Let $ f_1,f_2,f_3,f_4 $ be four polynomials with real coefficients, having the property that $$ f_1 (1) =f_2 (0), \quad f_2 (1) =f_3 (0),\quad f_3 (1) =f_4 (0),\quad f_4 (1) =f_1 (0) . $$ Prove that there exists a polynomial $ f\in\mathbb{R}[X,Y] $ such that $$ f(X,0)=f_1(X),\quad f(1,Y) =f_2(Y) ,\quad f(1-X,1) =f_3(X),\quad f(0,1-Y)=f_4(Y) . $$

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.) Let $\triangle ABC$ be a triangle with area $20$. Orthogonal circles $\omega_B$ and $\omega_C$ are drawn with $\omega_B$ centered at $B$ and $\omega_C$ centered at $C$. Points $T_B$ and $T_C$ are placed on $\omega_B$ and $\omega_C$ respectively such that $AT_B$ is tangent to $\omega_B$ and $AT_C$ is tangent to $\omega_C$. If $AT_B = 7$ and $AT_C = 11$, what is $\tan\angle BAC$?

We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?

The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3+a_2^3+\cdots + a_{10}^3$.

A pawn is put on each of $2n$ arbitrary selected cells of an $n\times n$ board ($n>1$). Prove that there are four cells that are marked with pawns and whose centers form a parallelogram.

$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$

Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[f(f(x) + 2x + 20) = 15. \] Call an integer $n$ $\textit{good}$ if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n) = m.$ Find the sum of all good integers $x.$

Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, $23=9+8+6.)$

For a natural number $n$ denote by Map $( n )$ the set of all functions $f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} . $ For $ f , g \in $ Map$( n ) , f \circ g $ denotes the function in Map $( n )$ that sends $x \rightarrow f ( g ( x ) ) . $ \\ \\ $(a)$ Let $ f \in$ Map $( n ) . $ If for all $x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x , $ show that $ f \circ f \neq f $ \\$(b)$ Count the number of functions $ f \in$ Map $( n )$ such that $ f \circ f = f $

Find the sum of all positive $30$-digit palindromes. The leading digit is not allowed to be $0$.

Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ be nine points in space such that $ABCDE$, $ABFGH$, and $GFCDI$ are each regular pentagons with side length $1$. Determine the lengths of the sides of triangle $EHI$.

Let the triangle $ABC$ with $BC$ the smallest side. Let $P$ on ($AB$) such that angle $PCB$ equals angle $BAC$. and $Q$ on side ($AC$) such that angle $QBC$ equals angle $BAC$. Show that the line passing through the circumenters of triangles $ABC$ and $APQ$ is perpendicular on $BC$.

Find the sum of the three smallest positive integers $N$ such that $N$ has a units digit of $1,$ $N^2$ has a tens digit of $2,$ and $N^3$ has a hundreds digit of $3.$

We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that $$P(x)=\sum_{}{x^{n_i}}+1$$ Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

Let $n$ be a positive integer. Starting with the sequence $1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}$, form a new sequence of $n -1$ entries $\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}$, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n -2$ entries and continue until the final sequence consists of a single number $x_n$. Show that $x_n < \frac{2}{n}$.

Evaluate \[\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.\] Own

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ \plus{} }$: \[ x\sqrt y \plus{} y\sqrt z \plus{} z\sqrt x\leq K\sqrt {(x \plus{} y)(y \plus{} z)(z \plus{} x)} \]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all $x,y \in \mathbb{R}$.