There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.

For a natural number $n$ let $W (n)$ be the number of possibilities, to distribute weights with the masses $1, 2,..., n$ all of them between the two bowls of a beam balance so that they are in balance/ Show that $W (100)$ is really larger than $W (99)$.

Find the number of subsets $X$ of $\{1,2,\dots,10\}$ such that $X$ contains at least two elements and such that no two elements of $X$ differ by $1$.

Solve the system of simultaneous equations \[\sqrt x - \frac 1y - 2w + 3z = 1,\]\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

A rhombus with sidelength $1$ has an inscribed circle with radius $\frac{1}{3}.$ If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

Let $A$ be a given finite set with some of its subsets called pretty. Let a subset be called small, if it's a subset of a pretty set. Let a subset be called big, if it has a pretty subset. (A set can be small and big simultaneously, and a set can be neither small nor big.) Let $a$ denote the number of elements of $A$, and let $p$, $s$ and $b$ denote the number of pretty, small and big sets, respectively. Prove that $2^a\cdot p\le s\cdot b$. [i]Proposed by András Imolay, Budapest[/i]

Show that $$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$ for every integer $n\geq 0.$

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A schoolgirl forgot to write a multiplication sign between two $3$-digit numbers and wrote them as one number. This $6$-digit result proved to be $3$ times greater than the product (obtained by multiplication). Find these numbers. (A Kovaldzhi,

Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

Let $ABCD$ be a square, and let $E$ and $F$ be points on sides $\overline{AB}$ and $\overline{CD}$, respectively, such that $AE:EB=AF:FD=2:1$. Let $G$ be the intersection of $\overline{AF}$ and $\overline{DE}$, and let $H$ be the intersection of $\overline{BF}$ and $\overline{CE}$. Find the ratio of the area of quadrilateral $EGFH$ to the area of square $ABCD$. [asy] size(5cm,0); draw((0,0)--(3,0)); draw((3,0)--(3,3)); draw((3,3)--(0,3)); draw((0,3)--(0,0)); draw((0,0)--(2,3)); draw((1,0)--(3,3)); draw((0,3)--(1,0)); draw((2,3)--(3,0)); label("$A$",(0,3),NW); label("$B$",(3,3),NE); label("$C$",(3,0),SE); label("$D$",(0,0),SW); label("$E$",(2,3),N); label("$F$",(1,0),S); label("$G$",(0.66666667,1),E); label("$H$",(2.33333333,2),W); [/asy]

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\tfrac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle? [asy] size(5cm); fill((0,0)--(2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(4,0)--cycle,lightgray*0.5+mediumgray*0.5); draw((0,0)--(4,0)--(2,2*sqrt(3))--cycle); //center: 2,1.155 draw((2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(2,2*sqrt(3)-0.7697)--cycle); dot((0,0)^^(4,0)^^(2,2*sqrt(3))^^(2/3,1.155/3)^^(4-(4-2)/3,1.155/3)^^(2,2*sqrt(3)-0.7697)); draw((0,0)--(2/3,1.155/3)); draw((4,0)--(4-(4-2)/3,1.155/3)); draw((2,2*sqrt(3))--(2,2*sqrt(3)-0.7697)); [/asy] $\textbf{(A) } 1:3\qquad\textbf{(B) } 3:8\qquad\textbf{(C) } 5:12\qquad\textbf{(D) } 7:16\qquad\textbf{(E) } 4:9$

The triangle $ABC$ with $AB < AC$ has an altitude $AD$. The points $E$ and $A$ lie on opposite sides of $BC$, with $E$ on the circumcircle of $ABC$. Furthermore, $AD = DE$ and $\angle ADO=\angle CDE$, where $O$ is the circumcentre of $ABC$. Determine $\angle BAC$.

Let $\mathbb R$ be the real numbers. Set $S = \{1, -1\}$ and define a function $\operatorname{sign} : \mathbb R \to S$ by \[ \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} \] Fix an odd integer $n$. Determine whether one can find $n^2+n$ real numbers $a_{ij}, b_i \in S$ (here $1 \le i, j \le n$) with the following property: Suppose we take any choice of $x_1, x_2, \dots, x_n \in S$ and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right), \quad \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then $z=x_1 x_2 \dots x_n$.

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0< a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}$. This process cotinues as long as Ann wishes. Amongst every $100$ consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than $100$ from the initial position of $A{}$. Can Ann achieve her goal?

Determine all integers solutions of the equation $x + x^3 = 5y^2$.

Many distinct points are marked in the plane. A student draws all the segments determined by those points, and then draws a line [i]r[/i] that does not pass through any of the marked points, but cuts exactly $60$ drawn segments. How many segments were not cut by [i]r[/i]? Give all possibilites.

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$. [i]Laurențiu Panaitopol[/i]

Let $f:(0;\infty) -- (0;\infty)$ such that $f(x^y)=(f(x))^{f(y)}$. Prove $f(xy)=f(x)f(y)$ and $f(x+y)=f(x)+f(y)$ for all positive real $x,y$.

$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:

How many positive integers less than 2005 are relatively prime to 1001?