You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. Prove that you will never have 2003 pieces.

Real numbers $x, y > 1$ are chosen such that the three numbers \[\log_4x, \; 2\log_xy, \; \log_y2\] form a geometric progression in that order. If $x + y = 90$, then find the value of $xy$.

Find all real numbers $ x$ which satisfy: $ \frac{x^2}{(x\plus{}1\minus{}\sqrt{x\plus{}1})^2}<\frac{x^2\plus{}3x\plus{}18}{(x\plus{}1)^2}.$

The centre of circle $1$ lies on circle $2$. $A$ and $B$ are the intersection points of the circles. The tangent line to circle $2$ at point $B$ intersects circle $1$ at point $C$. Prove that $AB = BC$. (V. Prasovov, Moscow)

Two noncongruent circles $k_1$ and $k_2$ are exterior to each other. Their common tangents intersect the line through their centers at points $A$ and $B$. Let $P$ be any point of $k_1$. Prove that there is a diameter of $k_2$ with one endpoint on line $PA$ and the other on $PB$.

[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable $\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$ tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]

Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.

A nonempty set $A$ is called an [i]$n$-level-good [/i]set if $ A \subseteq \{1,2,3,\ldots,n\}$ and $|A| \le \min_{x\in A} x$ (where $|A|$ denotes the number of elements in $A$ and $\min_{x\in A} x$ denotes the minimum of the elements in $A$). Let $a_n$ be the number of $n$-level-good sets. Prove that for all positive integers $n$ we have $a_{n+2}=a_{n+1}+a_{n}+1$.

There is $500$ dollars in a bank. Two bank operations are allowed: to withdraw $300$ dollars from the bank or to deposit $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done? (AK Tolpygo)

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

When the base of a triangle is increased $10\%$ and the altitude to this base is decreased $10\%$, the change in area is $\text{(A)} \ 1\%~ \text{increase} \qquad \text{(B)} \ \frac12 \%~ \text{increase} \qquad \text{(C)} \ 0\% \qquad \text{(D)} \ \frac12 \% ~\text{decrease} \qquad \text{(E)} \ 1\% ~\text{decrease}$

Of $4n$ points in a row, $2n$ are colored white and $2n$ are colored black. Swot tha tthere are $2n$ consecutive points of which exactly $n$ are white and $n$ are black.

On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$. (M. Volchkevich)

A regular hexagon of side $5$ is cut into unit equilateral triangles by lines parallel to the sides of the hexagon. We call the vertices of these triangles knots. If more than half of all knots are marked, show that there exist five marked knots that lie on a circle.

Find all positive integers $n$ that can be [i]uniquely[/i] expressed as a sum of five or fewer squares.

For $x >0$ , prove that $$\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}$$ and for all $n \ge 2$ prove that $$1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2$$

In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.

Let $AB$ be the diameter of a circle with center $O$ and radius $5$. Extend $AB$ past $A$ to a point $C$ such that $BC = 18$, and let $D$ be a point on the circle such that $CD$ lies tangent to the circle. Next, draw $E$ on $CD$ such that $OE \parallel BD$. Compute $DE$.

Let $f:\mathbb{N}\to \mathbb{N}$ be a bijection satisfying $f(ab)=f(a)f(b)$ for all $a,b\in \mathbb{N}$. Determine the minimum possible value of $f(n)/n$, taken over all possible $f$ and all $n\leq 2019$.

There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?

Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are there? [i]Proposed by Evan Chen[/i]

You start with a single piece of chalk of length $1$. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have $8$ pieces of chalk. What is the probability that they all have length $\frac18$ ?

On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points. $a)$ Prove that there exist two students with exactly the same answers to problems $b)$ Prove that there exist at least $4$ students with same number of points

Of the $500$ balls in a large bag, $80\%$ are red and the rest are blue. How many of the red balls must be removed so that $75\%$ of the remaining balls are red? $ \textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 150 $

Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$, $y\geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$. Determine the area of $R$.