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 A′DE have the same radii.
2023 Indonesia MO
29 August 2023 - Day 1
Determine all functions f:R→R 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.
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 X−d 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 n≥14, Asep can win in at most (n−5)/4 steps.
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
Let a and b be positive integers such that gcd(a,b)+lcm(a,b) is a multiple of a+1. If b≤a, show that b is a perfect square.
Determine the number of permutations a1,a2,…,an of 1,2,…,n such that for every positive integer k with 1≤k≤n, there exists an integer r with 0≤r≤n−k which satisfies 1+2+⋯+k=ar+1+ar+2+⋯+ar+k.
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 ED∥BM. Show that the circumcircle of CDE is tangent to ω.
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).