If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]

Students in a class of $n$ students had to solve $2^{n-1}$ problems on an exam. It turned out that for each pair of distinct problems: • there is at least one student who has solved both • there is at least one student who has solved one of them, but not the other. Show that there is a problem solved by all the students in the class.

On a checkered square $10 \times 10$ the cells of the upper left $5 \times 5$ square are black and all the other cells are white. What is the maximal $n$ such that the original square can be dissected (along the borders of the cells) into $n$ polygons such that in each of them the number of black cells is three times less than the number of white cells? (The polygons need not be congruent or even equal in area.)

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .

A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?

A badminton club consists of $2n$ members who are n couples. The club plans to arrange a round of mixed doubles matches where spouses neither play together nor against each other. Requirements are: $\cdot$ Each pair of members of the same gender meets exactly once as opponents in a mixed doubles match. $\cdot$ Any two members of the opposite gender who are not spouses meet exactly once as partners and also as opponents in a mixed doubles match. Given that $(n,6)=1$, can you arrange a round of mixed doubles matches that meets the above specifications and requirements?

From point $ O $ a car starts on a straight road and travels with constant speed $ v $. A cyclist who is located at a distance $ a $ from point $ O $ and at a distance $ b $ from the road wants to deliver a letter to this car. What is the minimum speed a cyclist should ride to reach his goal?

By placing parentheses in the expression $1:2:3$ we can get two different number values: $(1 : 2) : 3 = \frac16$ and $1 : (2 : 3) = \frac32$. Now brackets are placed in the expression $1:2:3:4:5:6:7:8$. Multiple bracket pairs are allowed, whether or not in nest form. (a) What is the largest numerical value we can get, and what is the smallest? (b) How many different number values can be obtained?

If $p$ is a prime greater than $3$, show that at least one of the numbers \[\frac{3}{p^2} , \frac{4}{p^2} , \cdots, \frac{p-2}{p^2}\] is expressible in the form $\frac{1}{x} + \frac{1}{y}$, where $x$ and $y$ are positive integers.

Kylie randomly selects two vertices of a rectangle. What is the probability that the two chosen vertices are adjacent? $\textbf{(A) } \dfrac13\qquad\textbf{(B) } \dfrac12\qquad\textbf{(C) } \dfrac23\qquad\textbf{(D) } \dfrac56\qquad\textbf{(E) } 1$

Find the largest integer $n$ for which there exist $n$ different integers such that none of them are divisible by either of $7,11$ or $13$, but the sum of any two of them is divisible by at least one of $7,11$ and $13$.

Bob is practicing addition in base $2.$ Each time he adds two numbers in base $2,$ he counts the number of carries. For example, when summing the numbers $1001$ and $1011$ in base $2,$ \[\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\ 0&1&0&0&1\\0&1&0&1&1 \\ \hline 1&0&1&0&0 \end{array}\] there are three carries (shown on the top row). Suppose that Bob starts with the number $0,$ and adds $111~($i.e. $7$ in base $2)$ to it one hundred times to obtain the number $1010111100~($i.e. $700$ in base $2).$ How many carries occur (in total) in these one hundred calculations? \[\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297\]

The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$ $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$

How many rational solutions are there to $5x^2+2y^2=1$?

Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.

A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]

Find all functions $f:\mathbb{N} \to \mathbb{N}$ so that for any distinct positive integers $x,y,z$ the value of $x+y+z$ is a perfect square if and only if $f(x)+f(y)+f(z)$ is a perfect square.

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.) [b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$. [b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point. Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.

A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$ for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.

Let $n$ be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer $l \geq 0$, there are exactly $n$ cards that have the number $l$ written on them. A move consists of picking $100$ cards from the infinite set of cards and discarding them. Find the least possible value of $n$ for which there is an infinitely long series of moves such that for each positive integer $k$, the sum of the numbers written on the $100$ chosen cards during the $k$-th move is equal to $k$.

Find all integers $k\ge 2$ such that for all integers $n\ge 2$, $n$ does not divide the greatest odd divisor of $k^n+1$.

Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$. What is the minimal number of connecting segments among the points, such that for each $n$-plet of points we can find four points, which are all connected to each other? [i]Proposed by Alexander Ivanov and Emil Kolev[/i]

One pupil has $7$ cards of paper. He takes a few of them and tears each in $7$ pieces. Then, he choses a few of the pieces of paper that he has and tears it again in $7$ pieces. He continues the same procedure many times with the pieces he has every time. Will he be able to have sometime $2009$ pieces of paper?