A trader owns horses of $3$ races, and exacly $b_j$ of each race (for $j=1,2,3$). He want to leave these horses heritage to his $3$ sons. He knowns that the boy $i$ for horse $j$ (for $i,j=1,2,3$) would pay $a_{ij}$ golds, such that for distinct $i,j$ holds holds $a_{ii}> a_{ij}$ and $a_{jj} >a_{ij}$. Prove that there exists a natural number $n$ such that whenever it holds $\min\{b_1,b_2,b_3\}>n$, trader can give the horses to their sons such that after getting the horses each son values his horses more than the other brother is getting, individually.

Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions: [b]1.[/b]Each plane contains at least $ 4$ of them [b]2.[/b]No four points are collinear.

There are $ 1993$ towns in a country, and at least $ 93$ roads going out of each town. It's known that every town can be reached from any other town. Prove that this can always be done with no more than $ 62$ transfers.

There are $n\ge 3$ students in a classroom. Every day, the teacher separates the students into at least two non-empty groups, and each pair of students from the same group will shake hands once. Suppose after $k$ days, each pair of students has shaken hands exactly once, and $k$ is as minimal as possible. Prove that $$\sqrt{n} \le k-1 \le 2\sqrt{n}$$ [i]Proposed by Wong Jer Ren[/i]

For a positive integer $n$, consider all nonincreasing functions $f : \{1,\hdots,n\}\to\{1,\hdots,n\}$. Some of them have a fixed point (i.e. a $c$ such that $f(c) = c$), some do not. Determine the difference between the sizes of the two sets of functions. [i]Remark.[/i] A function $f$ is [i]nonincreasing[/i] if $f(x) \geq f(y)$ holds for all $x \leq y$

p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$ p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$ Find the last seven digits of $A$, in order from millions to units. p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons. p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open? p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.

We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

Given natural number n. Suppose that $N$ is the maximum number of elephants that can be placed on a chessboard measuring $2 \times n$ so that no two elephants are mutually under attack. Determine the number of ways to put $N$ elephants on a chessboard sized $2 \times n$ so that no two elephants attack each other. Alternative Formulation: Determine the number of ways to put $2015$ elephants on a chessboard measuring $2 \times 2015$ so there are no two elephants attacking each othe PS. Elephant = Bishop

On a blackboard are written all the integers from $1$ to $2008$ inclusive. Two numbers are deleted and their difference is written. For example, if you erase $5$ and $241$, you write $236$. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be $2008$. What about $2007$? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.

$50$ gangsters simultaneously shoot at each other, and each shoots at the nearest gangster (if there are several of them, then at one of them) and kills him. Find the smallest possible number of people killed.

Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have?

On a planet there are $M$ countries and $N$ cities. There are two-way roads between some of the cities. It is given that: (1) In each county there are at least three cities; (2) For each country and each city in the country is connected by roads with at least half of the other cities in the countries; (3) Each city is connceted with exactly one other city ,that is not in its country; (4) There are at most two roads between cities from cities in two different countries; (5) If two countries contain less than $2M$ cities in total then there is a road between them. Prove that there is cycle of lenght at least $M+\frac{N}{2}$.

Consider $n\geq 3$ points in the plane, no three of which are collinear. For every convex polygon with vertices among the $n$ points, place $k\cdot 2^k$ coins in every one of its vertices, where $k$ is the number of points strictly in the interior of the polygon. Show that in total, no matter the configuration of the $n$ points, there are at most $n(n+1)\cdot 2^{n-3}$ placed coins. [i]Cristi Săvescu[/i]

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

Consider $n$ children in a playground, where $n\ge 2$. Every child has a coloured hat, and every pair of children is joined by a coloured ribbon. For every child, the colour of each ribbon held is different, and also different from the colour of that child’s hat. What is the minimum number of colours that needs to be used?

The squares of a $10 \times 10$ chessboard are labelled with $1,2,...,100 $ in the usual way: the $i$-th row contains the numbers $10i -9,10i - 8,...,10i$ in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.

Let $A_{1}A_{2}\ldots A_{2n}$ be a convex polygon and let $P$ be a point in its interior such that it doesn't lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines $PA_{1}$, $\ldots$, $PA_{2n}$ intersects it in its interior.

In a soccer championship $2004$ teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows: a) any two teams can play against one each other at most one game; b) from any $4$ teams, $3$ of them play against one each other. How many days are necessary to make such a championship, knowing that each team can play at most one game per day?

Eighteen soccer teams have played $8$ tours of a one-round tournament. Prove that there is a triple of teams, having not met each other yet.

a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$. The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the $x$ place in the $y$ row?" What is the minimal number of the questions asked by the second player before he founds out all the numbers? b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]