24
(d) 25

Answer: D

Problem 21

Find the sum of all real numbers b for which all the roots of the equation \(x^2+b x-3 b=0\) are integers.

(a) 4
(b) -8
(c) -12
(d) -24

Answer: D

Problem 22

A number x is selected randomly from the set of all real numbers such that a triangle with side lengths 5,8 , and x may be formed. What is the probability that the area of this triangle is greater than 12 ?

(a) \(\frac{3 \sqrt{15}-5}{10}\)
(b) \(\frac{3 \sqrt{15}-\sqrt{41}}{10}\)
(c) \(\frac{3 \sqrt{17}-5}{10}\)
(d) \(\frac{3 \sqrt{17}-\sqrt{41}}{10}\)

Answer: C

Problem 23

Two numbers a and b are chosen randomly from the set \({1,2, \ldots, 10}\) in order, and with replacement. What is the probability that the point \((a, b)\) lies above the graph of \(y=a x^3-b x^2 ?\)

(a) \(\frac{4}{25}\)
(b) \(\frac{9}{50}\)
(c) \(\frac{19}{100}\)
(d) \(\frac{1}{5}\)

Answer: C

Problem 24

For a real number \(t,\lfloor t\rfloor\) is the greatest integer less than or equal to t. How many integers n are there with \(4 \leq n \leq 2019\) such that \(\lfloor\sqrt{n}\rfloor\) divides n and \(\lfloor\sqrt{n+1}\rfloor\) divides \(n+1 ?\)

(a) 44
(b) 42
(c) 40
(d) 38

Answer: B

Problem 25

The number \(20^5+21\) has two prime factors which are three-digit numbers. Find the sum of these numbers.

(a) 1112
(b) 1092
(c) 1062
(d) 922

Answer: A

PART - III

Problem 26

Find the number of ordered triples of integers \((m, n, k)\) with \(0<k<100\) satisfying
\([
\frac{1}{2^m}-\frac{1}{2^n}=\frac{3}{k} .
]\)

Answer: 13

Problem 27

Triangle A B C has \(\angle B A C=60^{\circ}\) and circumradius 15 . Let O be the circumcenter of A B C and let P be a point inside A B C such that O P=3 and \(\angle B P C=120^{\circ}\). Determine the area of triangle B P C.

Answer: \(54 \sqrt{3}\)

Problem 28

A string of 6 digits, each taken from the set \({0,1,2}\), is to be formed. The string should not contain any of the substrings 012,120 , and 201 . How many such 6 -digit strings can be formed?

Answer: 492

Problem 29

Suppose a, b, and c are positive integers less than 11 such that

\(3 a+b+c \equiv a b c(\bmod 11)\)

\(a+3 b+c \equiv 2 a b c(\bmod 11)\)

\(a+b+3 c \equiv 4 a b c(\bmod 11)\)

What is the sum of all the possible values of a b c ?

Answer: 198

Problem 30

Find the minimum value of \(\frac{7 x^2-2 x y+3 y^2}{x^2-y^2}\) if x and y are positive real numbers such that \(x>y\).

Answer: \(2 \sqrt{6}+2\)

PART I

Problem 1

Find x if \(\frac{79}{125}\left(\frac{79+x}{125+x}\right)=1.\)

(a) 0
(b) -46
(c) -200
(d) -204

Answer : D

Problem 2

The line \(2 x+a y=5\) passes through (-2,-1) and (1, b). What is the value of b ?

(a) \(-\frac{1}{2}\)
(b) \(-\frac{1}{3}\)
(c) \(-\frac{1}{4}\)
(d) \(-\frac{1}{6}\)

Answer : B

Problem 3

Let ABCD be a parallelogram. Two squares are constructed from its adjacent sides, as shown in the figure below. If \(\angle BAD=56^{\circ}\), find \(\angle ABE+\angle ADH+\angle FC G\), the sum of the three highlighted angles.

