On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

The admission fee for an exhibition is $ \$25$ per adult and $ \$12$ per child. Last Tuesday, the exhibition collected $ \$1950$ in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to $ 1$?

One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into $n^2$ identical square cells. In each of these $2n^2$ cells is written a number from $1$ to $k$. Find the smallest $k$,for which the following properties hold simultaneously: (i) on the red side, any two numbers in different rows are distinct; (ii) on the blue side, any two numbers in different columns are different; (iii) for each of the $n^2$ squares of the partition, the number on the blue side is not equal to the number on the red side.

Let $n$ be a positive integer; and $S(n)$ denote the sum of all digits in the decimal representation of $n$. A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of $n$ is called the [i]truncation[/i] of $n$. The sum of all truncations of $n$ is denoted as $T(n)$. Prove that $S(n)+9T(n)=n$

Let $S$ be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in $S$ is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in $S$?

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.

Real numbers $x_1, x_2, x_3, \cdots , x_n$ satisfy $x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1$. Show that \[ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} . \]

There is a row with $2024$ cells. Ana and Beto take turns playing, with Ana going first. On each turn, the player selects an empty cell and places a digit in that space. Once all $2024$ cells are filled, the number obtained from reading left to right is considered, ignoring any leading zeros. Beto wins if the resulting number is a multiple of $99$, otherwise Ana wins. Determine which of the two players has a winning strategy and describe it.

The length of the interval of solutions of the inequality $ a\le 2x\plus{}3\le b$ is $ 10$. What is $ b\minus{}a$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 30$

Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B$. The probability that team $A$ finishes with more points than team $B$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The incenter of triangle $ABC$ is $ I$. the circumcircle of $ABC$ is tangent to $BC$, $CA$, $AB$ at $T, E, F$. $R$ is a point on $BC$ . Let the $C$-excenter of $\vartriangle CER$ be $L$. Prove that points $L,T,F$ are collinear if and only if $B,E,A,R$ are concyclic. [i]proposed by kyou46[/i]

Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides? [asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy] [asy] import three; picture mainframe; defaultpen(fontsize(11pt)); picture conePic(picture pic, real r, real h, real sh) { size(pic, 3cm); triple eye = (11, 0, 5); currentprojection = perspective(eye); real R = 1, y = 2; triple center = (0, 0, 0); triple radPt = (0, R, 0); triple negRadPt = (0, -R, 0); triple heightPt = (0, 0, y); draw(pic, arc(center, radPt, negRadPt, heightPt, CW)); draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8")); draw(pic, center--radPt, linetype("8 8")); draw(pic, center--heightPt, linetype("8 8")); draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270)); if (h != 0) { label(pic, (string) h, heightPt--center, dir(0)); } if (sh != 0) { label(pic, (string) sh, heightPt--radPt, dir(0)); } return pic; } picture pic1; pic1 = conePic(pic1, 6, 0, 10); picture pic2; pic2 = conePic(pic2, 6, 10, 0); picture pic3; pic3 = conePic(pic3, 7, 0, 10); picture pic4; pic4 = conePic(pic4, 7, 10, 0); picture pic5; pic5 = conePic(pic5, 8, 0, 10); picture aux1; picture aux2; picture aux3; add(aux1, pic1.fit(), (0,0), W); label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4)); label(aux1, "$\textbf{(B)}$", (0,0), 3E); add(aux1, pic2.fit(), (0,0), 35E); add(aux2, aux1.fit(), (0,0), W); label(aux2, "$\textbf{(C)}$", (0,0), 3E); add(aux2, pic3.fit(), (0,0), 35E); add(aux3, aux2.fit(), (0,0), W); label(aux3, "$\textbf{(D)}$", (0,0), 3E); add(aux3, pic4.fit(), (0,0), 35E); add(mainframe, aux3.fit(), (0,0), W); label(mainframe, "$\textbf{(E)}$", (0,0), 3E); add(mainframe, pic5.fit(), (0,0), 35E); add(mainframe.fit(), (0,0), N); [/asy]

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

Let $P$ be a point in the interior of a triangle $ABC$. The lines $AP, BP$ and $CP$ intersect again the circumcircles of the triangles $PBC, PCA$ and $PAB$ at $D, E$ and $F$ respectively. Prove that $P$ is the orthocenter of the triangle $DEF$ if and only if $P$ is the incenter of the triangle $ABC$. [i]Proposed by Romania[/i]

In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

Find pairs of positive integers $(n, k)$ satisfying \[(n + 1)^k - 1 = n!\]

Find the differentiable functions $ f:\mathbb{R}\longrightarrow (-\infty ,1) $ with the property $ f(1)=-1 $ and $$ f(x+y)=f(x)+f(y)-f(x)f(y) , $$ for any reals $ x,y. $ [i]Vasile Pop[/i]

A semicircle of radius $r$ is divided into $n + 1$ equal parts and any point $k$ of the division with the ends of the semicircle forms a triangle $A_k$. Calculate the limit, as $n$ tends to infinity, of the arithmetic mean of the areas of the triangles.

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Denote $$s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.$$ (a) Let $\lambda>1$ be a real number. Show that $s'<\lambda s+\frac\lambda{\lambda-1}$. (b) Deduce that $\sqrt{s'}<\sqrt s+1$.

The infinite product $$\sqrt[3]{10}\cdot\sqrt[3]{\sqrt[3]{10}}\cdot\sqrt[3]{\sqrt[3]{\sqrt[3]{10}}}\dots$$ evaluates to a real number. What is that number? $\textbf{(A) }\sqrt{10}\qquad\textbf{(B) }\sqrt[3]{100}\qquad\textbf{(C) }\sqrt[4]{1000}\qquad\textbf{(D) }10\qquad\textbf{(E) }10\sqrt[3]{10}$

Given a sequence $x_1,x_2,x_3,\cdots$ of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval $[0,1]$ which have denominator equal to $x_i$. Then he write down the number $a_i$ equal to the length of the smallest interval in $[0,1]$ which both two ends of that is a marked number. Find all sequences $x_1,x_2,x_3,\cdots$ with $x_5=5$ and such that for all $n\in \mathbb N$ we have $$ a_1+a_2+\cdots+a_n= 2-\dfrac{1}{x_n}. $$ Proposed by [i]Mojtaba Zare[/i]