A dragon gave a captured knight $100$ coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most (a) $50$ days? (b) $25$ days?

Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$. Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$

If \[x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2},\] where $ i^2 \equal{} \minus{} 1$, then which of the following is [i]not[/i] correct? $ \textbf{(A)}\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad \textbf{(B)}\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad \textbf{(C)}\ x^9 \plus{} y^9 \equal{} \minus{} 1$ $ \textbf{(D)}\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad \textbf{(E)}\ x^{13} \plus{} y^{13} \equal{} \minus{} 1$

The natural numbers $k \geqslant n$ are given. Peter has $n{}$ objects and $N{}$ special ways in which he likes to lay them out in a row from left to right. He noticed that for any non-empty subset $A{}$ of these objects containing $|A| \leqslant k$ objects, and any element $a\in A$, there are exactly $N/|A|$ special ways for which element $a{}$ is the leftmost in the set $A{}$. Prove that, under the same conditions on $A{}$ and $a{}$, for any integer $m =1,2,\ldots,|A|$ there are exactly $N/|A|$ special ways for which the element $a{}$ is the $m^{\text{th}}$ from the left in the set $A{}$.

Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.

Lucky and Jinx were given a paper with $2023$ points arranged as the vertices of a regular polygon. They were then tasked to color all the segments connecting these points such that no triangle formed with these points has all edges in the same color, nor in three different colors and no quadrilateral (not necessarily convex) has all edges in the same color. After the coloring it was determined that Jinx used at least two more colors than Lucky. How many colors did each of them use? [i]Authored by Ilija Jovcheski[/i]

Call a number [i]orz[/i] if it is a positive integer less than $2024$. Call a number [i]admitting[/i] if it can be expressed as $a^2-1$ where $a$ is a positive integer. Finally call a number [i]muztaba[/i] if it has exactly $4$ positive integer factors. Find the number of [i]muztaba admitting orz[/i] numbers.

There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions. (a) $ k \le 4r $ (b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $l$ such that \[ (m-1)! < l < (m+1)!+1 \]

Let $n\geq 3$ be an integer. Each vertex of a regular $n$-gon is labelled with a real number not exceeding $1$. For real numbers $a,b,c$ on any three consecutive vertices which are arranged clockwise in such an order, we have $c=|a-b|$. Determine the maximum value of the sum of all numbers in terms of $n$.

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A,B,\cdots,Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within $20$ years of each other, then your score is $0$. Otherwise, your score is $\max(3(k-3),0)$ where $k$ is the number of elements in your set. \[\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}\]

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum.

$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: $xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$

The roots of the equation $ 2\sqrt {x} \plus{} 2x^{ \minus{} \frac {1}{2}} \equal{} 5$ can be found by solving: $ \textbf{(A)}\ 16x^2 \minus{} 92x \plus{} 1 \equal{} 0 \qquad \textbf{(B)}\ 4x^2 \minus{} 25x \plus{} 4 \equal{} 0 \qquad \textbf{(C)}\ 4x^2 \minus{} 17x \plus{} 4 \equal{} 0 \\ \textbf{(D)}\ 2x^2 \minus{} 21x \plus{} 2 \equal{} 0 \qquad \textbf{(E)}\ 4x^2 \minus{} 25x \minus{} 4 \equal{} 0$

For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties: • $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$; • $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$. Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.

Determine two last digits of number $Q = 2^{2017} + 2017^2$

A triangle $\triangle ABC$ satisfies $AB = 13$, $BC = 14$, and $AC = 15$. Inside $\triangle ABC$ are three points $X$, $Y$, and $Z$ such that: [list] [*] $Y$ is the centroid of $\triangle ABX$ [*] $Z$ is the centroid of $\triangle BCY$ [*] $X$ is the centroid of $\triangle CAZ$ [/list] What is the area of $\triangle XYZ$? [i]Proposed by Adam Bertelli[/i]

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$