Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? [i]Yudi Satria, Jakarta[/i]

The integers $1, 2, \dots , 9$ are written on individual slips of paper and all are put into a bag. Ade chooses a slip at random, notes the integer on it, and replaces it in the bag. Bala then picks a slip at random and notes the integer written on it. Chioma then adds up Ade's and Bala's numbers. What is the probability that the unit's digit of this sum is less that $5$?

For positive integers $N$ and $m$, where $m \le N$, define $$a_{m,N} =\frac{1}{{N+1 \choose m}} \sum_{i=m-1}^{N-1} \frac{ {i \choose m-1}}{N - i}$$ Compute the smallest positive integer $N$ such that $$\sum^N_{m=1}a_{m,N} >\frac{2020N}{N +1}$$

Evaluate $\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$

We have $n{}$ coins, one of which is fake, which differs in weight from the real ones and a two-pan scale which works correctly if the weights on the pans are different, but can show any outcome if the weights on the pans are equal. For what $n{}$ can we determine which coin is fake and whether it is lighter or heavier than the real coins, in at most $k{}$ weightings? [i]Proposed by A. Zaslavsky[/i]

An empire has a finite number of cities. Every two cities are connected by a natural number of roads . Every street connects exactly two cities. Show that you have the kingdom can be divided into a maximum of three republics so that within each republic there are just many streets run away. (We say a road runs within a republic if the two cities that it connects, both belonging to this republic. The republics must meet each other be disjoint, and cover all cities of the empire in total.) [hide=original wording in German]Ein Reich hat endlich viele St¨adte. Je zwei St¨adte sind durch eine natu¨rliche Anzahl von Straßen verbunden. Jede Straße verbindet genau zwei St¨adte. Man zeige, dass man das Reich so in h¨ochstens drei Republiken zerteilen kann, dass innerhalb jeder Republik gerade viele Straßen verlaufen. (Wir sagen, eine Straße verl¨auft innerhalb einer Republik, wenn die zwei St¨adte, die sie verbindet, beide dieser Republik angeh¨oren. Die Republiken mu¨ssen zueinander disjunkt sein, und insgesamt alle St¨adte des Reiches abdecken.)[/hide]

Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img] Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. [img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]

Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$. (Walther Janous)

Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$ satisfying $x < y$ and $x + y = z$.

In a handbook of plants each plant is characterized by $100$ attributes (each attribute may either be present in a plant or not). Two plants are called [i]dissimilar [/i] if they differ by no less than $51$ attributes. (a) Prove that the handbook cannot describe more than $50$ pair-wise dissimilar plants. (b) Can it describe $50$ pairwise dissimilar plants? (Dima Tereshin)

Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way: Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$. For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$. Set $A''$ is constructed from $A'$ in the same manner. Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.

Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$. [b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy? [b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?

Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.

A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal. (A. Tolpygo, Kiev)

A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied: $(1)$ every member of the set $P$ knows every other member of the set $P$; $(2)$ no member of the set $P$ knows any other member of the set $P$.

Alma places spies on some of the squares on a $2020\times 2020$ game board. Now Bertha secretly chooses a quadradic area consisting of $1020 \times 1020$ squares and tells Alma which spies are standing on a square in the secret quadradic area. At least how many spies must Alma have placed in order for her to determine with certainty which area Bertha has chosen?

We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following: $1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell, $2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell. What is the minimum number of colours $k$ required to make such a colouring possible?

Joe makes two cubes of sidelengths $9$ and $10$ from $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. These cubes are dipped into white paint. Then two cubes of sidelengths $1$ and $12$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and these cubes are dipped into paint remover. Joe continues to alternately dip cubes of sides $9$ and $10$ into paint and cubes of sides $1$ and $12$ into paint remover ad nauseam. What is the limit of the expected number of painted unit cube faces immediately after dipping in paint remover?

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

Arthur and Renate play a game on a $7 \times 7$ board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?

An segment $S$ of length $50$ is covered by several segments of length $1$ , all of them contained in $S$. If any of these unit segments were removed, $S$ would no longer be completely covered. Find the maximum number of unit segments with this property. Clarification: Assume that the segments include their endpoints.

$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings? (A. Andjans)