Let $p$ be a prime number and let $m, n$ be integers greater than $1$ such that $n | m^{p(n-1)} - 1$. Prove that $gcd(m^{n-1} - 1, n) > 1$.

Let the number $x$. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$, $x^{2}\cdot x^{2}=x^{4}$, $x^{4}: x=x^{3}$, etc). Determine the minimal number of operations needed for calculating $x^{2006}$.

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$

In a classroom of at least four students, when any four of them take seats around a round table, there is always someone who either knows both of his neighbors, or does not know either of his neighbors. Prove that it is possible to divide the students into two groups so that in one of them, all students knows one another, and in the other, none of the students know each other. [i]Note: If $A$ knows $B$, then $B$ knows $A$ as well.[/i]

Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.

Let $ \mathcal{F}$ be a family of subsets of a ground set $ X$ such that $ \cup_{F \in \mathcal{F}}F=X$, and (a) if $ A,B \in \mathcal{F}$, then $ A \cup B \subseteq C$ for some $ C \in \mathcal{F};$ (b) if $ A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},$ and $ A_0 \subset A_1 \subset...,$ then, for some $ k \geq 0, \;A_n \cap B=A_k \cap B$ for all $ n \geq k$. Show that there exist pairwise disjoint sets ${ X_{\gamma} \;( \gamma \in \Gamma}\ )$, with $ X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},$ such that every $ X_{\gamma}$ is contained in some member of $ \mathcal{F}$, and every element of $ \mathcal{F}$ is contained in the union of finitely many $ X_{\gamma}$'s. [i]A. Hajnal[/i]

$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and: \[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\] and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.

On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ be on segment$ BC$ of $\vartriangle ABC$ so that $AM = 3$, $BM = 4$, and $CM = 5$. Find the largest possible area of $\vartriangle ABC$.

Let $O$ be the center of the circumcircle of an acute triangle $ABC$, let $P$ be any point inside the segment $BC$. Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$. (i) Consider the triangle $PQR$, show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$, $COP$, $PQR$ have the same radius.

Two circles $C_1$ and $C_2$ intersect at $S$. The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$. The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$. Another circle $C_3$ goes through $A, B, S$. The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$. Prove that the distance $PS$ is equal to the distance $QS$.

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

Find all the positive integers $x$ and $y$ that satisfy the equation $x(x - y) = 8y - 7$

Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\] for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$ [b](i)[/b] Prove that $S$ is an infinite set. [b](ii)[/b] Find the least positive integer in $S.$ [b](iii)[/b] If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \] show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]

Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be[i] legally transformed[/i] by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.

There is a bacterium in one of the cells of a $10 \times 10{}$ checkered board. At the first move, the bacterium shifts to a cell adjacent by side to the original one, and divides into two bacteria (both stay in the same cell). Then again, one of the bacteria on the board shifts to a cell adjacent by side and divides into two bacteria, and so on. Is it possible that after some number of such moves the number of bacteria in each cell of the board is the same? [i]Alexandr Gribalko[/i]

Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic. [b][i]Alternative formulation[/i][/b] Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression. [i]Proposed by Romania[/i]

Prove that, if $\tan (a/2)$ is rational (or else, if $ a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined), then $\cos a$ and $\sin a$ are rational. And, conversely, if $\cos a$ and $\sin a$ are rational, then $\tan (a/2)$ is rational unless $a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined.

Find all the natural numbers $x, y, z$ that satisfy simultaneously $$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$

The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who? [i]Remark.[/i] The answer may depend on initial position of the checker.

Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$.

How many multiples of $11$ of four digits, of the form $\overline{abcd}$, satisfy that $a\neq b, b\neq c$ and $c\neq a$?

Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation \[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]