Olympiad problems

2025 SEEMOUS, P1

Let $A$ be an $n\times n$ matrix with strictly positive elements and two vectors $u,v\in\mathbb{R}^n$, also with strictly positive elements, such that $$Au=v\text{ and }Av=u.$$ Prove that $u=v$.

2005 VTRMC, Problem 1

Tags: number theory

Find the largest positive integer $n$ with the property that $n+6(p^3+1)$ is prime whenever $p$ is a prime number such that $2\le p<n$. Justify your answer.

1962 Putnam, A1

Tags: point , geometry , convex

Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

1966 IMO Shortlist, 5

Tags: inequalities , trigonometry , geometric inequality

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2020 Moldova Team Selection Test, 3

Tags: combinatorics

Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?

2004 IMO, 6

Tags: induction , modular arithmetic , number theory , decimal representation

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

2014 Iranian Geometry Olympiad (junior), P1

Tags: geometry

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

1989 Putnam, A3

Tags: search , algebra

Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).

2024 Olimphíada, 4

Tags: geometry

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.

2001 IMO Shortlist, 6

Tags: inequalities

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]