Let $AG, CH$ be the angle bisectors of a triangle $ABC$. It is known that one of the intersections of the circles of triangles $ABG$ and $ACH$ lies on the side $BC$. Prove that the angle $BAC$ is $60 ^o$

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.

$\boxed{\text{G5}}$ The incircle of a triangle $ABC$ touches its sides $BC$,$CA$,$AB$ at the points $A_1$,$B_1$,$C_1$.Let the projections of the orthocenter $H_1$ of the triangle $A_{1}B_{1}C_{1}$ to the lines $AA_1$ and $BC$ be $P$ and $Q$,respectively. Show that $PQ$ bisects the line segment $B_{1}C_{1}$

Galatasaray and Fenerbahce have qualified last $16$ in the Europen Champions League. Aftar a random draw, eight matches are regulated in that knock-out phase. The winners of the eight matches will qualify for the next round - round of $8$. Knock-out phase continues until one team remains. If each team has equal chance to win, what is the propability of having a Galatasaray-Fenerbahce match? $ \textbf{(A)}\ \dfrac {1}{32} \qquad\textbf{(B)}\ \dfrac {1}{16} \qquad\textbf{(C)}\ \dfrac {1}{8} \qquad\textbf{(D)}\ \dfrac {1}{4} \qquad\textbf{(E)}\ \text{None of the preceding} $

An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

$ A,B,C,D$ are four "consecutive" points on the circumference of a circle and $ P, Q, R, S$ are points on the circumference which are respectively the midpoints of the arcs $ AB,BC,CD,DA$. Prove that $ PR$ is perpendicular to $ QS$.

Two sets of real numbers $A=\{a_1,a_2,\cdots,a_{100}\},B=\{b_1,b_2,\cdots,b_{50}\}$. Mapping $f:A\to B$, $\forall i(1\leq i\leq 50),\exists j(1\leq j\leq100),f(a_j)=b_i$, and $f(a_1)\leq f(a_2)\leq\cdots\leq f(a_{100})$ Then the number of different $f$ is $\text{(A)}\text{C}_{100}^{50}\qquad\text{(B)}\text{C}_{99}^{50}\qquad\text{(C)}\text{C}_{100}^{49}\qquad\text{(D)}\text{C}_{99}^{49}$

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

Is it possible for Pingu to choose $2025$ positive integers $a_1, ..., a_{2025}$ such that: 1. The sequence $a_i$ is increasing; 2. $\gcd(a_1,a_2)>\gcd(a_2,a_3)>...>\gcd(a_{2024},a_{2025})>\gcd(a_{2025},a_1)>1$? [i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]

Suppose that to every point of the plane a colour, either red or blue, is associated. (a) Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points $A,B$ and $C$ of the same colour such that $B$ is the midpoint of $AC$. (b) Show that there must be an equilateral triangle with all vertices of the same colour.

Robert has five beads in his hand, with the letters $C, M, I, M,$ and $C,$ and he wants to make a circular bracelet spelling "$CMIMC.$" However, the power went out, so Robert can no longer see the beads in his hand. Thus, he puts the five beads on the bracelet randomly, hoping that the bracelet, when possibly rotated or flipped, spells out "$CMIMC.$" What is the probability that this happens? (Robert doesn’t care whether some letters appear upside down or backwards.)

An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence \[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \] for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.

What is the smallest possible value of $x^2+2y^2-x-2y-xy$?

Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$, $A_{2}$ of the given set $A$, we fix the number of elements of $A_1 \cap A_2$. Find the sum of all the numbers obtained in the described way.

Let $n \geq 2$ be an integer. Switzerland and Liechtenstein are performing their annual festive show. There is a field divided into $n \times n$ squares, in which the bottom-left square contains a red house with $k$ Swiss gymnasts, and the top-right square contains a blue house with $k$ Liechtensteiner gymnasts. Every other square only has enough space for a single gymnast at a time. Each second either a Swiss gymnast or a Liechtensteiner gymnast moves. The Swiss gymnasts move to either the square immediately above or to the right and the Liechtensteiner gymnasts move either to the square immediately below or to the left. The goal is to move all the Swiss gymnasts to the blue house and all the Liechtensteiner gymnasts to the red house, with the caveat that a gymnast cannot enter a house until all the gymnasts of the other nationality have left. Determine the largest $k$ in terms of $n$ for which this is possible.

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$. A side of triangle $T_k$ is equal to $\frac{1}{k}$, $k = 1,2, . . . ,1979$. At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don’t intersect (points of contact are allowed)?

Square $ABCD$ has side length $4$. Side $AB$ is extended to point $E$ so that $AE$ has the same length as $AC$, as shown below. What is the length of $EC$? Express your answer as a decimal to the nearest hundredth. [asy] size(80); defaultpen(fontsize(8pt)); pair EE = (4sqrt(2),0); pair A = (0,0); pair B = (4,0); pair C = (4,4); pair D = (0,4); draw(A--B--C--D--cycle); draw(A--EE); draw(C--EE,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,N); label("$D$",D,N); label("$E$",EE,S); [/asy]

A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to [asy] draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); draw((2,0)--(2,2)); draw((0,2)--(1,3)); draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); draw((2,10/3)--(5/2,23/6)); draw((3,3)--(5/2,3)); [/asy] $\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25$

In the game of [i]Winners Make Zeros[/i], a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves: [list] [*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$. [*] If $m<n$, the player choose a positive integer $c$ such that $n-cm\geq 0$, and replaces $(m,n)$ with $(m,n-cm)$. [/list] When $m$ or $n$ becomes $0$, the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$, find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move.

For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.

In a party, there are $2n + 1$ people. It's well known that for every group of $n$ people, there exist a person(out of the group) who knows all them(the $n$ people of the group). Show that there exist a person who knows all the people in the party.

Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.

$N$ cells are marked on an $n\times n$ table so that at least one marked cel is among any four cells of the table which form the figure [img]https://cdn.artofproblemsolving.com/attachments/2/2/090c32eb52df31eb81b9a86c63610e4d6531eb.png[/img] (tbe figure may be rotated). Find the smallest possible value of $N$. (E. Barabanov)