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.

Find the largest number among the following numbers: $ \textbf{(A) }30^{30}\qquad\textbf{(B) }50^{10}\qquad\textbf{(C) }40^{20}\qquad\textbf{(D) }45^{15}\qquad\textbf{(E) }5^{60}$

In the tetrahedron $ABCD,$ on the continuation of the edges $AB, AC$ and $AD$, three points were marked for point $A{},$ located from $A{}$ at a distance equal to the semi-perimeter of the triangle $BCD.$ Similarly, we marked three points corresponding to vertices $B, C$ and $D.$ Prove that if there is a sphere touching all the edges of the tetrahedron $ABCD$, then the marked 12 points lie on the same sphere. [i]Proposed by V. Alexandrov[/i]

Let $r>s$ be positive integers. Let $P(x)$ and $Q(x)$ be distinct polynomials with real coefficients, non-constant(s), such that $P(x)^r-P(x)^s=Q(x)^r-Q(x)^s$ for every $x\in \mathbb{R}$. Prove that $(r,s)=(2,1)$.

The wire is bent in the form of a square with side $2$. Find the volume of the body consisting of all points in space located at a distance not exceeding $1$ from at least one point of the wire.

You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $\$1.02$, with at least one coin of each type. How many dimes must you have? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

Inside a $2\times 2$ square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a $1\times 2$ rectangle.