id int64 | day int64 | problem string |
|---|---|---|
0 | 1 | There are 2026 integers greater than 1 written on a blackboard, not necessarily distinct. In one move, Confucius chooses two integers m > 1 and n > 1 written in different positions and replaces them by gcd(m,n) and lcm(m,n)/gcd(m,n). He continues making moves for as long as a move is possible. (a) Prove that, regardles... |
1 | 1 | Let ABC be a triangle, and let M and N be the midpoints of AB and AC, respectively. Let K and L be chosen inside triangles BMC and BNC, respectively, such that K lies inside angle LBA, L lies inside angle ACK, and angle KBA = angle ACL, angle LBK = angle LNC, and angle LCK = angle BMK. If O is the circumcentre of trian... |
2 | 1 | Let n be a positive integer. Liu Bang and Xiang Yu have a stick of length 1. First Liu Bang marks at most n points on the stick, and then Xiang Yu marks at most n further points; all marked points are distinct. The stick is cut at every marked point. The players then alternately claim unclaimed pieces, with Liu Bang mo... |
3 | 2 | Shan-Yu and Mulan are playing a game. Let theta be an angle with 0 degrees < theta < 180 degrees known to both players. Initially, Shan-Yu makes a paper triangle T with measurements of his choice. Then they repeatedly perform the following steps: If T has at least one angle measuring exactly theta, then the game stops ... |
4 | 2 | Determine all functions f: R_{>0} -> R_{>0} such that sqrt((x^2 + f(y)^2)/2) >= (f(x) + y)/2 >= sqrt(x f(y)) for every x,y in R_{>0}. |
5 | 2 | Let a_1, a_2, a_3, ... be an infinite sequence of positive integers greater than 1. Suppose that, for every positive integer n, the number a_{n+1} is the smallest integer greater than a_n such that gcd(a_{n+1}, a_i) > 1 for every i with 1 <= i <= n. Prove that there exist positive integers T and L such that a_{n+T} = a... |
IMO 2026 Problems
The six IMO 2026 problem statements, indexed from 0 through 5 in contest order.
IDs 0–2 are from Day 1, and IDs 3–5 are from Day 2.
Schema
id: zero-based problem identifier (0corresponds to Problem 1).day: contest day (1or2).problem: complete English problem statement.
Source
Extracted from the problem statements in SignalPilot Labs' AutoFyn IMO 2026 results: https://github.com/SignalPilot-Labs/AutoFyn/tree/production/results/imo-2026
Official IMO 2026 edition page: https://www.imo-official.org/editions/2026/
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