Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?

There are $2n$ consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any $2n$ consecutive integers again. Alexandr Gribalko

Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$. [asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W);[/asy] $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \plus{} 16\sqrt {3} \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 32 \plus{} 16\sqrt {3} \qquad \textbf{(E)}\ 64$

Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that \[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\] for every positive integer $k$. [i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]

Let $RANDOM$ be a regular hexagon with side length $1.$ Points $I$ and $T$ lie on segments $\overline{RA}$ and $\overline{DO},$ respectively, such that $MI=MT$ and $\angle{TMI}=90^\circ.$ Compute the area of triangle $MIT.$

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that $f(x+yf(x)+y^2) = f(x)+2y$ for every $x,y\in\mathbb{R}^+$.

We call an ordered set of distinct natural numbers good if for any two numbers in it, the larger one is divided by the smaller one. Prove that the number $(n + 1)! – 1$ can be represented as $x_1 + 2x_2 + \ldots + nx_n$, where $\{ x_1, x_2, \ldots , x_n \}$ is a good set, by at least $n!$ ways.

Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this: $$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$ a) Represent the function graphically. b) Calculate $A_n =\int_0^1 f_n(x) dx$. c) Find, if it exists, $\lim_{n\to \infty} A_n$ .

The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine?

Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$

Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

Let $\omega= \cos \frac{2\pi}{727} + i \sin \frac{2\pi}{727}$. The imaginary part of the complex number $$\prod^{13}_{k=8} \left(1 + \omega^{3^{k-1}}+ \omega^{2\cdot 3^{k-1}}\right)$$ is equal to $\sin a$ for some angle $a$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ , inclusive. Find $a$.

Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$? $\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$

[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism. [b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [i]Marcel Țena[/i]

Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$

Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

A sequence $\{a_n\}$ satisfies $$a_0=0, a_1=a_2=1, a_{3n} = a_n, a_{3n+1}=a_{3n+2} = a_n+1$$ for all $n \geq 1$. Compute $a_{2021}$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 6)[/i]

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer. Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$

Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

Given an octagon with the equal angles. The lengths of all the sides are integers. Prove that the opposite sides are equal in pairs. [u]alternate wording[/u] Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

Let $\vartriangle ABC$ be a right triangle such that $AB = 3$, $BC = 4$, $AC = 5$. Let point $D$ be on $AC$ such that the incircles of $\vartriangle ABD$ and $\vartriangle BCD$ are mutually tangent. Find the length of $BD$.

There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.

A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?