Found problems: 160
2011 Iran Team Selection Test, 1
In acute triangle $ABC$ angle $B$ is greater than$C$. Let $M$ is midpoint of $BC$. $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$, prove that $TA=TM$.
2017 Iran Team Selection Test, 3
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.
Prove that $X,Y,Z$ are collinear.
[i]Proposed by Hooman Fattahi[/i]
2016 Iran MO (2nd Round), 3
A council has $6$ members and decisions are based on agreeing and disagreeing votes. We call a decision making method an [b]Acceptable way to decide[/b] if it satisfies the two following conditions:
[b]Ascending condition[/b]: If in some case, the final result is positive, it also stays positive if some one changes their disagreeing vote to agreeing vote.
[b]Symmetry condition[/b]: If all members change their votes, the result will also change.
[b] Weighted Voting[/b] for example, is an [b]Acceptable way to decide[/b]. In which members are allotted with non-negative weights like $\omega_1,\omega_2,\cdots , \omega_6$ and the final decision is made with comparing the weight sum of the votes for, and the votes against. For instance if $\omega_1=2$ and for all $i\ge2, \omega_i=1$, decision is based on the majority of the votes, and in case when votes are equal, the vote of the first member will be the decider.
Give an example of some [b]Acceptable way to decide[/b] method that cannot be represented as a [b]Weighted Voting[/b] method.
2012 Iran Team Selection Test, 2
Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$. Let $C$ be the circumcenter of the triangle $AOB$. Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$. $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$, cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$). Prove that $OP\perp QR$.
[i]Proposed by Mehdi E'tesami Fard, Ali Khezeli[/i]
2016 Iran MO (3rd Round), 1
The sequence $(a_n)$ is defined as:
$$a_1=1007$$
$$a_{i+1}\geq a_i+1$$
Prove the inequality:
$$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$
2016 Iran MO (3rd Round), 3
Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively.
$M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$.
Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$.
Show that: $DK=DL$
2017 Iran Team Selection Test, 1
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
2016 Iran MO (3rd Round), 3
A sequence $P=\left \{ a_{n} \right \}$ is called a $ \text{Permutation}$ of natural numbers (positive integers) if for any natural number $m,$ there exists a unique natural number $n$ such that $a_n=m.$
We also define $S_k(P)$ as:
$S_k(P)=a_{1}+a_{2}+\cdots +a_{k}$ (the sum of the first $k$ elements of the sequence).
Prove that there exists infinitely many distinct $ \text{Permutations}$ of natural numbers like $P_1,P_2, \cdots$ such that$:$
$$\forall k, \forall i<j: S_k(P_i)|S_k(P_j)$$
2018 Iran Team Selection Test, 4
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.
Prove that there exist infinitely many positive integers which they are not "useful but not optimized".
(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)
[i]Proposed by Mohsen Jamali[/i]
2017 Iran MO (3rd round), 1
Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that
$$P(Q(x))=P(x)^{2017}$$
for all real numbers $x$.