Philippine Mathematical Olympiad - Problems and Solution - 2023

PART I

Problem 1

1. How many four-digit numbers contain the digit 5 or 7 (or both)?
   (a) 5416
   (b) 5672
   (c) 5904
   (d) 6416

Answer: A

Problem 2

2. Let O(0,0) and A(0,1). Suppose a point B is chosen (uniformly) at random on the circle \(x^2+y^2=1\). What is the probability that OAB is a triangle whose area is at least \((\frac{1}{4}) \)?
   (a) \((\frac{1}{4})\)
   (b) \((\frac{1}{3})\)
   (c) \((\frac{1}{2})\)
   (d) \((\frac{2}{3})\)

Answer: D

Problem 3

3. Suppose \(a_1<a_2<\cdots<a_{25}\) are positive integers such that the average of \(a_1, a_2, \ldots, a_{24}\) is one-half the average of \(a_1, a_2, \ldots, a_{25}.\) What is the minimum possible value of \(a_{25}\) ?
   (a) 26
   (b) 275
   (c) 299
   (d) 325

Answer: D

Problem 4

4. Suppose that a real-valued function \(f(x)\) has domain \((-1,1)\). What is the domain of the function \(f\left(\frac{3-x}{3+x}\right)\) ?
   (a) \((0,+\infty)\)
   (b) \((-3,3)\)
   (c) \((-\infty,-3)\)
   (d) \((-\infty,-3) \cup(3,+\infty)\)

Answer: A

Problem 5

5. Aby chooses a positive divisor \(a\) of 120 (uniformly) at random. Brian then chooses a positive divisor b of a (uniformly) at random. What is the probability that b is odd?
   (a) \(\frac{2}{5}\)
   (b) \(\frac{13}{24}\)
   (c) \(\frac{15}{32}\)
   (d) \(\frac{25}{48}\)

Answer: D

Problem 6

6. Find the sum of the squares of all integers n for which \(\sqrt{\frac{4 n+25}{n-20}}\) is an integer.
   (a) 466
   (b) 475
   (c) 2306
   (d) 2531

Answer: D

Problem 7

7. Let \(k>1.\) The graphs of the functions \(f(x)=\) \(log (\left(\sqrt{x^2+k^3}+x\right))\) and \((g(x)=2 \log \left(\sqrt{x^2+k^3}-x\right))\) have a unique point of intersection (a, b). Find 2 a.
   (a) \(\sqrt{k^3-k+1}\)
   (b) \(k^{3 / 2}-k^{1 / 2}+1\)
   (c) \(k^2+k+1\)
   (d) \(k^2-k\)

Answer: D

Problem 8

8. The sides of a convex quadrilateral have lengths \(12 \mathrm{~cm}, 12 \mathrm{~cm}, 16 \mathrm{~cm},\) and \(16 \mathrm{~cm}\), and they are arranged so that there are no pairs of parallel sides. If one of the diagonals is 20 cm long, and the length of the other diagonal is a rational number, what is the length of the other diagonal?
   (a) \(\frac{48}{5} \mathrm{~cm}\)
   (b) \(\frac{84}{5} \mathrm{~cm}\)
   (c) \(\frac{96}{5} \mathrm{~cm}\)
   (d) \(\frac{108}{5} \mathrm{~cm}\)

Answer: C

Problem 9

9. How many numbers from 1 to \(10^4\) can be expressed both as a sum of five consecutive positive integers and as a sum of seven consecutive positive integers, but not as a sum of three consecutive positive integers?
   (a) 142
   (b) 190
   (c) 285
   (d) 2096

Answer: B

Problem 10

For positive real numbers a and b, the minimum value of
\( \left18 a+\frac{1}{3 b}\right\left3 b+\frac{1}{8 a}\right) \)
can be expressed as \(\frac{m}{n},\) where m and n are relatively prime positive integers. The value of m+n is
(a) 29
(b) 27
(c) 13
(d) 7

Answer: A

Problem 11

In \(\triangle ABC,\) let D be a point on BC such that BD: BC=1: 3. Given that AB=4, AC=5, and AD=3, find the area of \(\triangle ABD\).
(a) \(2 \sqrt{3}\)
(b) \(\sqrt{11}\)
(c) \(\sqrt{10}\)
(d) 3

