Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$

Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and ${H_{c}^{'} }$ are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$. Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$, we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?

Let $ x$ and $ y$ be two-digit integers such that $ y$ is obtained by reversing the digits of $ x$. The integers $ x$ and $ y$ satisfy $ x^2 \minus{} y^2 \equal{} m^2$ for some positive integer $ m$. What is $ x \plus{} y \plus{} m$? $ \textbf{(A)}\ 88\qquad \textbf{(B)}\ 112\qquad \textbf{(C)}\ 116\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 154$

Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that $f(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}$. Additionally, gcd$(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}$. Show that if $a,b$ are coprime natural numbers, that is, gcd$(a,b) = 1$ then $f(a),f(b)$ are also coprime.

You are given some $12$ non-zero, not necessarily distinct real numbers. Find all positive integers $k$ from $1$ to $12$, such that among these numbers you can always choose $k$ numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative. [i]Proposed by Anton Trygub[/i]

On a mathematical competition $ n$ problems were given. The final results showed that: (i) on each problem, exactly three contestants scored $ 7$ points; (ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems. Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.

Let $f$ be a function such that \[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\] Find the limit of $f$ in the point $x_0=1.$

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.

Rays $l_1$ and $l_2$ pass through a point $O$. Segments $OA_1$ and $OB_1$ on $l_1$, and $OA_2$ and $OB_2$ on $l_2$, are drawn so that $\frac{OA_1}{OA_2} \ne \frac{OB_1}{OB_2}$ . Find the set of all intersection points of lines $A_1A_2$ and $B_1B_2$ as $l_2$ rotates around $O$ while $l_1$ is fixed.

Let $a_1=3$, $a_2=7$, and $a_3=1$. Let $b_0=0$ and for all positive integers $n$, let $b_n=10b_{n-1}+a_n$. Compute $b_1\times b_2\times b_3$. [i]2020 CCA Math Bonanza Lightning Round #1.2[/i]

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.

Catherine has a plate containing $300$ circular crumbling mooncakes, arranged as follows: [asy] unitsize(10); for (int i = 0; i < 16; ++i){ for (int j = 0; j < 3; ++j){ draw(circle((sqrt(3)*i,j),0.5)); draw(circle((sqrt(3)*(i+0.5),j-0.5),0.5)); } } dot((16*sqrt(3)+.5,.75)); dot((16*sqrt(3)+1,.75)); dot((16*sqrt(3)+1.5,.75)); [/asy] (This continues for $100$ total columns). She wants to pick some of the mooncakes to eat, however whenever she takes a mooncake all adjacent mooncakes will be destroyed and cannot be eaten. Let $M$ be the maximal number of mooncakes she can eat, and let $n$ be the number of ways she can pick $M$ mooncakes to eat (Note: the order in which she picks mooncakes does not matter). Compute the ordered pair ($M$, $n$).

Let $n \ge 3$ be an odd integer. In a $2n \times 2n$ board, we colour $2(n-1)^2$ cells. What is the largest number of three-square corners that can surely be cut out of the uncoloured figure? [i]Proposed by G. Sharafetdinova[/i]

A right triangle has sides of integer length. One side has length 11. What is the area of the triangle?

Positive real numbers $a$ and $b$ have the property that \[ \sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100 \] and all four terms on the left are positive integers, where $\text{log}$ denotes the base 10 logarithm. What is $ab$? $\textbf{(A) } 10^{52} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200} $

Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are: $\textbf{(A)}\ x, z, y \qquad\textbf{(B)}\ x, y, z \qquad\textbf{(C)}\ y, x, z \qquad\textbf{(D)}\ y, z, x \qquad\textbf{(E)}\ z, x, y$

Suppose $a$, $b$, $c$, and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, \\ ab+cd &= 260, \\ ac+bd &= 365, \\ ad+bc &= 244. \end{aligned}\] Compute $abcd.$ [i]Proposed by Michael Tang[/i]

Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\] [i]Proposed by Otgonbayar Uuye. [/i]

Evaluate $2000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3$