[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for every real $x \ge \frac12$ there is an integer $n$ such that $|x - n^2| \le \sqrt{x-\frac{1}{4}}$.

Let $\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,$ where $[x]$ denotes the integer part of $x.$ Prove that [b]i.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m > 1;$ [b]ii.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m = 1.$

Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities $$a + b + c> 0,$$ $$ ab + bc + ca > 0$$ $$ abc > 0$$ then $a > 0, b > 0, c > 0$.

Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that $(a + 3b)(b + 4c)(c + 2a) \ge 60abc$. When does equality hold?

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.

Let $ n$ be a positive integer. Consider \[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \} \] as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$. [i]Author: Gerhard Wöginger, Netherlands [/i]

Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?

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

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

Let $a$ and $b$ be two complex numbers. Prove the inequality $$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$

The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all positive integer(s) $n$ such that $n^2+32n+8$ is a perfect square.

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17$

How many non-negative integral values of $x$ satisfy the equation $ \lfloor \frac{x}{5}\rfloor = \lfloor \frac{x}{7}\rfloor $

Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

Find all the integer solutions of the equation: $$\sqrt{x} + \sqrt{y} = \sqrt{x+2023}$$

Prove that for numbers $a,b,c$ in the interval $[0,1]$, \[\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.\]

Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let $$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$ If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.

Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x + f(y)) + f(x - f(y)) = x.$$

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.