Let $ABC$ be an acute, non-isosceles triangle with $AD$,$BE$, $CF$ are altitudes and $d$ is the tangent line of the circumcircle of triangle $ABC$ at $A$. The line through $H$ and parallel to $EF$ cuts $DE$, $DF$ at $Q, P$ respectively. Prove that $d$ is tangent to the ex-circle respect to vertex $D$ of triangle $DPQ$.

Let $ABC$ be an acute, non isosceles triangle with the orthocenter $H$, circumcenter $O$ and $AD$ is the diameter of $(O)$. Suppose that the circle $(AHD)$ meets the lines $AB, AC$ at $F$, respectively. Denote $J, K$ as orthocenter and nine- point center of $AEF$. Prove that $HJ \parallel BC$ and $KO = KH$.

Let $ABC$ be an acute, non-isosceles triangle inscribed in (O) and $BB'$, $CC'$ are altitudes. Denote $E, F$ as the intersections of $BB'$, $CC'$ with $(O)$ and $D, P, Q$ are projections of $A$ on $BC$, $CE$, $BF$. Prove that the perpendicular bisectors of $PQ$ bisects two segments $AO$, $BC$.

Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.

Let $ABCD$ be a rectangle with $P$ lies on the segment $AC$. Denote $Q$ as a point on minor arc $PB$ of $(PAB)$ such that $QB = QC$. Denote $R$ as a point on minor arc $PD$ of $(PAD)$ such that $RC = RD$. The lines $CB$, $CD$ meet $(CQR)$ again at $M, N$ respectively. Prove that $BM = DN$. by Tran Quang Hung

Let $A$ be a point lies outside circle $(O)$ and tangent lines $AB$, $AC$ of $(O)$. Consider points $D, E, M$ on $(O)$ such that $MD = ME$. The line $DE$ cuts $MB$, $MC$ at $R, S$. Take $X \in OB$, $Y \in OC$ such that $RX, SY \perp DE$. Prove that $XY \perp AM$.

Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$

Let $ABC$ be an non-isosceles triangle with incenter $I$, circumcenter $O$ and a point $D$ on segment $BC $such that $(BID) $cut segments $AB $ at$ E $and $(CID) $cuts segment $AC $at $F$ Circle $(DEF)$ cuts segments $AB$,$AC $again at $M,N$. Let $P$ The intersection of $IB$ and $DE $ , $Q$ The intersection of $IC$and $DF$ . Prove that $EN,FM,PQ $are parallel and the median of vertex $I$in triangle $IPQ$ bisects the arc $BAC$ of $(O)$.

Let $ABC$ be a triangle inscribed in circle $(O)$ with diamter $KL$ passes through the midpoint $M$ of $AB$ such that $L, C$ lie on the different sides respect to $AB$. A circle passes through $M, K $cuts $LC$ at$ P, Q $(point $P$ lies between$ Q, C$). The line $KQ $cuts $(LMQ)$ at $R$. Prove that $ARBP$ is cyclic and$ AB$ is the symmedian of triangle $APR$. Please help

Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.

Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$. Proposed by Iman Maghsoudi - Siamak Ahmadpour

Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, ex-center in angle $A$ is $J$. Denote $D$ as the tangent point of $(I)$ on $BC$ and the angle bisector of angle $A$ cuts $BC$, $(O)$ respectively at $E, F$. The circle $(DEF )$ meets $(O)$ again at $T$. Prove that $AT$ passes through an intersection of $(J)$ and $(DEF )$.

Let $ABCD$ be a quadrilateral with $\angle A = \angle B = 90^o$, $AB = AD$. Denote $E$ as the midpoint of $AD$, suppose that $CD = BC + AD$, $AD > BC$. Prove that $\angle ADC = 2\angle ABE$.

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

Let $ABC$ be convex quadrilateral and $X$ lying inside it such that $XA \cdot XC^2 = XB \cdot XD^2$ and $\angle AXD + \angle BXC = \angle CXD$. Prove that $\angle XAD + \angle XCD = \angle XBC + \angle XDC$.

