A triangle $ ABC$ is given. Let $ B_1$ be the reflection of $ B$ across the line $ AC$, $ C_1$ the reflection of $ C$ across the line $ AB$, and $ O_1$ the reflection of the circumcentre of $ ABC$ across the line $ BC$. Prove that the circumcentre of $ AB_1C_1$ lies on the line $ AO_1$. [i]Proposed by A. Akopyan[/i]

What fraction of the Earth's volume lies above the $45$ degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.

With $150$ white cubes of $1 \times 1 \times 1$ a prism of $6 \times 5 \times 5$ is assembled, its six faces are painted blue and then the prism is disassembled. Lucrecia must build a new prism, without holes, exclusively using cubes that have at least one blue face and so that the faces of Lucrecia's prism are all completely blue. Give the dimensions of the prism with the largest volume that Lucrecia can assemble.

For a natural number $ n\ge 2, $ prove that the $ \text{n-ary} $ direct product of the group of order $ 2 $ is abelian and isomorphic with the group of the power set of a set under symmetric difference.

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$. [i]Laurențiu Panaitopol[/i] and [i]Ioan Tomescu[/i]

Given an acute-angled $\vartriangle AB$C with altitude $AH$ ( $\angle BAC > 45^o > \angle AB$C). The perpendicular bisector of $AB$ intersects $BC$ at point $D$. Let $K$ be the midpoint of $BF$, where $F$ is the foot of the perpendicular from $C$ on $AD$. Point $H'$ is the symmetric to $H$ wrt $K$. Point $P$ lies on the line $AD$, such that $H'P \perp AB$. Prove that $AK = KP$.

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

Show that for all reals $x,y,z$, we have $$\left(x^2+3\right)\left(y^2+3\right)\left(z^2+3\right)\ge(xyz+x+y+z+4)^2.$$

Let $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$. Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}. $$ That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$, then $|x_{n_i+n_j}-L| < \epsilon.$

Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define \[ \rho (x,y)=yh_x -xh_y . \] Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $S$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $S$ is zero.

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

Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\tfrac{1}{3}a_{n-1}$ and is $\tfrac{1}{9}a_{n-1}$ with probability $\tfrac{1}{2}$. If the expected value of $\textstyle\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\tfrac{p}{q}$, what is $p+q$?

Q. An automatic cleaner of the disc shape has passed along a plain floor. For each point of its circular boundary there exists a straight line that has contained this point all the time. Is it necessarily true that the center of the disc stayed on some straight line all the time? ($9$ marks)

Let $a>0$ and $x_1,x_2,x_3$ be real numbers with $x_1+x_2+x_3=0$. Prove that $$\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.$$

Let $k$ be a constant number larger than $1$. Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$.

Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that: $a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$; $b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.

Find the interior angle between two sides of a regular octagon (degrees).

Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits. (Note: Four digits means that the thousand Unit digit must not be $0$.)

The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$. Which of the following is the smallest? $ \textbf{(A)}\ P( \minus{} 1)$ $ \textbf{(B)}\ \text{The product of the zeros of }P$ $ \textbf{(C)}\ \text{The product of the non \minus{} real zeros of }P$ $ \textbf{(D)}\ \text{The sum of the coefficients of }P$ $ \textbf{(E)}\ \text{The sum of the real zeros of }P$ [asy] size(170); defaultpen(linewidth(0.7)+fontsize(7));size(250); real f(real x) { real y=1/4; return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3; } draw(graph(f,-10.5,19.4)); draw((-13,0)--(22,0)^^(0,-10.5)--(0,15)); int i; filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white); for(i=-3; i<6; i=i+1) { if(i!=0) { draw((4*i,0)--(4*i,-0.2)); label(string(i), (4*i,-0.2), S); }} for(i=-5; i<6; i=i+1){ if(i!=0) { draw((0,2*i)--(-0.2,2*i)); label(string(2*i), (-0.2,2*i), W); }} label("0", origin, SE);[/asy]

The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that \[ c<2024 (a_1+a_2+\ldots+a_{2024}).\]

Given a fixed circle $C$ and a line L through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle with center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

At a length of $104$ miles, the Danyang-Kushan Bridge holds the title for being the longest bridge in the world. A car travels at a constant speed of $39$ miles per hour across the Danyang-Kushan Bridge. How long does it take the car to travel across the entire bridge? $\textbf{(A) }\text{2 hours, 12 minutes} \qquad \textbf{(B) }\text{2 hours, 20 minutes} \qquad \textbf{(C) }\text{2 hours, 25 minutes}$\\ $\textbf{(D) }\text{2 hours, 30 minutes} \qquad \textbf{(E) }\text{2 hours, 40 minutes}$

Let \( X \) be an arbitrary point on the side \( BC \) of triangle \( ABC \). The point \( M \) on the ray \( AB^\to \) beyond \( B \), the point \( N \) on the ray \( AC^\to \) beyond \( C \), and the point \( K \) inside \( ABC \) are such that \( \angle BMX = \angle CNX = \angle KBC = \angle KCB \). The line through \( A \), parallel to \( BC \), intersects the line \( KX \) at point \( P \). Prove that the points \( A \), \( P \), \( M \), \( N \) lie on a circle.