Olympiad problems

2025 Alborz Mathematical Olympiad, P2

Suppose that for polynomials $ P, Q, R $ with positive integer coefficients, the following two conditions hold: $\bullet$ The constant terms of $ P, Q, R $ are equal. $\bullet$ For all real numbers $ x $, the following relations hold: \[ P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))). \] Prove that for every real number $ x $, $ P(x) = Q(x) = R(x) $. Proposed by Soroush Behroozifar & Ali Nazarboland

2025 Alborz Mathematical Olympiad, P3

Tags: 3D geometry , combination , partition , AlborzMO , combinatorics

Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points? Proposed by Arvin Taheri

2024 Alborz Mathematical Olympiad, P4

Tags: geometry , excenter , collinear , AlborzMO

In triangle $ ABC $, let $ I $ be the $ A $-excenter. Points $ X $ and $ Y $ are placed on line $ BC $ such that $ B $ is between $ X $ and $ C $, and $ C $ is between $ Y $ and $ B $. Moreover, $ B $ and $ C $ are the contact points of $ BC $ with the $ A $-excircle of triangles $ BAY $ and $ AXC $, respectively. Let $ J $ be the $ A $-excenter of triangle $ AXY $, and let $ H' $ be the reflection of the orthocenter of triangle $ ABC $ with respect to its circumcenter. Prove that $ I $, $ J $, and $ H' $ are collinear. Proposed by Ali Nazarboland

2025 Alborz Mathematical Olympiad, P3

Tags: number theory , GCD , Sequence , AlborzMO

For every positive integer $ n $, do there exist pairwise distinct positive integers $ a_1, a_2, \dots, a_n $ that satisfy the following condition? For every $ 3 \leq m \leq n $, there exists an $ i \leq m-2 $ such that: $$ a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i). $$ Proposed by Alireza Jannati

2024 Alborz Mathematical Olympiad, P1

Tags: number theory , Divisors , complete residue system , AlborzMO

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari

2025 Alborz Mathematical Olympiad, P2

Tags: combination , Game Theory , circles , geometry , AlborzMO

In the Jordan Building (the Olympiad building of High School Mandegar Alborz), Ali and Khosro are playing a game. First, Ali selects 2025 points on the plane such that no three points are collinear and no four points are concyclic. Then, Khosro selects a point, followed by Ali selecting another point, and then Khosro selects one more point. The circumcircle of these three points is drawn, and the number of points inside the circle is denoted by $ t $. If Khosro's goal is to maximize $ t $ and Ali's goal is to minimize $ t $, and both play optimally, determine the value of $ t $. Proposed by Reza Tahernejad Karizi

2025 Alborz Mathematical Olympiad, P1

Tags: geometry , midpoint , Parallel Lines , AlborzMO

Let $ M $ and $ N $ be the midpoints of sides $ BC $ and $ AC $, respectively, in an acute-angled triangle $ ABC $. Suppose there exists a point $ P $ on the line segment $ AM $ such that $ \angle NPC = \angle MPC $. Let $ D $ be the intersection point of the line $ NP $ and the line parallel to $ CP $ passing through $ B $. Prove that $ AD = AB $. Proposed by Soroush Behroozifar

2024 Alborz Mathematical Olympiad, P2

Tags: function , functional equation , AlborzMO , algebra

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$ Proposed by Heidar Shushtari

2024 Alborz Mathematical Olympiad, P3

Tags: combinatorics , Game Theory , AlborzMO

A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is $ b $ and $ a $ is entered, the new password is $ c $, which is determined by $ b $ and $ a $ (naturally, the person does not know $ c $ or $ b $). Prove that regardless of the characteristics of this computer, the prisoner can free themselves. Proposed by Reza Tahernejad Karizi

2025 Alborz Mathematical Olympiad, P1

Tags: functional equation , number theory , positive integers , AlborzMO , algebra

Let $ \mathbb{Z^{+}} $ denote the set of all positive integers. Find all functions $ f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} $ such that for every pair of positive integers $ a $ and $ b $, there exists a positive integer $ c $ satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi