Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$. Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$, and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$. Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.

Three mutually externally tangent circles are internally tangent to a circle with radius $1$. If two of the inner circles have radius $\frac{1}{3}$, the largest possible radius of the third inner circle can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ where $c$ is squarefree and $\gcd(a,b,d)=1$. Find $a+b+c+d$.

The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle. (a) Draw a hexagon such that it has the largest possible number of points of self-intersection. (b) Prove that this number is indeed maximum. (NB Vassiliev)

In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

Let $S$ be the sum of all possible values of $a \cdot c$ such that $$a^3+3ab^2-72ab+432a=4c^3$$ if $a$, $b$, and $c$ are positive integers, $a+b > 11$, $a > b-13$, and $c \le 1000$. Find the sum of all distinct prime factors of $S$. [i]Proposed by Kevin Zhao[/i]

Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children's Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children's Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes?

Let $\omega$ be a unit circle with center $O$ and diameter $AB$. A point $C$ is chosen on $\omega$. Let $M$, $N$ be the midpoints of arc $AC$, $BC$, respectively, and let $AN,BM$ intersect at $I$. Suppose that $AM,BC,OI$ concur at a point. Find the area of $\triangle ABC$. [i]Proposed by Kevin You[/i]

Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively. Find the value of $PB \cdot PC$.

[u]Polyhedron Hopping[/u] [b]p1.[/b] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? [b]p2.[/b] In terms of $k$, for $k > 0$ how likely is he to be back where he started after $2k$ minutes? [b]p3.[/b] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After ve minutes, how likely is Sherry to be one edge away from where she started? [b]p4.[/b] In terms of $k$, for $k > 0$, how likely is it that after $k$ minutes Sherry is at the vertex opposite the vertex where she started?

Leaving his house at noon, Jim walks at a constant rate of $4$ miles per hour along a $4$ mile square route returning to his house at $1$ PM. At a randomly chosen time between noon and $1$ PM, Sally chooses a random location along Jim's route and begins running at a constant rate of $7$ miles per hour along Jim's route in the same direction that Jim is walking until she completes one $4$ mile circuit of the square route. The probability that Sally runs past Jim while he is walking is given by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $k \in \mathbb{Z}^+$ and set $n=2^k+1.$ Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1},\ldots,a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1},\ldots,g_{n-1},$ such that $n$ divides $g^{a_i}_i - a_{i+1}$ for every $i \in \{1,2,\ldots,n-1\},$ where we set $a_n = a_1.$ [i]Proposed by Vasily Astakhov, Russia[/i]

Nikees has a set $S$ of $n$ points on a plane and decides to colour them. All $\dbinom{n}{2}$ line segments are drawn and they have distinct lengths. Find the maximum number of colours that are used at least once, given that: (a) For each point $P$, the two endpoints of the longest line segment connecting $P$ must be of the same colour. (b) For each point $P$, the two endpoints of the shortest line segment connecting $P$ must be of the same colour.

Find all polynomials $P$ with integer coefficients such that for all positive integers $x,y$, $$\frac{P(x)-P(y)}{x^2+y^2}$$ evaluates to an integer (in particular, it can be zero).

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

Let $k\geq 4$ be an integer number. $P(x)\in\mathbb{Z}[x]$ such that $0\leq P(c)\leq k$ for all $c=0,1,...,k+1$. Prove that $P(0)=P(1)=...=P(k+1)$.

Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.

The sides $AB$ and $AC$ are tangent at points $P$ and $Q$, respectively, to the incircle of a triangle $ABC. R$ and $S$ are the midpoints of the sides $AC$ and $BC$, respectively, and $T$ is the intersection point of the lines $PQ$ and $RS$. Prove that $T$ lies on the bisector of the angle $B$ of the triangle. (M Evdokimov)

You are responsible for arranging a banquet for an agency. In the agency, some pairs of agents are enemies. A group of agents are called [i]avengers[/i], if and only if the number of agents in the group is odd and at least $3$, and it is possible to arrange all of them around a round table so that every two neighbors are enemies. You figure out a way to assign all agents to $11$ tables so that any two agents on the same tables are not enemies, and that’s the minimum number of tables you can get. Prove that there are at least $2^{10}-11$ avengers in the agency. This problem is adapted from 2015 IMO Shortlist C7.

For an acute triangle $ ABC $ let $ H $ be the point of intersection of the altitudes $ AA_1 $, $ BB_1 $, $ CC_1 $. Let $ M $ and $ N $ be the midpoints of the $ BC $ and $ AH $ segments, respectively. Show that $ MN $ is the perpendicular bisector of segment $ B_1C_1 $.

Given is a convex $2024$-gon $A_1A_2\ldots A_{2024}$ and $1000$ points inside it, so that no three points are collinear. Some pairs of the points are connected with segments so that the interior of the polygon is divided into triangles. Every point is assigned one number among $\{1, -1, 2, - 2\}$, so that the sum of the numbers written in $A_i$ and $A_{i+1012}$ is zero for all $i=1,2, \ldots, 1012$. Prove that there is a triangle, such that the sum of the numbers in some two of its vertices is zero. [hide=Remark on source of 11.3] It appears as Estonia TST 2004/5, so it will not be posted.

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\tfrac{5}{2}$ and that \[f(x)f(y)=f(x+y)+f(x-y)\] for all integers $x$ and $y$.