Olympiad problems

2021 Iran MO (3rd Round), 1

An acute triangle $ABC$ is given. Let $D$ be the foot of altitude dropped for $A$. Tangents from $D$ to circles with diameters $AB$ and $AC$ intersects with the said circles at $K$ and $L$, in respective. Point $S$ in the plane is given so that $\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ$. Prove that $A, K, L$ and $S$ lie on a circle.

2017 Iran MO (3rd round), 1

Tags: number theory , Divisibility , prime numbers , prime , Iran , IranMO

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2004 Iran MO (2nd round), 5

Tags: geometry , circumcircle , incenter , geometry solved , power of a point , Inversion , IranMO

The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.

2021 Iran MO (3rd Round), 2

Tags: geometry , geometric transformation , reflection , circumcircle , Iran , IranMO , tangent circles

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

2018 Iran MO (3rd Round), 3

Tags: number theory , functional equation , Iran , IranMO

Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.

2017 Iran MO (3rd round), 1

Tags: algebra , polynomial , Iran , IranMO

Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that $$P(Q(x))=P(x)^{2017}$$ for all real numbers $x$.