Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that $\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$ for any $x,y\not= 0$ and that $\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.

A set $S$ is called perfect if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all perfect subsets of the set $\{1,2,... ,n\}$

Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. The edges of $K_{n}(n \geq 3)$ are colored with $n$ colors, and every color is used. Show that there is a triangle whose sides have different colors.

Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.

Let $f(n)$ take in a nonnegative integer $n$ and return an integer between $0$ and $n-1$ at random (with the exception being $f(0)=0$ always). What is the expected value of $f(f(22))$?

In a store, there are 7 cases containing 128 apples altogether. Let $ N$ be the greatest number such that one can be certain to find a case with at least $ N$ apples. Then, the last digit of $ N$ is $ \text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.

A natural number $n$ is given and written in a row of $n$ numbers, each of which is equal to $0$ or $1$. Then $n - 1$ numbers are written below in a row - one number under each pair of adjacent numbers of the first row. At the same time, $0$ is written under a pair of identical numbers. and under a pair of different ones $1$. Then, under the second row, the third of $n- 2$ numbers is similarly written, etc., until we get a triangular table with $n$ rows. For a given $n$, find the largest possible number of units in such a table.

Flavian and Pavel play a game. Starting with Flavian, they take turns eliminating exactly one edge from a complete graph with $2024$ vertices. The first player to make a move that leaves no cycles loses. Determine who has a winning strategy. [i]Proposed by Pavel Ciurea[/i]

Hamza and Majid play a game on a horizontal $3 \times 2015$ white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal $1 \times 3$ rectangle. A legal move for Majid consists of painting three unit squares forming a vertical $3\times 1$ rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win? Lê Anh Vinh

Given three letters $X, Y, Z$, we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we can perform following operations: $T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for example, $T_1(XYZXXY) = (XYZXXYYZ).$ $T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for example, $T_2(XXYYZYYYX) = (XXYYZZX).$ $T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, $T_3(XXYZ) = (XXYZX).$ $T_4$: In a sequence containing one or more $Z$, we can replace the first $Z$ by $XY$, for example, $T_4(XXYYZZX) = (XXYYXYZX).$ $T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$ Using above operations, can we get $XYZZ$ from $XYZ \ ?$

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? [img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]