Olympiad problems

the 11th XMO, 1

Tags: geometry , xmo

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

the 12th XMO, Problem 4

Tags: graph theory , xmo , combinatorics

求最小的 $n,$ 使得对任意有 ${1000}$ 个顶点且每个顶点度均为 ${4}$ 的简单图 $G,$ 都一定可以从中取掉 ${n}$ 条边$,$ 使 ${G}$ 变为二部图$.$

the 12th XMO, Problem 3

Tags: number theory , xmo

Let $a_0=0,a_1\in\mathbb Z_+.$ For integer $n\geq 2,a_n$ is the smallest positive integer satisfy that for $\forall 0\leq i\neq j\leq n-1,a_n\nmid (a_i-a_j).$ (1) If $a_1=2023,$ calculate $a_{10000}.$ (2) If $a_t\leq\frac{a_1}2,$ find the maximum value of $\frac t{a_1}.$

the 12th XMO, Problem 2

Tags: xmo , algebra , inequalities

Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of $$\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}$$where $a_{23}=a_1.$

XMO (China) 2-15 - geometry, 11.1

Tags: geometry , xmo

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.