Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$. A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$, respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively. Prove that $ABCD$ is a parallelogram.

a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$. b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

Let be a sequence $ \left( b_n \right)_{n\ge 1} $ of integers, having the following properties: $ \text{(i)} $ the sequence $ \left( \frac{b_n}{n} \right)_{n\ge 1} $ is convergent. $ \text{(ii)} m-n|b_m-b_n, $ for any natural numbers $ m>n. $ Prove that there exists an index from which the sequence $ \left( b_n \right)_{n\ge 1} $ is an arithmetic one. [i]Cristinel Mortici[/i]

In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$ [list=1] [*] $3+\frac{\sqrt{3}}{3}$ [*] $2+\frac{4\sqrt{3}}{3}$ [*] $2+2\sqrt{2}$ [*] $\frac{3+3\sqrt{5}}{2}$ [/list]

Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$. Prove that $ZA=ZH$. [i]Vadzim Kamianetski[/i]

The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?

Let $a, b, c\in (0, 1)$ with $a + b + c = 1$. Prove that $$\frac{a^5+b^5}{a^3+b^3}+\frac{b^5+c^5}{b^3+c^3}+\frac{c^5+a^5}{c^3+a^3}\geq\frac{a}{8+b^3+c^3}+\frac{b}{8+c^3+a^3}+\frac{c}{8+a^3+b^3}.$$

Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$. Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$ otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

It is possible to place an even number of pears in a row such that the masses of any two neighbouring pears differ by at most $1$ gram. Prove that it is then possible to put the pears two in a bag and place the bags in a row such that the masses of any two neighbouring bags differ by at most $1$ gram.

Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.

In the rectangle $ABCD$, point $M$ is the midpoint of the side $BC$. The points $P$ and $Q$ lie on the diagonal $AC$ such that $\angle DPC = \angle DQM = 90^o$. Prove that $Q$ is the midpoint of the segment $AP$.

Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions. [list=disc] [*] $f(a)$ is not an integer for some rational number $a$. [*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers. [/list]

Polina and Yan have $n$ cards, on the first card on one side $1$ is written, on the other side $n+1$, on the second card on one side $2$ is written, on the other side $n+2$, etc. Polina laid all cards in a circle in some order. Yan wants to turn some cards such that the numbers on the top sides of adjacent cards were not coprime. For every positive integer $n \geq 3$ determine can Yan accomplish that regardless of the actions of Polina. [i]M. Shutro[/i]

Let $f: N \to N$ be a function satisfying the following: $\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$. $\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes. Determine all possible values of $f(2002)$. Justify your answers.

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation: $3^m5^k=n^3+125$

[asy] size(5cm); pen p=linewidth(3), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); transform dishift(real x) { return shift(x,x); } // Draw the table of latch of table path ell = ((0,0)--(0,-1)--(-0.1,-1)--(-0.1,-0.1)--(-1,-0.1)--(-1,0)--cycle); // the ell shape path corner = dishift(-0.85)*ell; // define the path path table = dishift(-1)*scale(5)*ell; // define the table by scaling the pulley filldraw(corner, ll_grey, light_grey+p); // base of pulley filldraw(table, ll_grey, grey+p); // table real block_size = 1.6; // template for block path block = unitsquare; pair block_center = (0.5,0.5); /* Resting block */ transform rest = shift(-5, -0.9) * scale(block_size); // transformation for resting block filldraw(rest * block, ll_grey, light_grey+p); // draw block draw(rest*(1,0.5)--dir(110), light_grey+p); // rope fr0m midpoint of right block to wheel label("$m$", rest * block_center, fontsize(16)); // label block /* Hanging block */ transform hang = shift(0.2,-4.1) * scale(block_size); // transformation for hanging block draw((1,0)--(1,-2.5), light_grey+p); // string of pulley filldraw(hang * block, ll_grey, light_grey+p); // fill it label("$M$",hang * block_center,fontsize(16)); // label the small m // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley [/asy] A pulley system of two blocks, shown above, is released from rest. The block on the table, which has mass $m=1.0 \, \text{kg}$ slides after the time of release and hits the pulley to come to a dead stop. There was originally a distance of $ 1.0 \, \text{m} $ between the block and the pulley, which the block fully covers during the slide. From the time of release to the time of hitting the pulley, the angle that the rope above the table makes with the horizontal axis is a, nearly constant, $10.0^\circ$. The hanging block has mass $ M = 2.0 \, \text{kg} $. The table has a coefficient of friction of $0.50$ with the block that sits on it. The pulley is frictionless. Also, assume that, during the entire slide, the block never leaves the ground. Let $t$ be the number of seconds in takes for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures. [i](Ahaan Rungta, 4 points)[/i]

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.