Philippine Mathematical Olympiad - Problems and Solution - 2019

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