The [i]weird [/i] mean of two numbers $ a$ and $ b$ is defined as $ \sqrt {\frac {2a^2 + 3b^2}{5}}$. $ 2009$ positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these $ 2009$ numbers must be equal.

In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$

A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$. Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.

The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$?

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Given the polynomial $p(x) = x^2 + x - 70$, do there exist integers $0<m<n$, so that $p(m)$ is divisible by $n$ and $p(m+1)$ is divisible by $n+1$? [i]Proposed by Nairy Sedrakyan[/i]

Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

We are given a convex quadrilateral $ABCD$ in the plane. ([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$? ([i]ii[/i]) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in ([i]i[/i]).

Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.

A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards? [asy] size(300); draw(circle((0,0),7)); path A=(1,2)--(1,-2)--(-1,-2)--(-1,2)--cycle; filldraw(shift(7*left)*A,lightgray); draw((-7,0)--(-7,5),EndArrow(size=21)); label(scale(1.5)*"A",(-8,2),2*N); label(scale(1.5)*"B",(-8,0),2*W); label(scale(1.5)*"C",(-7,-2),3*S); label(scale(1.5)*"D",(-6,0),2*E); [/asy] (A) A (B) B (C) C (D) D (E) Remains vertical

Set $u_0 = \frac{1}{4},$ and for $k \geq 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$ This sequence tends to a limit, call it $L.$ What is the least value of $k$ such that $$|u_k - L| \leq \frac{1}{2^{1000}}?$$ $\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 253 \qquad\textbf{(D)}\ 329 \qquad\textbf{(E)}\ 401$

Let $ p$ be an odd prime number and $ n$ a natural number. Then $ n$ is called $ p\minus{}partitionable$ if $ T\equal{}\{1,2,...,n \}$ can be partitioned into (disjoint) subsets $ T_1,T_2,...,T_p$ with equal sums of elements. For example, $ 6$ is $ 3$-partitionable since we can take $ T_1\equal{}\{ 1,6 \}$, $ T_2\equal{}\{ 2,5 \}$ and $ T_3\equal{}\{ 3,4 \}$. $ (a)$ Suppose that $ n$ is $ p$-partitionable. Prove that $ p$ divides $ n$ or $ n\plus{}1$. $ (b)$ Suppose that $ n$ is divisible by $ 2p$. Prove that $ n$ is $ p$-partitionable.

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.

All the rationals are coloured with $n$ colours so that, if rationals $a$ and $b$ are colored with different colours then $\frac{a+b}2$ is coloured with a colour different from both $a$ and $b$. Prove that every rational is coloured with the same colour.

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

Let $k$ be a positive integer. A positive integer $n$ is called a $k$-good number if it satisfies the following two conditions: 1. $n$ has exactly $2k$ digits in decimal representation (it cannot have leading zeros). 2. If the first $k$ digits and the last $k$ digits of $n$ are considered as two separate $k$-digit numbers (which may have leading zeros), the square of their sum is equal to $n$. For example, $2025$ is a $2$-good number because \[(20 + 25)^2 = 2025.\] Find all $3$-good numbers.

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

On a line a set of segments is given of total length less than $n$. Prove that every set of $n$ points of the line can be translated in some direction along the line for a distance smaller than $\frac{n}{2}$ so that none of the points remain on the segments.

Solve in nonnegative integers the following equation : $$21^x+4^y=z^2$$

Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$? [asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("$E$",(1,0),SE); label("$F$",(25,10),NW); label("$A$", (0,0), SW); label("$B$", (0,10), NW); label("$C$", (26,10), NE); label("$D$", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy] [i]Proposed by Nathan Xiong[/i]