Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] $ \textbf{(A)}\ B \equal{} W \qquad \textbf{(B)}\ W \equal{} R \qquad \textbf{(C)}\ B \equal{} R \qquad \textbf{(D)}\ 3B \equal{} 2R \qquad \textbf{(E)}\ 2R \equal{} W$

Show that among all quadrilaterals of a given perimeter the square has the largest area.

Let $p$ be a prime number and let $a_1,a_2,\dots,a_k$ be distinct integers chosen from $1,2,\dots,p-1$. For $1\le i \le k$, let $f_i^{(n)}$ denote the remainder of the integer $na_1$ upon division by $p$, so $0\le f_i^{(n)}<p$. Define $S=\{n:1\le n \le p-1,f_1^{(n)}<\dots<f_k^{(n)}\}$ Show that $S$ has less than $\frac{2p}{k+1}$ elements.

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having $0$ as a left-hand endpoint, such that for every positive $x\in I$ the average of $f$ over the closed interval $[0,x]$ is equal to $\sqrt{ f(0) f(x)}.$

Find all integer coefficient polynomials $Q$ such that [list] [*] $Q(n)\ge 1$ $\forall n\in \mathbb{Z}_+$. [*] $Q(mn)$ and $Q(m)Q(n)$ have the same number of prime divisors $\forall m,n\in\mathbb{Z}_+$. [/list]

We are given a finite number of functions of the form $y = c2^{-|x-d|}$. In each case $c$ and $d$ are parameters with $c > 0$. The function $f(x)$ is defined on the interval $[a, b]$ as follows: For each $x$ in $[a, b]$, $f(x)$ is the maximum value taken by any of the given functions $y$ (defined above) at that point $x$. It is known that $f(a) = f(b)$. Prove that the total length of the intervals in which the function $f$ is increasing is equal to the total length of the intervals in which it is decreasing (that is, both are equal to $(b- a)/2$ ). (NB Vasiliev)

Prove that the only way to cover a square of side $1$ with a finite number of circles that do not overlap, it is with only one circle of radius greater than or equal to $\frac{1}{\sqrt2}$. Circles can occupy part of the outside of the square and be of different radii.

A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$

Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.

Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out? [i]A. Golovanov[/i]

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

If $a=\log_8225$ and $b=\log_215$, then $\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$

A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.

Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.

Karina has a polynomial $p_1(x) = x^2 + x + k$, where $k$ is an integer. Noticing that $p_1$ has integer roots, she forms a new polynomial $p_2(x) = x^2 + a_1x + b_1$, where $a_1$ and $b_1$ are the roots of $p_1$ and $a_1 \ge b_1$. The polynomial $p_2$ also has integer roots, so she forms a new polynomial $p_3(x) = x^2 + a_2x + b_2$, where $a_2$ and $b_2$ are the roots of $p_2$ and $a_2 \ge b_2$. She continues this process until she reaches $p_7(x)$ and finds that it does not have integer roots. What is the largest possible value of $k$?

Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

What are the last two digits of the decimal representation of $21^{2006}$?

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)

Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. [i]Proposed by Walther Janous, Austria[/i]

A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them

Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other. [i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]

Consider $\vartriangle ABC$, right at $B$, let $I$ be its incenter and $F,D,E$ the points where the circle inscribed on sides AB, $BC$ and $AC$, respectively. If $M$ is the intersection point of $CI$ and $EF$, and $N$ is the intersection point of $DM$ and $AB$. Prove that $AN = ID$.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

Find the number of three-element subsets of $\{1, 2, 3,...,13\}$ that contain at least one element that is a multiple of $2$, at least one element that is a multiple of $3$, and at least one element that is a multiple of $5$ such as $\{2,3, 5\}$ or $\{6, 10,13\}$.