Determine whether it is possible to distribute $60$ zeros and $61$ ones in the cells of an $11 \times 11$ board, one number in each cell, so that the sum of the numbers in each row is odd, the sum of the numbers in each column is odd, and the sum of the numbers on each of the two diagonals is odd. What if the board is $12 \times 12$ and one wants to distribute $72$ zeros and $72$ ones?
Juan has 11 weights, all of different masses and all of integer masses. The sum of the masses of the 11 weights is 1810. With these weights one can obtain all integer masses from 1 to 1810.
Determine the possible values of the sixth weight, counting from the lightest to the heaviest.
Let $ABC$ be a triangle and $P$ a point on the bisector of angle $\hat{A}$ lying in the interior of triangle $ABC$. It is known that $PC = BC$ and $\hat{A}BP = 30^\circ$. Find the measure of angle $\hat{APC}$.
Rocío must write in a line 100 distinct integers chosen from $1$ to $199$ in such a way that every number, from the second one up to the second-to-last one, is greater than at least one of its two neighbours. Then she computes the sum of the numbers in the even positions, which we denote $P$, and the sum of the numbers in the odd positions, which we denote $I$. Determine the largest possible value of $P - I$ that Rocío can obtain.
Determine whether it is possible to divide a square of side $11$ into the following $5$ pieces: a square of side $1$ and four rectangles whose dimensions are $8$ distinct integers greater than $1$. What if the square to be divided has side $10$?
Germán wrote numbers in the cells of an $11 \times 11$ board in such a way that in each row the sum of the $11$ numbers equals $3$, in each column the sum of the $11$ numbers equals $3$, and in each $3 \times 3$ square the sum of the $9$ numbers equals $1$.
Fede chooses $2008$ positive integers such that the product of these $2008$ numbers ends in $75$, and writes them on the blackboard. Then Iván, without seeing Fede's numbers, chooses a positive integer $k$ less than $2008$. If in Fede's list there are $k$ numbers whose product ends in $75$, Iván wins. Otherwise, Fede wins.
Determine all values of $k$ with which Iván is sure to win, no matter how well Fede plays.
On the blackboard a multiplication of 26 positive integers is written, that is, 26 integers separated by $\times$ signs. Lucía changes two of the $\times$ signs into $+$ signs and computes the result of the new expression. She repeats this procedure for every possible choice of two $\times$ signs in the initial expression. (The expression Lucía computes always has two $+$ signs and 23 $\times$ signs.) Of all the numbers Lucía obtains, exactly 115 are odd. Knowing that in the expression on the blackboard the odd numbers appear only in blocks of 2 and blocks of 3 (that is, there is no isolated odd number and there are never four consecutive odd numbers), compute how many of the 26 numbers on the blackboard can be odd.
Let $P$ be the number obtained by multiplying the factorials of the first 2008 positive integers:
$$
P = (1!)(2!)(3!) \ldots (2007!)(2008!)
$$
Determine whether it is possible to cancel one of these factorials so that the product of the 2007 remaining factorials is a perfect square.
**CLARIFICATION:** The factorial of a positive integer is the product of all integers from 1 up to that number. For example, $5!=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$; $12!=1 \cdot 2 \cdot 3 \cdot ... \cdot 12 = 479001600$. An integer is called a perfect square if it is the square of an integer. For example, 16 and 10000 are perfect squares, because $16 = 4^2$ and $10000 = 100^2$.
Beto chose $101$ positive integers and wrote them in a line. Prove that one can place parentheses, addition signs and multiplication signs between the numbers of Beto's list so that the resulting expression makes sense and, when the indicated operations are carried out, a number divisible by $16!$ is obtained. It is forbidden to change the order of the numbers in the list.
In each cell of a $60 \times 60$ board a number of absolute value less than or equal to $1$ is written. The sum of all the numbers on the board is equal to $600$. Prove that the board contains a $12 \times 12$ square in which the sum of the $144$ numbers in its cells has absolute value less than or equal to $24$.
On a circle with centre $O$, let $A$ and $B$ be points of the circle such that $AB = 120^\circ$ (that is, the arc $AB$ measures $120^\circ$). The point $C$ lies on the minor arc $AB$ and the point $D$ lies on the chord $AB$. It is known that $AD = 2$, $BD = 1$ and $CD = \sqrt{2}$. Compute the area of triangle $ABC$.
Find all perfect powers that end with the digits 2, 0, 0, 8, in that order.
CLARIFICATION: A perfect power is a number of the form $a^k$ where $a$ and $k$ are positive integers and $k \geq 2$. For example, $6^2$; $2^7$; $100^3$.
Consider an $a \times b$ board, with $a$ and $b$ integers greater than or equal to $2$. Initially its cells are coloured white and black like a chessboard. The allowed operation consists of choosing two cells with a common side and recolouring them as follows: a white cell becomes black; a black cell becomes green; a green cell becomes white.
Determine for which values of $a$ and $b$ it is possible, by a sequence of allowed operations, to make all the cells that were initially white end up black and all the cells that were initially black end up white.
**CLARIFICATION:** Initially there are no green cells, but they appear after the first operation.
Let $A$, $B$ and $C$ be three points such that $B$ is the midpoint of segment $AC$, and let $P$ be a point such that $\angle PBC = 60^\circ$. The equilateral triangle $PCQ$ is constructed so that $B$ and $Q$ lie in different half-planes with respect to $PC$, and the equilateral triangle $APR$ so that $B$ and $R$ lie in the same half-plane with respect to $AP$. Let $X$ be the intersection point of the lines $BQ$ and $PC$; let $Y$ be the intersection point of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.
Ana and Beto play on a board with $11$ rows and $9$ columns. First Ana divides the board into $33$ zones. Each zone consists of $3$ adjacent cells aligned vertically or horizontally, as shown in the figure.
Then Beto writes in each cell one of the numbers $0$, $1$, $2$, $3$, $4$, $5$, so that the sum of the numbers in each zone equals $5$. Beto wins if the sum of the numbers written in each of the $9$ columns of the board is a prime number. Otherwise, Ana wins. Prove that Beto has a winning strategy.
Given a sequence $S$ of $1001$ positive real numbers, not necessarily distinct, and given a set $A$ of distinct positive integers, the allowed operation is: for some $k \in A$, select $k$ numbers of $S$, compute the average of the $k$ numbers (arithmetic mean) and replace each of the $k$ selected numbers by that average.
If $A$ is a set such that for every $S$ one can achieve, by means of a sequence of allowed operations, that the numbers are all equal, determine the smallest possible value of the largest element of $A$.
Pablo has a certain number of rectangles whose areas add up to $3$ and whose sides are all less than or equal to $1$. Prove that with these rectangles it is possible to cover a square of side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square.
**Note:** The rectangles may overlap and may stick out of the square.
