Let $a,b$ be positive even integers. A rectangle with side lengths $a$ and $b$ is split into $a \cdot b$ unit squares. Anja and Bernd take turns and in each turn they color a square that is made of those unit squares. The person that can't color anymore, loses. Anja starts. Find all pairs $(a,b)$, such that she can win for sure. [b]Extension:[/b] Solve the problem for positive integers $a,b$ that don't necessarily have to be even. [b]Note:[/b] The [i]extension[/i] actually was proposed at first. But since this is a homework competition that goes over three months and some cases were weird, the problem was changed to even integers.

Consider finitely many points in the plane such that no three are collinear. Prove that we can paint the points with two colors such that there is no half-plane that contains exactly three points such that those three points have the same color.

In a triangle $ABC, AB = c, BC = a, CA = b$, and $a < b < c$. Points $B'$ and $A'$ are chosen on the rays $BC$ and $AC$ respectively so that $BB'= AA'= c$. Points $C''$ and $B''$ are chosen on the rays $CA$ and $BA$ so that $CC'' = BB'' = a$. Find the ratio of the segment $A'B'$ to the segment $C'' B''$. (R Zhenodarov)

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

Joy has a square board of size $n \times n$. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has $0$ points. Every step, after colouring a cell $c$, he takes the largest possible set $S$ that creates a "$+$" sign where all cells are coloured and $c$ lies in the centre. Then, he gets the size of set $S$ as points. After colouring the whole $n \times n$ board, what is the maximum possible amount of points he can get?

An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$

Prove that $\ln n \geq k\ln 2$, where $n$ is a natural number and $k$ is the number of distinct primes that divide $n$.

[u]Calculus Round[/u] [b]p1.[/b] Evaluate $$\int^2_{-2}(x^3 + 2x + 1)dx$$ [b]p2.[/b] Find $$\lim_{x \to 0} \frac{ln(1 + x + x^3) - x}{x^2}$$ [b]p3.[/b] Find the largest possible value for the slope of a tangent line to the curve $f(x) = \frac{1}{3+x^2}$ . [b]p4.[/b] An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point when the water reaches a height of $10$ meters. [b]p5.[/b] Compute $$\int^{\frac{\pi}{2}}_0 \sin (2016x) \cos (2015x) dx$$ [b]p6.[/b] Let $f(x)$ be a function defined for $x > 1$ such that $f''(x) = \frac{x}{\sqrt{x^2-1}}$ and $f'(2) =\sqrt3$. Compute the length of the graph of $f(x)$ on the domain $x \in (1, 2]$. [b]p7.[/b] Let the function $f : [1, \infty) \to R$ be defuned as $f(x) = x^{2 ln(x)}$. Compute $$\int^{\sqrt{e}}_1 (f(x) + f^{-1}(x))dx$$ [b]p8.[/b] Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$. [b]p9.[/b] Compute $$\int^e_1 \frac{ln (x)}{(1 + ln (x))^2} dx$$ [b]p10.[/b] For $x \ge 0$, let $R$ be the region in the plane bounded by the graphs of the line $\ell$ : $y = 4x$ and $y = x^3$. Let $V$ be the volume of the solid formed by revolving $R$ about line $\ell$. Then $V$ can be expressed in the form $\frac{\pi \cdot 2^a}{b\sqrt{c}}$ , where $a$, $b$, and $c$ are positive integers, $b$ is odd, and $c$ is not divisible by the square of a prime. Compute $a + b + c$. [u]Calculus Tiebreaker[/u] [b]Tie 1.[/b] Let $f(x) = x + x(\log x)^2$. Find $x$ such that $xf'(x) = 2f(x)$. [b]Tie 2.[/b] Compute $$\int^{\frac{\sqrt2}{2}}_{-1} \sqrt{1 - x^2} dx$$ [b]Tie 3.[/b] An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle. PS. You should use hide for answers.

For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.\]

a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.

For any positive numbers $a, b, c$ prove the inequalities \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]

Every point in the plane was colored in red or blue. Prove that one the two following statements is true: $\bullet$ there exist two red points at distance $1$ from each other; $\bullet$ there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i $ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$.

It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$\\ a) Compute the angles of $ABC$, if $k=2\pi$ b) Prove it is not possible for k to be $3$

Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.

Let $a, b, c$ be the roots of $x^3-7x^2-6x+5=0$. Compute $(a+b)(a+c)(b+c)$.

Let $r_1,r_2,r$, with $r_1 < r_2 < r$, be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$, respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$.

Let $ k$ and $ s$ be positive integers. For sets of real numbers $ \{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $ \{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy \[ \sum^s_{i\equal{}1} \alpha^j_i \equal{} \sum^s_{i\equal{}1} \beta^j_i \quad \forall j \equal{} \{1,2 \ldots, k\}\] we write \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}.\] Prove that if \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}\] and $ s \leq k,$ then there exists a permutation $ \pi$ of $ \{1, 2, \ldots , s\}$ such that \[ \beta_i \equal{} \alpha_{\pi(i)} \quad \forall i \equal{} 1,2, \ldots, s.\]

An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called [i]central[/i] if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$. \\Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.

Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$

Find the maximum number of cells that can be coloured from a $4\times 3000$ board such that no tetromino is formed. [i]Proposed by Arian Zamani, Matin Yousefi[/i] [b]Rated 5[/b]

Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed? $\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$ $\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$ $\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$ $\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$ $\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.

On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.

Find all integers $n\geq 2$ for which there exist the real numbers $a_k, 1\leq k \leq n$, which are satisfying the following conditions: \[\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.\]