(a) \(348^{\circ}\)
(b) \(384^{\circ}\)
(c) \(416^{\circ}\)
(d) \(432^{\circ}\)

Answer : C

Problem 4

For how many integers x from 1 to 60 , inclusive, is the fraction \(\frac{x}{60}\) already in lowest terms?

(a) 15
(b) 16
(c) 17
(d) 18

Answer : B

Problem 5

Let r and s be the roots of the polynomial \(3 x^2-4 x+2\). Which of the following is a polynomial with roots \(\frac{r}{s}\) and \(\frac{s}{r}\) ?

(a) \(3 x^2+2 x+3\)
(b) \(3 x^2+2 x-3\)
(c) \(3 x^2-2 x+3\)
(d) \(3 x^2-2 x-3\)

Answer : C

Problem 6

If the difference between two numbers is a and the difference between their squares is b, where \(a, b>0\), what is the sum of their squares?

(a) \(\frac{a^2+b^2}{a}\)
(b) \(2\left(\frac{a+b}{a}\right)^2\)
(c) \(\left(a+\frac{b}{a}\right)^2\)
(d) \(\frac{a^4+b^2}{2 a^2}\)

Answer : D

Problem 7

Evaluate the sum
\([
\sum_{n=3}^{2017} \sin \left(\frac{(n !) \pi}{36}\right) .
]\)

(a) 0
(b) \(\frac{1}{2}\)
(c) \(-\frac{1}{2}\)
(d) 1

Answer : B

Problem 8

In \(\triangle ABC,D\) is the midpoint of BC. If the sides AB,BC, and CA have lengths 4,8 , and 6 , respectively, then what is the numerical value of \(AD^2 ?\)

(a) 8
(b) 10
(c) 12
(d) 13

Answer : B

Problem 9

Let A be a positive integer whose leftmost digit is 5 and let B be the number formed by reversing the digits of A. If A is divisible by 11,15,21, and 45 , then B is not always divisible by

(a) 11
(b) 15
(c) 21
(d) 45

Answer : C

Problem 10

In \(\triangle ABC\), the segments AD and AE trisect \(\angle BAC\). Moreover, it is also known that AB= 6, AD=3, AE=2.7, AC=3.8 and DE=1.8. The length of BC is closest to which of the following?

(a) 8
(b) 8.2
(c) 8.4
(d) 8.6

Answer : A

Problem 11

Let \({a_n} \) be a sequence of real numbers defined by the recursion \(a_{n+2}=a_{n+1}-a_n\) for all positive integers n. If \(a_{2013}=2015\), find the value of \(a_{2017}-a_{2019}+a_{2021}.\)

(a) 2015
(b) -2015
(c) 4030
(d) -4030

Answer : D

Problem 12

A lattice point is a point whose coordinates are integers. How many lattice points are strictly inside the triangle formed by the points (0,0),(0,7), and (8,0) ?

(a) 21
(b) 22
(c) 24
(d) 28

Answer : A

Problem 13

Find the sum of the solutions to the logarithmic equation
\([
x^{\log x}=10^{2-3 \log x+2(\log x)^2},
]\)
where \(\log x\) is the \(\log a\) rithm of x to the base 10 .

(a) 10
(b) 100
(c) 110
(d) 111

Answer : C

Problem 14

Triangle ABC has AB=10 and AC=14. A point P is randomly chosen in the interior or on the boundary of triangle ABC. What is probability that P is closer to AB than to AC ?

(a) 1 / 4
(b) 1 / 3
(c) 5 / 7
(d) 5 / 12

Answer : D

Problem 15

Suppose that \({a_n}\) is a nonconstant arithmetic sequence such that \(a_1=1\) and the terms \(a_3, a_{15}, a_{24}\) form a geometric sequence in that order. Find the smallest index n for which \(a_n<0.\)

(a) 50
(b) 51
(c) 52
(d) 53

Answer : C

PART II

Problem 1

Two red balls, two blue balls, and two green balls are lined up into a single row. How many ways can you arrange these balls such that no two adjacent balls are of the same color?

