There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

[b]8.[/b] Let the function $f(x)$ be periodic with the period $1$, non-negative, concave in the interval $(0,1)$ and continuous at the point $0$. Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$. [b](R. 6)[/b]

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$). Show that there is an interval $I$ such that $f(I)=[a,b]$.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

In rectangle $ ABCD$, we have $ AB\equal{}8$, $ BC\equal{}9$, $ H$ is on $ \overline{BC}$ with $ BH\equal{}6$, $ E$ is on $ \overline{AD}$ with $ DE\equal{}4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$. [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep); draw(A--G); label("$F$",F,W); label("$G$",G,W); label("$C$",C,WSW); label("$H$",H,NNE); label("$6$",(6,8),N); label("$B$",B,NE); label("$A$",A,SW); label("$E$",Ep,S); label("$4$",(2,0),S); label("$D$",D,S);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?

1400 real numbers are given. Prove that one can choose three of them like $x,y,z$ such that : $$\left|\frac{(x-y)(y-z)(z-x)}{x^4+y^4+z^4+1}\right| < 0.009$$

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Points $M,N,P,Q$ are taken on the sides $AB,BC,CD,DA$ respectively of a square $ABCD$ such that $AM=BN=CP=DQ=\frac1nAB$. Find the ratio of the area of the square determined by the lines $MN,NP,PQ,QM$ to the ratio of $ABCD$.

Initially, each unit square of an $n \times n$ grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously: [list] [*] For a red square, if there is a yellow square that has a common edge with it, then color it yellow. [*] For a yellow square, if there is a blue square that has a common edge with it, then color it blue. [*] For a blue square, if there is a red square that has a common edge with it, then color it red. [/list] It is known that after several rounds, all unit squares of this $n \times n$ grid have the same color. Prove that the grid has became monochromatic no later than the end of the $(2n-2)$-th round.

Is there a set $S\subset\mathbb{R}^3$ of $2006$ points such that not all its points are coplanar, no three of the points are collinear, and for any $A,B\in S$ we can find points $C,D\in S$ for which $AB||CD$?

Let $ n$, $ k$, $ p$ be positive integers with $ 2 \le k \le \frac {n}{p \plus{} 1}$. Let $ n$ distinct points on a circle be given. These points are colored blue and red so that exactly $ k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $ p$ red points. How many such colorings are there?

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$ Prove that (A) $\lambda _n$ is an integer (B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .

Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2+(\log y)^2=\log(x^2)+\log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$.

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

In an acute triangle $ABC$ with $AB<AC$, the foot of altitudes from $A,B,C$ to the sides $BC,CA,AB$ are $D,E,F$, respectively. $H$ is the orthocenter. $M$ is the midpoint of segment $BC$. Lines $MH$ and $EF$ intersect at $K$. Let the tangents drawn to circumcircle $(ABC)$ from $B$ and $C$ intersect at $T$. Prove that $T;D;K$ are colinear

Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.

In the triangle $ ABC $, the lines $ CP $, $ AP $, $ BP $ are drawn through the internal point $ P $ and intersect the sides $ AB $, $ BC $, $ CA $ at points $ K $, $ L $, $ M$, respectively. Prove that if circles can be inscribed in the quadrilaterals $ AKPM $ and $ KBLP $, then a circle can also be inscribed in the quadrilateral $ LCMP $.

A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap, each vertex of one polygon coincides with a vertex of another polygon, and no point of the plane is left uncovered. A unit polygon is a polygon with all sides of length one. It is quite easy to tile the plane with infinitely many unit squares. Likewise, it is easy to tile the plane with infinitely many unit equilateral triangles. (a) Prove that there is a tiling of the plane with infinitely many unit squares and infinitely many unit equilateral triangles in the same tiling. (b) Prove that it is impossible to find a tiling of the plane with infinitely many unit squares and finitely many (and at least one) unit equilateral triangles in the same tiling.

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?

In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?