Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$. [i]Jakob Jurij Snoj, Slovenia[/i]

In a tournament, every team plays exactly once against every other team. One won match earns $3$ points for the winner and $0$ for the loser. With a draw both teams receive $1$ point each. At the end of the tournament it appears that all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ point. The second to last team has not lost a match. a) How many teams participated in the tournament? b) How many points did the team score in second place in the final ranking?

Let $x$ and $y$ be real numbers such that \[ x+\frac{1}{y} = 20 \,\,\, \text{and} \,\,\, y+\frac{1}{x} = 24. \]Find $\frac{x}{y}.$

Determine all integers $n$ for which \[\sqrt{\frac{25}{2}+\sqrt{\frac{625}{4}-n}}+\sqrt{\frac{25}{2}-\sqrt{\frac{625}{4}-n}}\] is an integer.

A natural number $n$ is called [i]nice[/i] if it enjoys the following properties: • The expression is made ​​up of $4$ decimal digits; • the first and third digits of $n$ are equal; • the second and fourth digits of $n$ are equal; • the product of the digits of $n$ divides $n^2$. Determine all nice numbers.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency. Nguyễn Văn Linh

The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.

At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made $ 48$ free throws. How many free throws did she make at the first practice? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$

On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

Let $a$ and $b$ be natural numbers. We consider the set $M$ of the points of the plane with an integer $x$-coordinate from $1$ to $a$ and integer $y$-coordinate from $1$ to $b$. For two points $P = (x, y)$ and $Q = (\tilde x, \tilde y)$ in M we write $P\le Q$ if $x\le \tilde x$ and $y \le \tilde y$, we say $P$ is [i]less [/i] than $Q$ when $P\le Q$ and $P \ne Q$. A subset $S$ of $M$ is now called [i]cute [/i] if for every point $P \in S$ it also contains all smaller points. From an arbitrary subset $S$ of $M$ we can now create new subsets in four ways to construct: (a) the complement $K (S) = \overline{S}$, (b) the subset $\min (S)$ of its minima, i.e. those points for which there is no smaller in $S$ occurs, (c) the cute set $P (S)$ of all those points in M that are less than or equal to some point are from $S$, (d) you do all these things one after the other and get a set $Z (S) = P (\min (K (S)))$. Let $S$ be cute. Prove that $$\underset{a+b\,\, times\,\, Z}{Z(Z(...(Z(S))...))=S}$$

The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$, a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m\minus{}1$ players in $ S$ and one player who lost all her games against the other $ m \minus{} 1$ players in $ S$. For a given integer $ m \ge 4$, determine the minimum value of $ n$ (as a function of $ m$) such that the following holds: in every $ n$-player round robin chess tournament with property $ P(m)$, the final scores of the $ n$ players are all distinct.

Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency. ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$. Nguyễn Minh Hà

Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$. [i]Proposed by Qu Zhenhua[/i]

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$? $\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57$

$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$. $ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$.

155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.

What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$? [i]2016 CCA Math Bonanza Lightning #2.4[/i]

Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$ [i]Marius Mînea[/i]

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.