A circle passing through vertices $A$ and $B$ of triangle $ABC$ intersects the sides $AC$ and $BC$ again at points $P$ and $Q$, respectively. Given that the median from vertex $C$ bisect the arc $PQ$ of the circle. Prove that $ABC$ is an isosceles triangle.

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. [img]https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif[/img] We will say that a distribution of numbers is [i]graceful [/i] if all the numbers from $1$ to $n$ appear in the paths, where $n$ is the number of paths. The following is an example of graceful distribution: [img]https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif[/img] Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

For some positive integers $a$ and $b$, $(x^a+abx^{a-1}+13)^{b}(x^3+3bx^2+37)^{a}=x^{42}+126x^{41}+\cdots$. Find the ordered pair $(a, b)$.

The numbers 1 through 10 can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram below shows an arrangement with sum 16. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum. [asy] int i; pair[] A={dir(18+72*0), dir(18+72*1), dir(18+72*2),dir(18+72*3), dir(18+72*4), dir(18+72*0)}; pair O=origin; int[] v = {7,1,10,4,3}; int[] s = {8, 5, 2, 9, 6}; for(i=0; i<5; i=i+1) { label(string(v[i]), A[i], dir(O--A[i])); label(string(s[i]), A[i]--A[i+1], dir(-90)*dir(A[i]--A[i+1])); } draw(rotate(0)*polygon(5));[/asy]

Let ABC be a right triangle with $\angle B = 90^{\circ}$.Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.

Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$. Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$

On the legs $AC, BC$ of a right triangle $\vartriangle ABC$ select points $M$ and $N$, respectively, so that $\angle MBC = \angle NAC$. The perpendiculars from points $M$ and $C$ on the line $AN$ intersect $AB$ at points $K$ and $L$, respectively. Prove that $KL=LB$. (O. Clurman)

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?

Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$? $ \textbf{a)}\ 2 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 5 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{None of above} $

Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.

Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

Let $B$ be a $1 \times 2 \times 4$ box (rectangular parallelepiped). Let $R$ be the set of points that are within distance 3 of some point in $B$. (Note that $R$ contains $B$.) What is the volume of $R$?

a) Find all real numbers $k$ such that there exists a positive constant $c_k$ satisfying $$(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k$$ for all positive real numbers. b) Given the numbers $k$ found, determine the largest number $c_k$.

Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.

At each vertex of a regular $14$-gon, lies a coin. Initially $7$ coins are heads, and $7$ coins are tails. Determine the minimum number $t$ such that it’s always possible to turn over at most $t$ of the coins so that in the resulting $14$-gon, no two adjacent coins are both heads and no two adjacent coins are both tails.

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

The image shown below is a cross with length 2. If length of a cross of length $k$ it is called a $k$-cross. (Each $k$-cross ahs $6k+1$ squares.) [img]http://aycu08.webshots.com/image/4127/2003057947601864020_th.jpg[/img] a) Prove that space can be tiled with $1$-crosses. b) Prove that space can be tiled with $2$-crosses. c) Prove that for $k\geq5$ space can not be tiled with $k$-crosses.