2023 Indonesia MO

29 August 2023 - Day 1

1

An acute triangle ABC has BC as its longest side. Points D,E respectively lie on AC,AB such that BA=BD and CA=CE. The point A is the reflection of A against line BC. Prove that the circumcircles of ABC and ADE have the same radii.

2

Determine all functions f:RR such that the following equation holds for every real x,y: f(f(x)+y)=x+f(f(y)).Note: x denotes the greatest integer not greater than x.

3

A natural number n is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is X. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor d of X, and replace X with X+d if Neneng chose the sign "up" or Xd if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero. Prove that if n14, Asep can win in at most (n5)/4 steps.

4

Determine whether or not there exists a natural number N which satisfies the following three criteria: 1. N is divisible by 22023, but not by 22024, 2. N only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of N are odd.

30 August 2023 - Day 2

5

Let a and b be positive integers such that gcd(a,b)+lcm(a,b) is a multiple of a+1. If ba, show that b is a perfect square.

6

Determine the number of permutations a1,a2,,an of 1,2,,n such that for every positive integer k with 1kn, there exists an integer r with 0rnk which satisfies 1+2++k=ar+1+ar+2++ar+k.

7

Given a triangle ABC with ACB=90. Let ω be the circumcircle of triangle ABC. The tangents of ω at B and C intersect at P. Let M be the midpoint of PB. Line CM intersects ω at N and line PN intersects AB at E. Point D is on CM such that EDBM. Show that the circumcircle of CDE is tangent to ω.

8

Let a,b,c be three distinct positive integers. Define S(a,b,c) as the set of all rational roots of px2+qx+r=0 for every permutation (p,q,r) of (a,b,c). For example, S(1,2,3)={1,2,1/2} because the equation x2+3x+2 has roots 1 and 2, the equation 2x2+3x+1=0 has roots 1 and 1/2, and for all the other permutations of (1,2,3), the quadratic equations formed don't have any rational roots. Determine the maximum number of elements in S(a,b,c).