Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

Alice and Bob are racing. Alice runs at a rate of $2\text{ m/s}$. Bob starts $10\text{ m}$ ahead of Alice and runs at a rate of $1.5\text{ m/s}$. How many seconds after the race starts will Alice pass Bob?

Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

On a straight road, points $1, 2, \ldots, n$ are marked. The distance between any two adjacent points is 1. A "placement" refers to the arrangement of $n$ cars, numbered with the same numbers, at the marked points (there can be multiple cars at one point). The "distance" between two placements is defined as the minimum total length of sections that need to be paved so that cars from the first placement can drive on the asphalt, forming the second one (cars can change places on the road). Prove that for any $\alpha<1$, there exists an integer number $n$ for which there are $100^n$ placements, the pairwise distances between which are greater than $\alpha n$. [i]Proposed by Ilya Bogdanov[/i]

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that, $$ \frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}. $$

Let $\Gamma$ be a circle with the center $O$ and let $A$, $A ^ \prime $ be two diametrically opposite points in $\Gamma$. Let $P$ be the midpoint of $OA ^ \prime$ and $\ell$ a line that passes through $P$, different from the line $AA ^ \prime$ and different from the line perpendicular on $AA ^ \prime$. Let $B$ and $C$ be the intersection points of $\ell$ with $\Gamma$, let $H$ be the foot of the altitude from $A$ on $BC$, let $M$ be the midpoint of $BC$, and let $D$ be the intersection of the line $A ^ \prime M$ with $AH$. Show that the angle $\angle ADO = 90 ^ \circ $.

Given a finite set $K_0$ of points (in the plane or space). The sequence of sets $K_1, K_2, ... , K_n, ...$ is constructed according to the rule: [i]we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$.[/i] a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$? b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices. c) How many faces contain the minimal convex polyhedron containing $K_1$? d) What is the volume of the above mentioned polyhedron? e) What is the volume of the minimal convex polyhedron containing $K_n$?

For $2n$ numbers in a row, Bob could perform the following operation: $$S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).$$ Let $T$ be the order of this operation. In other words, $T$ is the smallest positive integer such that $S_i=S_{i+T}$. Prove that $T<2n$.

A regular hexagon is split into $6$ congruent equilateral triangles by drawing in the $3$ main diagonals. Each triangle is colored $1$ of $4$ distinct colors. Rotations and reflections of the figure are considered nondistinct. Find the number of possible distinct colorings.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ-\frac{3}{2}\alpha$. [i]Proposed by Sergey Berlov[/i]

Given the distinct points $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_1+x_2, y_1+y_2)$. Line segments are drawn connecting these points to each other and to the origin $0$. Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure $OPRQ$, depending upon the location of the points $P, Q,$ and $R$, can be: $ \textbf{(A)}\ \text{(1) only}\qquad\textbf{(B)}\ \text{(2) only}\qquad\textbf{(C)}\ \text{(3) only}\qquad\textbf{(D)}\ \text{(1) or (2) only}\qquad\textbf{(E)}\ \text{all three} $

Consider a cube $C$ and two planes $\sigma, \tau$, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.

Find all polynomials $P(x)$ with integer coefficients such that $a^2+b^2-c^2$ divides $P(a)+P(b)-P(c)$, for all integers $a,b,c$.