Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$, $$f(x)f''(x)\le f'(x)^2.$$

On $\triangle ABC$ if there existed a point $D$ in $AC$ such that $\angle CBD=\angle ABD+60$ and $\angle BDC=30$ and $AB \cdot BC=BD^2$, then find the angles inside the triangle $\triangle ABC$

Initially we have a $2 \times 2$ table with at least one grain of wheat on each cell. In each step we may perform one of the following two kinds of moves: $i.$ If there is at least one grain on every cell of a row, we can take away one grain from each cell in that row. $ii.$ We can double the number of grains on each cell of an arbitrary column. a) Show that it is possible to reach the empty table using the above moves, starting from the position down below. b) Show that it is possible to reach the empty table from any starting position. c) Prove that the same is true for the $8 \times 8$ tables as well.

$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$? Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?

Let $A$ and $B$ be sets with $2011^2$ and $2010$ elements, respectively. Show that there is a function $f:A \times A \to B$ satisfying the condition $f(x,y)=f(y,x)$ for all $(x,y) \in A \times A$ such that for every function $g:A \to B$ there exists $(a_1,a_2) \in A \times A$ with $g(a_1)=f(a_1,a_2)=g(a_2)$ and $a_1 \neq a_2.$

(a) Some vertices of a dodecahedron are to be marked so that each face contains a marked vertex. What is the smallest number of marked vertices for which this is possible? (b) Answer the same question, but for an icosahedron. (G. Galperin, Moscow) (Recall that a dodecahedron has $12$ pentagonal faces which meet in threes at each vertex, while an icosahedron has $20$ triangular faces which meet in fives at each vertex.)

4. A square can be divided into four congruent figures as shown: [asy] size(2cm); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); [/asy] For how many $n$ with $1 \le n \le 100$ can a unit square be divided into $n$ congruent figures? 5. If $x + 2y - 3z = 7$ and $2x - y + 2z = 6$, determine $8x + y$. 6. Let $ABCD$ be a rectangle, and let $E$ and $F$ be points on segment $AB$ such that $AE = EF = FB$. If $CE$ intersects the line $AD$ at $P$, and $PF$ intersects $BC$ at $Q$, determine the ratio of $BQ$ to $CQ$.

There are some markets in a city. Some of them are joined by one-way streets, such that for any market there are exactly two streets to leave it. Prove that the city may be partitioned into $1014$ districts such that streets join only markets from different districts, and by the same one-way for any two districts (either only from first to second, or vice-versa).

$ABCD$ is a trapezium such that $AB\parallel DC$ and $\frac{AB}{DC}=\alpha >1$. Suppose $P$ and $Q$ are points on $AC$ and $BD$ respectively, such that \[\frac{AP}{AC}=\frac{BQ}{BD}=\frac{\alpha -1}{\alpha+1}\] Prove that $PQCD$ is a parallelogram.

Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$, if at least one of player $A$'s losers is not a $B$'s loser. Determine, with proof, all possible values of $n$, such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.

All cells of $2015 \times 2015$ table colored in one of $4$ colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all $4$ colors in less than $51\%$ of total number of ways.

Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?

Let $\triangle ABC$ have $AB = 1$ cm, $BC = 2$ cm and $AC = \sqrt{3}$ cm. Points $D$, $E$ and $F$ lie on segments $AB$, $AC$ and $BC$ respectively are such that $AE = BD$ and $BF = AD$. The angle bisector of $\angle BAC$ intersects the circumcircle of $\triangle ADE$ for the second time at $M$ and the angle bisector of $\angle ABC$ intersects the circumcircle of $\triangle BDF$ at $N$. Determine the length of $MN$.

What is the sum of all positive integers less than $30$ divisible by $2, 3$, or $5$?

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

Prove that if a natural number $n$ is greater than $4$ and is not a prime number, then the produxt of the consecutive natural numbers from $1$ to $n-1$ is divisible by $ n$.

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in her set $S$? [b]p2.[/b] Jake has $2021$ balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs? [b]p3.[/b] Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to $(2, 2)$, but she doesn’t know how to get there. So each second, she rides one unit in the positive $x$ or $y$-direction, each with probability $\frac12$ . If the probability that she makes it to $(2, 2)$ during her ride can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, then find $a + b$. [b]p4.[/b] Triangle $ABC$ with $AB = BC = 6$ and $\angle ABC = 120^o$ is rotated about $A$, and suppose that the images of points $B$ and $C$ under this rotation are $B'$ and $C'$, respectively. Suppose that $A$, $B'$ and $C$ are collinear in that order. If the area of triangle $B'CC'$ can be expressed as $a - b\sqrt{c}$ for positive integers $a, b, c$ with csquarefree, find $a + b + c$. [b]p5.[/b] Find the sum of all possible values of $a + b + c + d$ if $a, b, c, $d are positive integers satisfying $$ab + cd = 100,$$ $$ac + bd = 500.$$ [b]p6.[/b] Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points $(1, 2)$ and $(2, 0)$ and a chute-ladder between points $(1, 3)$ and $(4, 0)$, whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from $(0, 0)$ to $(4, 4)$? [b]p7.[/b] There are $8$ identical cubes that each belong to $8$ different people. Each person randomly picks a cube. The probability that exactly $3$ people picked their own cube can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$. Find $a + b$. [b]p8.[/b] Suppose that $p(R) = Rx^2 + 4x$ for all $R$. There exist finitely many integer values of $R$ such that $p(R)$ intersects the graph of $x^3 + 2021x^2 + 2x + 1$ at some point $(j, k)$ for integers $j$ and $k$. Find the sum of all possible values of $R$. [b]p9.[/b] Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$. [b]p10.[/b] In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n \times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [b]p11.[/b] Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2}$$ is an integer. [b]p12.[/b] Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt3$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b}-c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

(Game) At the beginning of the game, the organisers place paper disks on the table, grouped into piles which may contain various numbers of disks. The two players take turns. On a player’s turn, their opponent selects two piles (one if there is only one pile left), and the player must remove some number of disks from one of the piles selected. This means that at least one disk has to be removed, and removing all disks in the pile is also permitted. The player removing the last disk from the table wins. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.