A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. [i]N. Agakhanov[/i]

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

Let $ABC$ be a triangle such that $|AB|=|AC|$ and let $D,E$ be points on the minor arcs $\overarc{AB}$ and $\overarc{AC}$ respectively. The lines $AD$ and $BC$ intersect at $F$ and the line $AE$ intersects the circumcircle of $\triangle FDE$ a second time at $G$. Prove that the line $AC$ is tangent to the circumcircle of $\triangle ECG$.

Find all quadruples $(a, b, c, d)$ of positive integers satisfying $\gcd(a, b, c, d) = 1$ and \[ a | b + c, ~ b | c + d, ~ c | d + a, ~ d | a + b. \] [i]Vítězslav Kala (Czech Republic)[/i]

You start with a single piece of chalk of length $1$. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have $8$ pieces of chalk. What is the probability that they all have length $\frac18$ ?

Karl's rectangular vegetable garden is $20$ by $45$ feet, and Makenna's is $25$ by $40$ feet. Which garden is larger in area? $\textbf{(A)}$ Karl's garden is larger by 100 square feet. $\textbf{(B)}$ Karl's garden is larger by 25 square feet. $\textbf{(C)}$ The gardens are the same size. $\textbf{(D)}$ Makenna's garden is larger by 25 square feet. $\textbf{(E)}$ Makenna's garden is larger by 100 square feet.

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

Let $n$ be an integer greater than $1$ and let $S$ be a finite set containing more than $n+1$ elements.Consider the collection of all sets $A$ of subsets of $S$ satisfying the following two conditions : [b](a)[/b] Each member of $A$ contains at least $n$ elements of $S$. [b](b)[/b] Each element of $S$ is contained in at least $n$ members of $A$. Determine $\max_A \min_B |B|$ , as $B$ runs through all subsets of $A$ whose members cover $S$ , and $A$ runs through the above collection.

$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that: $\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$

For each real number $x,$ let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x)\ge L$ for all $x\in [0,1)$? (As usual, $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$

Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$

The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple? [i]Proposed by CDMX[/i]

A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$: $$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$ Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define: $$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$ Investigate a) whether $M$ is a semigroup; b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.

Lennart and Eddy are playing a betting game. Lennart starts with $7$ dollars and Eddy starts with $3$ dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet $3$ dollars. A fair coin is then tossed. If it lands heads, Lennart wins all the money bet; if it lands tails, Eddy wins all the money bet. They continue playing this game until one person has no money. What is the probability that Eddy ends with $10$ dollars?

Let $p$ be a real number and $c\neq 0$ such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p,c)$.

A country has $2023$ cities and there are flights between these cities. Each flight connects two cities in both directions. We know that you can get from any city to any other using these flights, and from each city there are flights to at most $4$ other cities. What is the maximum possible number of cities in the country from which there is a flight to only one city?

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$