Let $A,B,M,C,D$ be distinct points on a line such that $AB=BM=MC=CD=6.$ Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radius $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively such that $O_1,O_2$ lie on the same side of line $AD.$ Let $P$ be the point such that $PB\perp O_1M$ and $PC\perp O_2M.$ Determine the value of $PO_2^2-PO_1^2.$ [i]Proposed by Ray Li[/i]

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes? [i](3 points)[/i]

Let $m, n, k$ and $l$ be positive integers with $n \ne 1$ such that $n^k + mn^l + 1$ divides $n^{k+l }- 1$. Prove that either $m = 1$ and $l = 2k$, or $l | k$ and $m =\frac{n^{k-l} - 1}{n^l - 1}$.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

Let $n\ge 5$ be an integer. Consider $2n-1$ subsets $A_1, A_2, A_3, \ldots , A_{2n-1}$ of the set $\{ 1, 2, 3,\ldots , n \}$, these subsets have the property that each of them has $2$ elements (that is that is, for $1 \le i \le 2n-1$ it is true that $A_i$ has $2$ elements). Show that it is always possible to select $n$ of these subsets in such a way that the union of these $n$ subsets has at most $\frac{2}{3}n + 1$ elements in total.

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.)

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

How many complete directed graphs with vertex set $V=\{1,2,3,4,5,6\}$ contain no $3$-cycles? A graph is $\textit{directed}$ if all edges have a direction (e.g. from $u$ to $v$ rather than between $u$ and $v$), and $\textit{complete}$ if every pair of vertices has an edge between them. Further, a $\textit{3-cycle}$ in a directed graph is a triple $(u,v,w)$ of vertices such that there are edges from $u$ to $v$, $v$ to $w$, and $w$ to $u$.

The number $1234$ is written on the board. A play consists of subtracting a non-zero digit of that number from that number, and replacing the number by the result. The player who writes the number (not digit) zero wins. Determine if there is a winning strategy for one of the two players who play consecutively. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ the probability that you land on the starting face after $n$ turns. Supposed that $F(8) = \frac{43}{256}$ , and F(10) can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.

Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.

Juanita wrote the numbers from $1$ to $13$ , calculated the sum of all the digits he had written and obtained $$1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+(1+2)+(1+3)=55.$$ His brother Ariel wrote the numbers from $1$ to $100$ and calculated the sum of all the digits written. Find the value of Ariel's sum.

Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent. [i]Proposed by Romania, Radu-Andrew Lecoiu[/i]

A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?

An exam at a university consists of one question randomly selected from the$ n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$. What is the largest lower bound of the probabilities $p_n$?

In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$ [i] Proposed by Iman Maghsoudi[/i]

Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties: $\bullet$ [b]Ascension:[/b]If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable. $\bullet$ [b]Symmetry:[/b] If all of the members change their vote, the final decision will change too. Prove that the council uses a weighted decision-making method ; that is , nonnegative weights $\omega _1 , \omega _2 , \cdots ,\omega _5$ can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest. Remark. The statement isn't true at all if you replace $5$ with arbitrary $n$ . In fact , finding a counter example for $n=6$ , was appeared in the same year's [url=https://artofproblemsolving.com/community/c6h1459567p8417532]Iran MO 2nd round P6[/url]

Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$ let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$ Prove that the following statements are equivalent: $\textbf{i) }$ $|\det A | = 1$ $\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$ [i]Romeo Raicu[/i]

There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ($A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?

Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.

a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively. Prove that: $M,O,N,K$ are on the same circle. b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$