[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer 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] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/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. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root. Proposed by N. Agakhanov

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

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.

Six numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)

[b]p1.[/b] Show that for every $n \ge 6$, a square in the plane may be divided into $n$ smaller squares, not necessarily all of the same size. [b]p2.[/b] Let $n$ be the $4018$-digit number $111... 11222...2225$, where there are $2008$ ones and $2009$ twos. Prove that $n$ is a perfect square. (Giving the square root of $n$ is not sufficient. You must also prove that its square is $n$.) [b]p3.[/b] Let $n$ be a positive integer. A game is played as follows. The game begins with $n$ stones on the table. The two players, denoted Player I and Player II (Player I goes first), alternate in removing from the table a nonzero square number of stones. (For example, if $n = 26$ then in the first turn Player I can remove $1$ or $4$ or $9$ or $16$ or $25$ stones.) The player who takes the last stone wins. Determine if the following sentence is TRUE or FALSE and prove your answer: There are infinitely many starting values n such that Player II has a winning strategy. (Saying that Player II has a winning strategy means that no matter how Player I plays, Player II can respond with moves that lead to a win for Player II.) [b]p4.[/b] Consider a convex quadrilateral $ABCD$. Divide side $AB$ into $8$ equal segments $AP_1$, $P_1P_2$, $...$ , $P_7B$. Divide side $DC$ into $8$ equal segments $DQ_1$, $Q_1Q_2$, $...$ , $Q_7C$. Similarly, divide each of sides $AD$ and $BC$ into $8$ equal segments. Draw lines to form an $8 \times 8$ “checkerboard” as shown in the picture. Color the squares alternately black and white. (a) Show that each of the $7$ interior lines $P_iQ_i$ is divided into $8$ equal segments. (b) Show that the total area of the black regions equals the total area of the white regions. [img]https://cdn.artofproblemsolving.com/attachments/1/4/027f02e26613555181ed93d1085b0e2de43fb6.png[/img] [b]p5.[/b] Prove that exactly one of the following two statements is true: A. There is a power of $10$ that has exactly $2008$ digits in base $2$. B. There is a power of $10$ that has exactly $2008$ digits in base $5$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, \[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$. In both cases, describe all triples for which equality holds.

Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $

We denote $N_{2010}=\{1,2,\cdots,2010\}$ [b](a)[/b]How many non empty subsets does this set have? [b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? [b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $2f(x^2+y^2)=(x+f(y))^2+f(x-f(y))^2$ for all $x,y\in\mathbb{R}$.

Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.

Show that $x^4 > x - \frac12$ for all real $x$.

Let $a$ and $b$ be positive integers satisfying \[ \frac a{a-2} = \frac{b+2021}{b+2008} \] Find the maximum value $\dfrac ab$ can attain.