Given is a grid 8x8. Every square is colored in black or white, so that in every 3x3, the number of white squares is even. What is the minimum number of black squares

Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$

Natural numbers $1, 2, 3,.., 100$ are contained in the union of $N$ geometric progressions (not necessarily with integer denominations). Prove that $N \ge 31$

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card.

Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that $$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$ Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle. [i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]

Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$. Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying \[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \] for all $x,y \in S$. Let $N$ be the product of all possible nonzero values of $f(81)$. Find the remainder when when $N$ is divided by $p$. [i]Proposed by Yang Liu and Ryan Alweiss[/i]

Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

Adam has a circle of radius $1$ centered at the origin. - First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces. - Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis. - Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle. What is the product of the lengths of all $18$ segments Adam drew? [img]https://cdn.discordapp.com/attachments/813077401265242143/816190774257516594/circle2.png[/img] [i]Proposed by Adam Bertelli[/i]

Let $a$ be a positive real number. Prove that a) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a$. b) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a$. (As usual, $\sigma(n) = \sum_{d\mid n} d$ and $\varphi(n)$ is the number of integers $1\le m\le n$ which are coprime with $n$.) Proposed by [i]Deniz Can Karaçelebi[/i]

Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$.

In a garden, which is organized as a $2024\times 2024$ board, we plant three types of flowers: roses, daisies, and orchids. We want to plant flowers such that the following conditions are satisfied: (i) Each grid is planted with at most one type of flower. Some grids can be left blank and not planted. (ii) For each planted grid $A$, there exist exactly $3$ other planted grids in the same column or row such that those $3$ grids are planted with flowers of different types from $A$'s. (iii) Each flower is planted in at least $1$ grid. What is the maximal number of the grids that can be planted with flowers?

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed below. What is the least number of circles needed to make the total shaded area at least $2023\pi$?

Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.

9.6, 10.6 Construct for each vertex of the trapezium a symmetric point wrt to the diagonal, which doesn't contain this vertex. Prove that if four new points form a quadrilateral then it is a trapezium. ([i]L. Emel'yanov[/i])