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, incenter $I$ and $(I)$ tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. Suppose that $EF$ cuts $(O)$ at $P, Q$. Prove that $(PQD)$ bisects segment $BC$.

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Tangent lines to $(O)$ at $B,C$ meet at $T$. A line passes through $T$ cuts segments $AB$ at $D$ and cuts ray $CA$ at $E$. Take $M$ as midpoint of $DE$ and suppose that $MA$ cuts $(O)$ again at $K$. Prove that $(MKT)$ is tangent to $(O)$.

Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$ ($D \in BC$), $M$ is the midpoint of $AD$ and $O$ is the circumcenter. Line $AO$ meets $BC$ at $K$ and circle of center $K$, radius $KA$ cuts $AB,AC$ at $E, F$ respectively. Prove that $AO$ bisects $EF$.

Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$

Let $ABC$ be a non-isosceles triangle with altitudes $AD$, $BE$, $CF$ with orthocenter $H$. Suppose that $DF \cap HB = M$, $DE \cap HC = N$ and $T$ is the circumcenter of triangle $HBC$. Prove that $AT\perp MN$.

Let $ABC$ be triangle with the symmedian point $L$ and circumradius $R$. Construct parallelograms $ ADLE$, $BHLK$, $CILJ$ such that $D,H \in AB$, $K, I \in BC$, $J,E \in CA$ Suppose that $DE$, $HK$, $IJ$ pairwise intersect at $X, Y,Z$. Prove that inradius of $XYZ$ is $\frac{R}{2}$ .

Let $ABC$ be triangle with $M$ is the midpoint of $BC$ and $X, Y$ are excenters with respect to angle $B,C$. Prove that $MX$, $MY$ intersect $AB$, $AC$ at four points that are vertices of circumscribed quadrilateral.

The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right

Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?

Each of $N$ people have chosen some $5$ elements from a $23$-element set so that any two people share at most $3$ chosen elements. Does this mean that $N \le 2020$? Answer the same question with $25$ instead of $23$.

Find all positive integer $n\ge 3$ such that it is possible to mark the vertices of a regular $n$- gon with the number from 1 to n so that for any three vertices $A, B$ and $C$ with $AB = AC$, the number in $A$ is greater or smaller than both numbers in $B, C$.

Prove that it is impossible to fill the cells of an $8 \times 8$ table with the numbers from $ 1$ to $64$ (each number must be used once) so that for each $2\times 2$ square, the difference between products of the numbers on it’s diagonals will be equal to $ 1$.

For a positive integer $k$, denote by $f(k)$ the number of positive integer $m$ such that the remainder of $km$ modulo $2019^3$ is greater than $m$. Find the amount of different numbers among $f(1), f(2), ..., f(2019^3)$.

Let $n$ be a positive integer. What is the smallest value of $m$ with $m > n$ such that the set $M = \{n, n + 1, ..., m\}$ can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

Call a positive integer $x$ to be remote from squares and cubes if each integer $k$ satisfies both $|x - k^2| > 10^6$ and $|x - k^3| > 10^6$. Prove that there exist infinitely many positive integer $n$ such that $2^n$ is remote from squares and cubes.

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

Let $P (x)$ be a non constant integer polynomial and positive integer $n$. The sequence $a_0, a_1, ...$ is defined by $a_0 = n$ and $a_k = P (a_{k-1})$ for $k \ge 1$. Given that for each positive integer $b$, the sequence contains a $b$-th power of some positive integer greater than $1$. Prove that deg $P = 1$

There are $330$ seats in the first row of the auditorium. Some of these seats are occupied by $25$ viewers. Prove that among the pairwise distances between these viewers, there are two equal.

Given $n \ge 2$ distinct positive integers $a_1, a_2, ..., a_n$ none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

Given $m, n$ such that $m > n^{n-1}$ and the number $m+1$, $m+2$,$ ...$, $m+n$ are composite. Prove that there exist distinct primes $p_1, p_2, ..., p_n$ such that $m + k$ is divisible by $p_k$ for each $k = 1, 2, ...$