[b]p1.[/b] A right triangle has a hypotenuse of length $25$ and a leg of length $16$. Compute the length of the other leg of this triangle. [b]p2.[/b] Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If a necklace can be transformed into another necklace through a series of rotations and reflections, then the two necklaces are considered to be the same. [b]p3.[/b] Find the sum of the digits in the decimal representation of $10^{2016} - 2016$. [b]p4.[/b] Let $x$ be a real number satisfying $$x^1 \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5 \cdot x^6 = 8^7.$$ Compute the value of $x^7$. [b]p5.[/b] What is the smallest possible perimeter of an acute, scalene triangle with integer side lengths? [b]p6.[/b] Call a sequence $a_1, a_2, a_3,..., a_n$ mountainous if there exists an index $t$ between $1$ and $n$ inclusive such that $$a_1 \le a_2\le ... \le a_t \,\,\,\, and \,\,\,\, a_t \ge a_{t+1} \ge ... \ge a_n$$ In how many ways can Bishal arrange the ten numbers $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$, $5$, and $5$ into a mountainous sequence? (Two possible mountainous sequences are $1$, $1$, $2$, $3$, $4$, $4$, $5$, $5$, $3$, $2$ and $5$, $5$, $4$, $4$, $3$, $3$, $2$, $2$, $1$, $1$.) [b]p7.[/b] Find the sum of the areas of all (non self-intersecting) quadrilaterals whose vertices are the four points $(-3,-6)$, $(7,-1)$, $(-2, 9)$, and $(0, 0)$. [b]p8.[/b] Mohammed Zhang's favorite function is $f(x) =\sqrt{x^2 - 4x + 5} +\sqrt{x^2 + 4x + 8}$. Find the minumum possible value of $f(x)$ over all real numbers $x$. [b]p9.[/b] A segment $AB$ with length $1$ lies on a plane. Find the area of the set of points $P$ in the plane for which $\angle APB$ is the second smallest angle in triangle $ABP$. [b]p10.[/b] A binary string is a dipalindrome if it can be produced by writing two non-empty palindromic strings one after the other. For example, $10100100$ is a dipalindrome because both $101$ and $00100$ are palindromes. How many binary strings of length $18$ are both palindromes and dipalindromes? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality $$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$

What is the largest number of acute angles that a convex polygon can have?

Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$.

Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$. Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.

Find all the integers $x$ in $[20, 50]$ such that $6x + 5 \equiv -19 \mod 10,$ that is, $10$ divides $(6x + 15) + 19.$

Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by \[x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) \quad n\geq 1\] for every positive integer $m$ is a divisor of some non-zero element of $(x_n )$

In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.

Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that \[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear. [i]Valentin Vornicu[/i]

Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?

Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.

When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.

Let $ABC$ be a triangle with side lengths $AB=6, BC=7, CA=8$ and circumcircle $\omega.$ Denote $M$ to be the midpoint of $BC.$ Let $P$ be the intersection of the tangent to $\omega$ at $A$ and $BC.$ The line parallel to $BC$ passing through $A$ intersects $\omega$ at another point $D.$ The tangent to $\omega$ passing through $P$ that is not $PA$ intersects $DM$ at a point $Q.$ Denote $J$ to be the intersection of $(BMQ)$ and $AQ.$ Extend $BJ$ to intersect $AC$ at $E.$ Compute $\tfrac{BJ}{JE}.$ [i]Lightning 4.4[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$