Philippine Mathematical Olympiad - Problems and Solution - 2017

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