There is a regular hexagon that is cut direct to $6n^2$ equilateral triangles (Fig.). There are arranged $2n$ rooks, neither of which beats each other (the rooks hit in directions parallel to sides of the hexagon). Prove that if we consider chess coloring all $6n^2$ equilateral triangles, then the number of rooks that stand on black triangles will be equal to the number of rooks standing on white triangles. [img]https://cdn.artofproblemsolving.com/attachments/d/0/43ce6c5c966f60a8ec893d5d8cd31e33c43fc0.png[/img] [hide=original wording] Є правильний шестикутник, що розрізаний прямими на 6n^2 правильних трикутників (рис. 2). У них розставлені 2n тур, ніякі дві з яких не б'ють одна одну (тура б'є в напрямках, що паралельні до сторін шестикутника). Доведіть, що якщо розглянути шахове розфарбування всіх 6n^2 правильних трикутників, то тоді кількість тур, що стоять на чорних трикутниках, буде рівна кількості тур, що стоять на білих трикутниках. [/hide]

Given two lines in a plane, intersecting at an acute angle. In the direction of one of the straight lines, compression is performed with a coefficient of 1/2. Prove that there is a point from which the distance to the point of intersection of the lines increases. Note: What is meant here is a transformation in which each point moves parallel to one straight line so that its distance to the second straight line is halved, while it remains the same side from the second straight line. [hide=original wording] На плоскости даны две прямые, пересекающиеся под острым углом. В направлении одной из прямых производится сжатие 1 с коэффициентом 1/2. Доказать, что найдется точка, расстояние от которой до точки пересечения прямых увеличится. Здесь имеется в виду преобразование, при котором каждая точка перемещается параллельно одной прямой так, что её расстояние до второй прямой уменьшается вдвое, причём она остаётся по ту же самую сторону от второй прямой[/hide]

The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle? $ \textbf{(A) }40 \qquad \textbf{(B) }126 \qquad \textbf{(C) }154 \qquad \textbf{(D) }176 \qquad \textbf{(E) }208 \qquad $

Construct a circle equidistant from four points on a plane. How many solutions are there?

Let H be a Hilbert space over the field of real numbers $\Bbb R$. Find all $f: H \to \Bbb R$ continuous functions for which $$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$ $$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$ is satisfied for any $x , y , z \in H$.

Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible measures of the angles of the triangle $ABC$ ?

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$. A point $P$ (other than $B$) is chosen on the segment $BC$ in such a way that $AB=AP$. Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$, and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$. Let $K$ be the intersection of lines $AP$ and $DE$. Let $F$ be a point on $BC$ (other than $P$) such that $KP=KF$. Show that $C,\ D,\ E$ and $F$ lie on the same circle.

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares? [i]Proposed by Oleksii Masalitin[/i]

Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $? $ \textbf{(A)} \ (\log 2016, \log 2017) $ $ \textbf{(B)} \ (\log 2017, \log 2018) $ $ \textbf{(C)} \ (\log 2018, \log 2019) $ $ \textbf{(D)} \ (\log 2019, \log 2020) $ $ \textbf{(E)} \ (\log 2020, \log 2021) $

It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.

Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.

Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$

Last year $100$ adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was $4$. What was the total number of cats and kittens received by the shelter last year? $ \textbf{(A)}\ 150 \qquad\textbf{(B)}\ 200 \qquad\textbf{(C)}\ 250 \qquad\textbf{(D)}\ 300 \qquad\textbf{(E)}\ 400 $

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?

suppose that $S_{\mathbb N}$ is the set of all permutations of natural numbers. finite permutations are a subset of $S_{\mathbb N}$ that behave like the identity permutation from somewhere. in other words bijective functions like $\pi: \mathbb N \longrightarrow \mathbb N$ that only for finite natural numbers $i$, $\pi(i)\neq i$. prove that we cannot put probability measure that is countably additive on $\wp(S_{\mathbb N})$ (family of all the subsets of $S_{\mathbb N}$) that is invarient under finite permutations.

Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$.

In the diagram below, a group of equilateral triangles are joined together by their sides. A parallelogram in the diagram is defined as a parallelogram whose vertices are all at the intersection of two grid lines and whose sides all travel along the grid lines. Find the number of distinct parallelograms in the diagram below. [asy] size(3cm); pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(1, 1.73); B=(2, 3.46); C=(3, 5.19); D=(4, 6.92); E=(5, 8.65); F=(6, 10.38); L=(13, 1.73); K=(12, 3.46); J=(11, 5.19); I=(10, 6.92); H=(9, 8.65); G=(8, 10.38); M=(2,0); N=(4,0); O=(6,0); P=(8,0); Q=(10,0); R=(12,0); draw(A--M); draw(B--N); draw(C--O); draw(D--P); draw(E--Q); draw(F--R); draw(A--L); draw(B--K); draw(C--J); draw(D--I); draw(E--H); draw(F--G); draw(M--G); draw(N--H); draw(O--I); draw(P--J); draw(Q--K); draw(R--L); draw(A--F); draw(G--L); draw(M--R); [/asy] [i]Proposed by Awesome_guy[/i]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $ \forall x\notin\{-1,1\}$ holds: \[\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}\]

At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? $\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

A parabola is inscribed in equilateral triangle $ABC$ of side length $1$ in the sense that $AC$ and $BC$ are tangent to the parabola at $A$ and $B$, respectively. Find the area between $AB$ and the parabola.