Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

Let $a,b,c$ be positive integers greater than $1$ such that $$p=ab+bc+ac$$ is prime. A) Prove that $a^2, b^2, c^2$ all have different reminder $mod\ p$. B) Prove that $a^3, b^3, c^3$ all have different reminder $mod\ p$.

Zaineb makes a large necklace from beads labeled $290, 291, \ldots, 2023$. She uses each bead exactly once, arranging the beads in the necklace any order she likes. Prove that no matter how the beads are arranged, there must be three beads in a row whose labels are the side lengths of a triangle.

A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.

You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations: 1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta. 2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up. 3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively. 4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair. 5. Remove four consecutive beads of one color. Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'', b) ``cyan, yellow, cyan, magenta, cyan'', c) ``magenta, magenta, cyan, cyan, cyan'', d) ``yellow, cyan, cyan, cyan''. [i]Proposed by Sammy Luo[/i]

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

Prove that for real $x\ge 1$, holds the inequality $$\frac{2^x +3^x }{3^x +4^x} \le \frac57$$

It is known that odd prime numbers $x, y z$ satisfy $$x|(y^5+1),y|(z^5+1),z|(x^5+1).$$ Find the minimum value of the product $xyz$.

What is the probability of having no $B$ before the first $A$ in a random permutation of the word $\text{ABRAKADABRA}$? $ \textbf{(A)}\ \dfrac 23 \qquad\textbf{(B)}\ \dfrac 57 \qquad\textbf{(C)}\ \dfrac 56 \qquad\textbf{(D)}\ \dfrac 67 \qquad\textbf{(E)}\ \text{None of the preceding} $

Prove that the number $19^{1976} + 76^{1976}$: $(a)$ is divisible by the (Fermat) prime number $F_4 = 2^{2^4} + 1$; $(b)$ is divisible by at least four distinct primes other than $F_4$.

Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.

Let $K$ be a convex planar set, symmetric about a point $O$, and let $X, Y , Z$ be three points in $K$. Show that $K$ contains the head of one of the vectors $\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}$.

Given the polynomial $ P(x) = \frac{1}{2} - \frac{1}{3}x + \frac{1}{6}x^2 $. Let $ Q(x) = \sum_{k=0}^{m} b_k x^k $ be a polynomial given by $$ Q(x) = P(x) \cdot P(x^3) \cdot P(x^9) \cdot P(x^{27}) \cdot P(x^{81}). $$ Calculate $ \sum_{k=0}^m |b_k| $.

Let $ABC$ be an acute triangle with orthocenter $ H $ and $AB<AC.$ Let $\Omega_1$ be a circle with diameter $AC$ and $\Omega_2$ a circle with diameter $ AB.$ Line $BH$ intersects $\Omega_1$ in points $ D $ and $E$ such that $E$ is not on segment $BH.$ Line $ CH $ intersects $\Omega_2$ in points $ F $ and $G$ such that $G$ is not on segment $CH.$ Prove that the lines $EG, DF$ and $BC$ are concurrent.

Find all positive integers $k$ that can be expressed as the sum of $50$ fractions such that the numerators are the $50$ natural numbers from $1$ to $50$ and the denominators are positive integers, that is, $k = \dfrac{1}{a_1} + \dfrac{2}{a_2} + \ldots + \dfrac{50}{a_{50}}$ with a$_1 , a_2 , \ldots , a_n$ positive integers.

In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$. a) Prove that $T$ is the incircle of triangle $XYZ$. b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$. c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.

A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$.

$ a)$ Two players play a cooperative game. They can discuss a strategy prior to the game, however, they cannot communicate and have no information about the other player during the game. The game master chooses one of the players in each round. The player on turn has to guess the number of the current round. Players keep note of the number of rounds they were chosen, however, they have no information about the other player's rounds. If the player's guess is correct, the players are awarded a point. Player's are not notified whether they've scored or not. The players win the game upon collecting 100 points. Does there exist a strategy with which they can surely win the game in a finite number of rounds? $b)$ How does this game change, if in each round the player on turn has two guesses instead of one, and they are awarded a point if one of the guesses is correct (while keeping all the other rules of the game the same)? [i]Proposed by Gábor Szűcs, Budapest[/i]

Given a $\triangle ABC$ with its area equal to $1$, suppose that the vertices of quadrilateral $P_1P_2P_3P_4$ all lie on the sides of $\triangle ABC$. Show that among the four triangles $\triangle P_1P_2P_3, \triangle P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4$ there is at least one whose area is not larger than $1/4$.

Determine all pairs $(a, b)$ of integers for which $a^{2}+b^{2}+3$ is divisible by $ab$.

Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$

The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)

Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1}=9786453210$. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $M$.

Prove: In an acute triangle $ ABC$ angle bisector $ w_{\alpha},$ median $ s_b$ and the altitude $ h_c$ intersect in one point if $ w_{\alpha},$ side $ BC$ and the circle around foot of the altitude $ h_c$ have vertex $ A$ as a common point.