Problem 1: Let ABC be a triangle with CA = CB and ∠ACB = 120◦, and let, M be the midpoint of AB. Let P be a variable point on ...

Problem 1. The positive integers a0, a1, a2, …, a3030 satisfy 2an+2 = an+1 + 4an for n = 0, 1, 2, …, 3028. Prove that at least one of ...

Problem 1: The number 2021 is fantabulous. For any positive integer m, if any element of the set {m, 2m + 1, 3m} is fantabulous, then all the elements are ...

Problem 1: We call a 5-tuple of integers arrangeable if its elements can be labeled a, b, c, d, e in some order so that a – b + c ...

APMO

APMO Problems and Solutions 2018

Problem 1: Let H be the orthocenter of the triangle ABC. Let M and N be the midpoints of the sides AB and AC, respectively. Assume that H lies inside ...

Problem 1. Let ℤ+ be the set of positive integers. Determine all functions f : ℤ+→ ℤ+ such that a2 + f(a)f(b) is divisible by f(a) + b for all ...

Problem 1. Let Γ be the circumcircle of ∆ABC. Let D be a point on the side BC. The tangent to Γ at A intersects the parallel line to BA ...

Angle: In Euclidean geometry, an angle is a figure formed by two rays with the same endpoint, denoted by “∠”. In the figure, two rays OP and OQ make ...

Problem 1. For each integer a0 > 1, define the sequence a0, a1, a2, … … by: Determine all values of a0 for which there is a number A such ...