Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions. (i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$ (iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$, respectively such that $\angle BAM = \angle MNC$. Suppose that the lines $MN$ and $AB$ intersects at $P$. Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$

There are relatively prime positive integers $m$ and $n$ so that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$ . Find $m + n$.

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

Albrecht is travelling in his car on the motorway at a constant speed. The journey is very long so Marvin who is sitting next to Albrecht gets bored and decides to calculate the speed of the car. He was a bit careless but he noted that at noon they passed milestone $XY$ (where $X$ and $Y$ are digits), at $12:42$ milestone $YX$ and at $1$pm they arrived at milestone $X0Y$. What did Marvin deduce, what is the speed of the car?

Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that \[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2} \]

Let $f$ be a polynomial of degree $98$, such that $f (k) =\frac{1}{k}$ for $k=1,2,3,\ldots,99$. Determine $f(100)$.

Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$ $$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\ 1\; \text{otherwise}. \end{cases}$$

If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.

Given a triangle $ABC$ with symmedians $BE,CF(E,F$ are on the sides $CA,AB,$ respectively$)$ intersecting at [i]Lemoine[/i] point $L.$ Prove that: $AB=AC$ in each case: a) $LB=LC.$ b) $BE=CF.$

Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Let \( H \) be the orthocenter, and \( O \) be the circumcenter of \( \triangle ABC \). The line \( AH \) intersects the circumcircle of \( \triangle ABC \) at point \( N \) for the second time. The circumcircle of \( \triangle BOC \), with center at point \( Q \), intersects the line \( OH \) at point \( X \) for the second time. Prove that the points \( O, Q, N, X \) lie on the same circle. [i]Proposed by Matthew Kurskyi[/i]

In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.

$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square. [i]* * *[/i]

In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

The graphs of the following equations divide the $xy$ plane into some number of regions. $4 + (x + 2)y =x^2$ $(x + 2)^2 + y^2 =16$ Find the area of the second smallest region.

In $ n$-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue. [i]A. Csaszar[/i]

let $ ABC $ an equilateral triangle with circum circle $ w $ let $ P $ a point on arc $ BC $ ( point $ A $ is on the other side ) pass a tangent line $ d $ through point $ P $ such that $ P \cap AB = F $ and $ AC \cap d = L $ let $ O $ the center of the circle $ w $ prove that $ \angle LOF > 90^{0} $

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]