Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a [i]Perfect Set [/i]if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.

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There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)

Let $ABC$ be a triangle with $AB > AC$. Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$. Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$. Prove that $BQ = 2AP.$

Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.

Angeline flips three fair coins, and if there are any tails, she then flips all coins that landed tails each one more time. The probability that all coins are now heads can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the equation $$f(x^2+yf(x))=(1-x)f(y-x)$$ holds for all $x,y\in\mathbb{R}$.

Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$.

Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy $11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$.

Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges' trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13,13,13,$ and $3$, and for each $i\geq 2$, let $P_i=\text{clip}(P_{i-1}).$ This iterative clipping process approaches a limiting shape $P_\infty=\lim_{i\to\infty}P_i$. If the difference of the areas of $P_{10}$ and $P_\infty$ is written as a fraction $\tfrac xy$ in lowest terms, calculate the number of positive integer factors of $x\cdot y$.

On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

Find all triples of real numbers $(a, b, c)$ satisfying $$a+b+c=14, \quad a^2+b^2+c^2=84,\quad a^3+b^3+c^3=584.$$

Let $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. The greatest common divisor of $b$ and $81$ is $\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }9\qquad\text{(D) }27\qquad\text{(E) }81$

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

Three integers $A, B, C$ are written on a whiteboard. Every move, Mr. Doba can either subtract $1$ from all numbers on the board, or choose two numbers on the board and subtract $1$ from both of them whilst leaving the third untouched. For how many ordered triples $(A, B, C)$ with $1 \le A < B < C\le 20$ is it possible for Mr. Doba to turn all three of the numbers on the board to $0$?

Given $25$ different positive numbers. Prove that you can choose two of them such, that none of the other numbers equals neither to the sum nor to the difference between the chosen numbers.

For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if (i) $ A_1 \cap A_2 \cap \cdots \cap A_n \equal{} A$, (ii) $ A_i \cap A_j \neq \emptyset$. Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A \equal{} \{1, 2, \ldots, m\}$, there exists a set $ A_i$ ($ 1 \leq i \leq 14$) such that there are two elements $ a, b$ of $ A_i$ such that $ b < a \leq \frac {4}{3}b$.

A quadrilateral $ABCD$ is inscribed into a circle with center $O.$ Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drawn to that circle at these points meet the line $AB$ in points $E$ and $F.$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$ respectively. Prove that $XV=YU.$

Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively. A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$ Prove that: $ X,\,Y,\,Z$ are collinear.

$ABCD$ is a cyclic quadrilateral inscribed in circle $O$. Let $O_1$ be the $A$-excenter of $ABC$ and $O_2$ the $A$-excenter of $ABD$. Show that $A, B, O_1, O_2$ is concyclic, and $O$ passes through the center of $(ABO_1O_2)$. Recall that for concyclic $X, Y, Z, W$, the notation $(XYZW)$ denotes the circumcircle of $XYZW$.

Each cell of the 4x4 board has a grasshopper. When a grasshopper jumps, it moves to one of the adjacent cells (down, up, right, or left). The grasshopper cannot move diagonally or go off the board. At most how many cells can remain empty after each grasshopper jumps once?