Answer: B

Problem 12

A five-digit perfect square number \(\overline{ABCDE}\), with A and D both nonzero, is such that the two-digit number \(\overline{DE}\) divides the three-digit number \(\overline{A B C}\). If \(\overline{DE}\) is also a perfect square, what is the largest possible value of \(\overline{ABC} / \overline{DE}\)?
(a) 23
(b) 24
(c) 25
(d) 26

Answer: D

Problem 13

Consider the sequence \(\left{a_n\right},\) where a_1=1, and for \(n \geq 2,\) we have \(a_n=n^{a_{n-1}}.\) What is the remainder when a_{2022} is divided by 23 ?
(a) 11
(b) 12
(c) 21
(d) 22

Answer: C

Problem 14

How many ways are there to divide a \(5 \times 5\) square into three rectangles, all of whose sides are integers? Assume that two configurations which are obtained by either a rotation and/or a reflection are considered the same.
(a) 10
(b) 12
(c) 14
(d) 16

Answer: B

Problem 15

Let \(a_1\) be a positive integer less than 200. Define a sequence \(\left{a_n\right} by 3 a_{n+1}-1=2 a_n for n \geq 1\). Let A be the set of all indices m such that a_m is an integer but \(a_{m+1}\) is not. What is the largest possible element of A ?
(a) 5
(b) 6
(c) 7
(d) 8

Answer: A

PART II

Problem 1

Let \(S={1,2, \ldots, 2023}\). Suppose that for every two-element subset of S, we get the positive difference between the two elements. The average of all of these differences can be expressed as a fraction a / b, where a and b are relatively prime integers. Find the sum of the digits of a+b.

Answer: \(11 \quad(2024 / 3)\)

Problem 2

Let x be the number of six-letter words consisting of three vowels and three consonants which can be formed from the letters of the word "ANTIDERIVATIVE". What is \( |x / 1000| \)?

Answer: 42

Problem 3

Let \(f(x)=\cos (2 \pi x / 3)\). What is the maximum value of \([f(x+1)+f(x+14)+f(x+2023)]^2\) ?

Answer: 3

Problem 4

A function \( f: \mathbb{N} \cup{0} \rightarrow \mathbb{N} \cup{0})\) is defined by \((f(0)=0)\) and \(f(n)=1+f\left(n-3^{\left\lfloor\log _3 n\right\rfloor}\right)\) for all integers \( n \geq 1\). Find the value of \(f\left(10^4\right).\)

Answer: 8

Problem 5

Let \(\triangle ABC\) be equilateral with side length 6. Suppose Pis a point on the same plane as \(\triangle ABC\) satisfying \(PB=2 PC\). The smallest possible length of segment PA can be expressed in the form \(a+b \sqrt{c}\), where a, b, c are integers, and c is not divisible by any square greater than 1 . What is the value of \(a+b+c ?\)

Answer: \(11 \quad(2 \sqrt{13}-4)\)

Problem 6

In chess, a rook may move any number of squares only either horizontally or vertically. In how many ways can a rook from the bottom left corner of an $8 \times 8$ chessboard reach the top right corner in exactly 4 moves? (The rook must not be on the top right corner prior to the 4 th move.)

Answer: 532

Problem 7

In acute triangle ABC, points D and E are the feet of the altitudes from points B and C respectively. Lines BD and CE intersect at point H. The circle with diameter DE again intersects sides AB and AC at points F and G, respectively. Lines FG and AH intersect at point K. Suppose that \(BC=25, BD=20\), and \(BE=7\). The length of AK can be expressed as \(a / b\) where a and b are relatively prime positive integers. Find a-b.

Answer: \(191 \quad(216 / 25)\)

Problem 8

Determine the largest perfect square less than 1000 that cannot be expressed as \(\lfloor x\rfloor+\lfloor 2 x\rfloor+) (\lfloor 3 x\rfloor+\lfloor 6 x\rfloor\) for some positive real number x.

Answer: 784

Problem 9

A string of three decimal digits is chosen at random. The probability that there exists a perfect cube ending in those three digits can be expressed as a / b, where a and b are relatively prime positive integers. Find a+b.

Answer: \(301 \quad(101 / 200)\)

Problem 10

Point D is the foot of the altitude from A of an acute triangle ABC to side BC. The perpendicular bisector of BC meets lines AC and AB at E and P, respectively. The line through E parallel to BC meets line DP at X, and lines AX and BE meet at Q. Given that AX=14 and XQ=6, find AP.

Answer: 35