Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality $$f(x) + f(2x^2 - 1) = 2x + a?$$

Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]

If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is: $\text{(A)} \ 2 \qquad \text{(B)} \ -2-\frac{3i\sqrt{3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ -\frac{3i\sqrt{3}}{4} \qquad \text{(E)} \ -2$

Let $D=\{ (x,y)|x^2+y^2\le \pi \}$. Find $\iint\limits_D(sin x^2cosx^2+x\sqrt{x^2+y^2})dxdy$.

Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.

Let m be a positive integer. An $m$-[i]pattern [/i] is a sequence of $m$ symbols of strict inequalities. An $m$-pattern is said to be [i]realized [/i] by a sequence of $m + 1$ real numbers when the numbers meet each of the inequalities in the given order. (For example, the $5$-pattern $ <, <,>, < ,>$ is realized by the sequence of numbers $1, 4, 7, -3, 1, 0$.) Given $m$, which is the least integer $n$ for which there exists any number sequence $x_1,... , x_n$ such that each $m$-pattern is realized by a subsequence $x_{i_1},... , x_{i_{m + 1}}$ with $1 \le i_1 <... < i_{m + 1} \le n$?

Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be on the lines $AB$ and $AD$, respectively, so that $B$ lies between $A$ and $E$, and $D$ lies between $A$ and $F$. Suppose that $\angle BCE = 20^o$ and $\angle DCF = 25^o$. Find the area of triangle $\vartriangle EAF$.

On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?

In $\triangle ABC$, $AB = 4$, $BC = 5$, and $CA = 6$. Circular arcs $p$, $q$, $r$ of measure $60^\circ$ are drawn from $A$ to $B$, from $A$ to $C$, and from $B$ to $C$, respectively, so that $p$, $q$ lie completely outside $\triangle ABC$ but $r$ does not. Let $X$, $Y$, $Z$ be the midpoints of $p$, $q$, $r$, respectively. If $\sin \angle XZY = \dfrac{a\sqrt{b}+c}{d}$, where $a, b, c, d$ are positive integers, $\gcd(a,c,d)=1$, and $b$ is not divisible by the square of a prime, compute $a+b+c+d$. [i]Proposed by Michael Tang[/i]

Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

Define a size-$n$ tromino to be the shape you get when you remove one quadrant from a $2n \times 2n$ square. In the figure below, a size-$1$ tromino is on the left and a size-$2$ tromino is on the right. [center][img]http://i.imgur.com/2065v7Y.png[/img][/center] We say that a shape can be tiled with size-$1$ trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-$1$ trominos. For example, a $23$ rectangle can be tiled with size-$1$ trominos as shown below, but a $33$ square cannot be tiled with size-$1$ trominos. [center][img]http://i.imgur.com/UBPeeRw.png[/img][/center] a) Can a size-$5$ tromino be tiled by size-$1$ trominos? b) Can a size-$2013$ tromino be tiled by size-$1$ trominos? Justify your answers.

An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$ B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.

(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that $$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$ (b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that $p(2018) = p(2019).$ Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$.

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?

Alice and Bob are builders; Charlie is a destroyer. Alice can build a car in $20$ hours and Bob can build a car in $10$ hours, while Charlie destroys a car in $40$ hours. If Alice and Bob are working together on a car Charlie is destroying, how many hours will it take for Alice and Bob to finish building the car?

Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? $\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$

For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? $ \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 20 \qquad \mathrm {(C) \ } 39 \qquad \mathrm{(D) \ } 49 \qquad \mathrm{(E) \ } 59$

Suppose that $p(n)$ is the number of ways to express $n$ as a sum of some naturall numbers (the two representations $4=1+1+2$ and $4=1+2+1$ are considered the same). Prove that for an infinite number of $n$'s $p(n)$ is even and for an infinite number of $n$'s $p(n)$ is odd.