(a) 15
(b) 30
(c) 60
(d) 90

Answer : B

Problem 2

What is the sum of the last two digits of \(403^{\left(10^{10}+6\right)} ?\)

(a) 9
(b) 10
(c) 11
(d) 12

Answer : C

Problem 3

How many strictly increasing finite sequences (having one or more terms) of positive integers less than or equal to 2017 with an odd number of terms are there?

(a) \(2^{2016}\)
(b) \(\frac{4034 !}{(2017 !)^2}\)
(c) \(2^{2017}-2017^2\)
(d) \(2^{2018}-1\)

Answer : A

Problem 4

If one of the legs of a right triangle has length 17 and the lengths of the other two sides are integers, then what is the radius of the circle inscribed in that triangle?

(a) 8
(b) 14
(c) 11
(d) 10

Answer : A

Problem 5

Let N be the smallest three-digit positive number with exactly 8 positive even divisors. What is the sum of the digits of N ?

(a) 4
(b) 9
(c) 12
(d) 13

Answer : B

Problem 6

Let a, b, c be randomly chosen (in order, and with replacement) from the set \({1,2,3, \ldots, 999}\). If each choice is equally likely, what is the probability that \(a^2+b c\) is divisible by 3 ?

(a) \(\frac{1}{3}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{7}{27}\)
(d) \(\frac{8}{27}\)

Answer : A

Problem 7

Folding a rectangular sheet of paper with length \(\ell\) and width w in half along one of its diagonals, as shown in the figure below, reduces its "visible" area (the area of the pentagon below) by \(30 \%\). What is \(\frac{\ell}{w}\) ?

(a) \(\frac{4}{3}\)
(b) \(\frac{2}{\sqrt{3}}\)
(c) \(\sqrt{5}\)
(d) \(\frac{\sqrt{5}}{2}\)

Answer : C

Problem 8

Find the sum of all positive integers k such that k(k+15) is a perfect square.

(a) 63
(b) 65
(c) 67
(d) 69

Answer : C

Problem 9

Let \(f(n)=\frac{n}{3^r}\) where n is an integer, and r is the largest nonnegative integer such that n is divisible by \(3^r\). Find the number of distinct values of f(n) where \(1 \leq n \leq 2017\).

(a) 1344
(b) 1345
(c) 1346
(d) 1347

Answer : B

Problem 10

If A,B, and C are the angles of a triangle such that
\([
5 \sin A+12 \cos B=15
]\)
and
\([
12 \sin B+5 \cos A=2,
]\)
then the measure of angle C is

(a) \(150^{\circ}\)
(b) \(135^{\circ}\)
(c) \(45^{\circ}\)
(d) \(30^{\circ}\)

Answer : D

PART III

Problem 1

How many three-digit numbers are there such that the sum of two of its digits is the largest digit?

Answer : \(279(\text { or } 126)^1\)

Problem 2

In the figure, a quarter circle, a semicircle and a circle are mutually tangent inside a square of side length 2. Find the radius of the circle.

Answer : \(\frac{2}{9}\)

Problem 3

Find the minimum value of
\([
\frac{18}{a+b}+\frac{12}{a b}+8 a+5 b,
]\)
where a and b are positive real numbers.

Answer : 30

Problem 4

Suppose \(\frac{\tan x}{\tan y}=\frac{1}{3}\) and \(\frac{\sin 2 x}{\sin 2 y}=\frac{3}{4}\), where \(0<x, y<\frac{\pi}{2}\). What is the value of \(\frac{\tan 2 x}{\tan 2 y}\) ?

Answer : \(-\frac{3}{11}\)

Problem 5

Find the largest positive real number x such that
\([
\frac{2}{x}=\frac{1}{\lfloor x\rfloor}+\frac{1}{\lfloor 2 x\rfloor},
]\)
where \(\lfloor x\rfloor\) denotes the greatest integer less than or equal to x.

Answer : \(\frac{20}{7}\)