Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

At Mallard High School there are three intermural sports leagues: football, basketball, and baseball. There are 427 students participating in these sports: 128 play on football teams, 291 play on basketball teams, and 318 play on baseball teams. If exactly 36 students participate in all three of the sports, how many students participate in exactly two of the sports?

In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Circles with centres $O_1$ and $O_2$ intersect in two points, let one of which be $A$. The common tangent of these circles touches them respectively in points $P$ and $Q$. It is known that points $O_1, A$ and $Q$ are on a common straight line and points $O_2, A$ and $P$ are on a common straight line. Prove that the radii of the circles are equal.

$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$. Prove that $$ \sum (\frac{x^2-1}{x})^2 \geq 4$$ [i]Proposed by Amin Fathpour[/i]

A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)

Ankit wants to create a pseudo-random number generator using modular arithmetic. To do so he starts with a seed $x_0$ and a function $f(x) = 2x + 25$ (mod $31$). To compute the $k$-th pseudo random number, he calls $g(k)$ dened as follows: $$g(k) = \begin{cases} x_0 \,\,\, \text{if} \,\,\, k = 0 \\ f(g(k- 1)) \,\,\, \text{if} \,\,\, k > 0 \end{cases}$$ If $x_0$ is $2017$, compute $\sum^{2017}_{j=0} g(j)$ (mod $31$).

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? $\textbf{(A) }\text{The second height is 10\% less than the first.}$ $\textbf{(B) }\text{The first height is 10\% more than the second.}$ $\textbf{(C) }\text{The second height is 21\% less than the first.}$ $\textbf{(D) }\text{The first height is 21\% more than the second.}$ $\textbf{(E) }\text{The second height is 80\% of the first.}$

Let $ABC$ be a non-isosceles triangle with altitudes $AD$, $BE$, $CF$ with orthocenter $H$. Suppose that $DF \cap HB = M$, $DE \cap HC = N$ and $T$ is the circumcenter of triangle $HBC$. Prove that $AT\perp MN$.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$ a) Prove that $EF=\frac{BC}{2}.$ b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

Which of the following numbers is a perfect square? $\textbf{(A) }4^45^56^6\qquad\textbf{(B) }4^45^66^5\qquad\textbf{(C) }4^55^46^6\qquad\textbf{(D) }4^65^46^5\qquad\textbf{(E) }4^65^56^4$

Suppose $p > 3$ is a prime number and \[S = \sum_{2 \le i < j < k \le p-1} ijk\] Prove that $S+1$ is divisible by $p$.

Let $O=(0,0)$ be the origin of the $xy$-plane. We say a lattice triangle $ABC$ is [i]marine[/i] if it has centroid $O$ and area $\tfrac{3}{2}$. Let $P$ be any point in the plane which is not a lattice point. Prove that $P$ lies in the interior of some marine triangle if and only if the line segment $\overline{OP}$ does not pass through any lattice points besides $O$. (A [i]lattice point[/i] is a point whose $x$-coordinate and $y$-coordinate are both integers. A [i]lattice triangle[/i] is a triangle whose vertices are lattice points.)

Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves. [i](4 points)[/i]

Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.

Prove that there exist infinitely many positive integers which cannot be presented in the form $x_1^3+x_2^5+x_3^7+x_4^9+x_5^{11}$ where $x_1,x_2,x_3,x_4,x_5$ are positive integers. (V. Bernik)

Let $F$ be a family of n subsets of a set $K$ with $5$ elements, such that any two subsets in $F$ have a common element. Find the minimal value of $n$ such that no matter how we choose $F$ with the properties above, there exists exactly one element of $K$ which belongs to all the sets in $F$.

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer? $\textbf{(A)}\ 17 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 19 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 21$

Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and $$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$ $$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$ $$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$ $$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$ $$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$ $$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$ The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

A positive integer $d$ is called [i]nice[/i] iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ . a) Is 29 nice? b) Is 2006 nice? c) Prove that infinitely many nice numbers exist.

Prove that exists triangle that can be partitions in $2050$ congruent triangles.