In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a [b]snake of length $m$[/b] if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

In Sikinia, there are $2024$ cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all $2023$ other cities. It is known, however, that there is a positive integer $n$ with the following property: For any $n$ cities in Sikinia, there is another city which is directly connected to all these cities. Determine the largest possible value of $n$.

Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: \[|A+qA|\ge (q+1)|A|-C_q.\] [i]Proposed by Antal Balog.[/i]

Prove that, for any natural number $n$, the expression $$A = 2903^n-803^n-464^n+261^n$$ is divisible by $1897$.

The graph of the equation $\tan(x+y) = \tan(x)+2\tan(y),$ with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions.

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by $a_{1}=1$ and \[a_{n+1}= \frac{a_{n}}{2}+\frac{1}{4a_{n}}\; (n \in \mathbb{N}).\] Prove that $\sqrt{\frac{2}{2a_{n}^{2}-1}}$ is a positive integer for $n>1$.

Prove that there is a positive integer $m$ such that the number $5^{2021}m$ has no even digits (in its decimal representation).

Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.

$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$ [i](K. Ivanov )[/i]

Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$. [i]Raja Oktovin, Pekanbaru[/i]

On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

A [i]magic[/i] octagon is an octagon whose sides follow the lines of the checkerboard's checkers and the side lengths are $1, 2, 3, 4, 5, 6, 7, 8$ (in any order). What is the largest possible area of the magic octagon? [hide=original wording]Burvju astoņstūris ar astoņstūris, kura malas iet pa rūtiņu lapas rūtiņu līnijām un malu garumi ir 1, 2,3, 4, 5, 6, 7, 8 (jebkādā secībā). Kāds ir lielākais iespējamais burvju astoņstūra laukums?[/hide]

Two $10 \times 24$ rectangles are inscribed in a circle as shown. Find the shaded area. [img]https://cdn.artofproblemsolving.com/attachments/1/7/c97fb0e6f45a52fa751777da6ebc519839e379.png[/img]

A piece of paper is folded in half. A second fold is made at an angle $ \phi$ ($ 0^\circ < \phi < 90^\circ$) to the first, and a cut is made as shown below. [img]12881[/img] When the piece of paper is unfolded, the resulting hole is a polygon. Let $ O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $ O$, and also that $ \angle XOY \equal{} 144^\circ$, where $ X$ and $ Y$ are the the vertices of the hole adjacent to $ O$. Find the value(s) of $ \phi$ (in degrees).

$29$ quadratic polynomials $f_1(x), \ldots, f_{29}(x)$ and $15$ real numbers $x_1<x_2<\ldots<x_{15}$ are given. Prove that for some two given polynomials $f_i(x)$ and $f_j(x)$ the following inequality holds: $$\sum_{k=1}^{14} (f_i(x_{k+1})-f_i(x_k))(f_j(x_{k+1})-f_j(x_k))>0$$ [i]A. Voidelevich[/i]

Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that $$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$

Compute $$\int_0^{\frac{\pi}{2}}\frac{e^x(\sin x+\cos x-2)}{(\cos x-2)^2}dx.$$

$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.

A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? [asy] size(200); defaultpen(linewidth(0.8)); draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300)); draw((82,-204) -- (133,-490) -- (342, -662)); draw((652,-626) -- (600,-727)); draw((447,-458) -- (652,-626) -- (859,-410)); draw((133,-490) -- (184, -388)); draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

A teacher gave a test to a class in which $ 10\%$ of the students are juniors and $ 90\%$ are seniors. The average score on the test was $ 84$. The juniors all received the same score, and the average score of the seniors was $ 83$. What score did each of the juniors receive on the test? $ \textbf{(A)}\ 85 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 93 \qquad \textbf{(D)}\ 94 \qquad \textbf{(E)}\ 98$

In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]