The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$. Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.

Two squares of area $38$ are given. Each of the squares is divided into $38$ connected pieces of unit area by simple curves. Then the two squares are patched together. Show that one can sting the patched squares with $38$ needles so that every piece of each square is stung exactly once.

Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m,n)$ for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares. Prove that $a$ divides $2b$. [i]Holden Mui[/i]

Find the number of nonnegative integers $n$ for which $(n^2 - 3n +1)^2 + 1$ is a prime number.

Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$. [i]Proposed by tkhalid[/i]

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

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 $ad - bc$.