The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i] (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.) [b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$. [b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point. Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

A [i]snake of length $k$[/i] is an animal which occupies an ordered $k$-tuple $(s_1, \dots, s_k)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. If the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can [i]move[/i] to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has [i]turned around[/i] if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead. Determine whether there exists an integer $n > 1$ such that: one can place some snake of length $0.9n^2$ in an $n \times n$ grid which can turn around. [i]Nikolai Beluhov[/i]

Let $ABC$ be a triangle and a circle $\omega$ through $C$ and its incenter $I$ meets $CA, CB$ at $P, Q$. The circumcircles $(CPQ)$ and $(ABC)$ meet at $L$. The angle bisector of $\angle ALB$ meets $AB$ at $K$. Show that, as $\omega$ varies, $\angle PKQ$ is constant.

The MathMatters competition consists of 10 players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked 10th, while the winner plays $P_8$. The loser of that game is ranked 9th, while the winner plays $P_7$. They keep repeating this process until someone plays $P_1$: the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible?

Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.

a) Find all multiplicative functions $f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*}$ (i.e. that $\forall x,y\in\mathbb Z_{p}^{*}$, $f(xy)=f(x)f(y)$.) b) How many bijective multiplicative does exist on $\mathbb Z_{p}^{*}$ c) Let $A$ be set of all multiplicative functions on $\mathbb Z_{p}^{*}$, and $VB$ be set of all bijective multiplicative functions on $\mathbb Z_{p}^{*}$. For each $x\in \mathbb Z_{p}^{*}$, calculate the following sums :\[\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)\]

Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$, $$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$

Moor and nine friends are seated around a circular table. Moor starts out holding a bottle, and whoever holds the bottle passes it to the person on his left or right with equal probability until everyone has held the bottle. Compute the expected distance between Moor and the last person to receive the bottle, where distance is the fewest number of times the bottle needs to be passed in order to go back to Moor.

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.

Two different circles $\omega_1$ ,$\omega_2$, with centers $O_1, O_2$ respectively intersect at the points $A, B$. The line $O_1B$ intersects $\omega_2$ at the point $F (F \ne B)$, and the line $O_2B$ intersects $\omega_1$ at the point $E (E\ne B)$. A line was drawn through the point $B$, parallel to the $EF$, which intersects $\omega_1$ at the point $M (M \ne B)$, and $\omega_2$ at the point $N (N\ne B)$. Prove that the lines $ME, AB$ and $NF$ intersect at one point.

Define a a sequence $ {<{a_n}>}^{\infty}_{n\equal{}1}$ as follows $ a_n\equal{}0$, if number of positive divisors of $ n$ is [i]odd[/i] $ a_n\equal{}1$, if number of positive divisors of $ n$ is [i]even[/i] (The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x\equal{}0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.

Let $n,k$ be positive integers such that $n^k>(k+1)!$ and consider the set \[ M=\{(x_1,x_2,\ldots,x_n)\dvd x_i\in\{1,2,\ldots,n\},\ i=\overline{1,k}\}. \] Prove that if $A\subset M$ has $(k+1)!+1$ elements, then there are two elements $\{\alpha,\beta\}\subset A$, $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)$, $\beta=(\beta_1,\beta_2,\ldots,\beta_n)$ such that \[ (k+1)! \left| (\beta_1-\alpha_1)(\beta_2-\alpha_2)\cdots (\beta_k-\alpha_k) \right. .\]

Let $\mathcal{C}$ be a circle and let $P$ and $Q$ be points inside $\mathcal{C}$. Prove that there are infinitely many circle through $P$ and $Q$ that are completely contained inside of $\mathcal{C}$.

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of $N$ and all possible ways of voting.

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$, such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$. Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$, and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$. Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$.

Together, the two positive integers $a$ and $b$ have $9$ digits and contain each of the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. For which possible values of $a$ and $b$ is the fraction $a/b$ closest to $1$?

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.