A square sheet of $9 \times 9$ is ruled into $1 \times 1$ unit squares. The sheet is cut with the goal of dividing it into the $1 \times 1$ unit squares.
Each cut must be straight and follow a line of the grid.
After each cut, it is allowed to rearrange the pieces conveniently into a stack so that the next cut divides several pieces simultaneously (in each piece the cut must be straight and follow a line of the grid). Folding the paper is forbidden.
What is the least number of cuts needed to achieve the goal?
(For the number found, indicate which cuts are made and explain why it is impossible to achieve the goal with fewer cuts.)
On the table there are $21$ cards, one with each of the integers from $1$ to $21$ inclusive.
Xavier selects $4$ cards and shows them to Ana. Then Ana takes one card away from Xavier (whichever she wants). If the sum of the numbers on the $3$ cards Xavier keeps is a multiple of $3$, Ana wins. Otherwise, Xavier wins.
Determine in how many ways Xavier can choose the $4$ cards so as to be sure of winning, no matter how well Ana plays.
(Two choices of the same $4$ cards in a different order are considered the same choice.)
Let $ABC$ be a triangle with $\hat{B} = 40°$. It is known that there is a point $P$ on the bisector of angle $\hat{B}$ such that $BP = BC$ and $\hat{B}AP = 20°$. Determine the measures of the angles $\hat{A}$ and $\hat{C}$.
In a contest each participant drew a $99 \times 100$ grid board and wrote a $1$ or a $-1$ in each cell, as they chose. Next, each participant wrote beside each row the result of multiplying the $100$ numbers of that row, and below each column the result of multiplying the $99$ numbers of that column. Finally, they added the $99$ row results and the $100$ column results and obtained their final number.
If in this contest all participants obtained different final numbers, determine the maximum possible number of participants and, for the maximum number found, give the final numbers of all the participants.
There is a bag with 99 balls of different colours (each ball has a single colour and the number of colours is unknown). If 21 balls are drawn at random from the bag, there are always four or more of the same colour. Decide whether it is necessarily true that the bag contains 18 or more balls of the same colour. And 17 or more balls of the same colour?
Axel and Franco play the following game. First Axel thinks of a natural number $N$. From then on, in each move, Franco chooses 4 distinct numbers $a, b, c, d$ from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ and tells them to Axel. Then Axel announces one of the sums $N+a, N+b, N+c, N+d$, of his choice. (For example, if Axel thought of 2007 and in one move Franco chooses 1, 3, 4, 6, Axel must announce one of the numbers 2008, 2010, 2011, 2013, of his choice.)
Franco's goal is to know the number $N$ with certainty. Determine the minimum number of moves that always allows Franco to achieve his goal.
In each cell of a $1 \times 2007$ board of consecutive cells one must write an integer from $1$ to $2007$, without repeating numbers. Then the following $2007$ numbers are considered: the number in the first cell from the left; the sum of the numbers in the first two cells (from the left); the sum of the numbers in the first three cells; ...; the sum of the numbers in the first $2006$ cells and the sum of the numbers in all the cells. For each of these $2007$ numbers that leaves remainder $5$ when divided by $6$, one earns $1$ peso. What is the maximum amount of money that can be earned?
In a country there are $100$ electoral districts, all with the same number of voters. Each district elects one deputy to the national parliament among $3$ candidates representing the $3$ political parties $A$, $B$ and $C$. At the national level, parties $A$, $B$ and $C$ have the support of exactly $60\%$, $30\%$ and $10\%$ of the voters, but the distribution of supporters by district can be arbitrary.
The elections have two rounds. If in a district in the first round one of the three candidates obtains more than half of the votes, he wins the election and there is no second round. Otherwise, the two candidates with the most votes compete in the second round, where again the one who obtains more than half of the votes wins.
Each voter votes for the candidate of their favourite party. If that candidate is not in the second round, the voters of $A$, $B$ and $C$ vote for the candidates of $C$, $C$ and $B$, respectively.
What is the minimum number of districts that party $A$ wins? What is the maximum number of districts that $C$ could win?
**CLARIFICATION:** Assume that in no district does a candidate obtain exactly $50\%$ of the votes, and also that there are no candidates with the same number of votes.
Let $ABC$ be a triangle with $\hat{A} = 45°$ such that the bisector of $\hat{A}$, the median from $B$ and the altitude from $C$ are concurrent. Compute the measure of the angle $\hat{B}$.
In the plane, $n$ distinct lines are drawn. Each of them intersects exactly 2007 of the other lines. Find all values of $n$ for which this is possible.
We have a rectangular $9 \times 2007$ board divided into $1 \times 1$ unit squares. Initially all the unit squares are white. In each step, 4 white unit squares lying at the intersection of two rows and two columns of the board are coloured black. When there are no longer 4 white unit squares at the intersection of two rows and two columns of the board, the process stops. What is the minimum number of unit squares that will remain uncoloured? And if the board were $99 \times 2007$?
Let $ABCD$ be a parallelogram with sides $AB$, $BC$, $CD$, $AD$, such that $AB \gt AD$ and $\frac{AC}{BD} = 3$. Let $r$ be the reflection of the line $AD$ with respect to $AC$ and let $s$ be the reflection of the line $BC$ with respect to $BD$. If $r$ and $s$ meet at $P$, compute the value of $\frac{PA}{PB}$.
We are given 10 real numbers $a_1, a_2, ..., a_{10}$, and we form the 45 sums of two of these numbers $a_i + a_j$, $1 \le i \lt j \le 10$. It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the 45 sums there are $k$ that are not integers and $45 - k$ that are integers.
We say that a positive integer is *lucky* if the sum of its digits is divisible by $31$. What is the largest possible difference between two consecutive lucky numbers?
Julián chooses $2007$ points in the plane, no $3$ of them collinear, and draws in red all the segments joining two of these points. Next, Roberto draws several lines. His goal is that every red segment is cut at an interior point by (at least) one of the lines. Determine the smallest $l$ such that, no matter how Julián chooses the $2007$ points, with $l$ suitably chosen lines Roberto surely achieves his goal.
Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1, 2, ..., 250\}$ but does not see the number received by the other. The goal is for each friend to discover the other's number. The procedure they must follow is to announce, in turns, positive integers, not necessarily distinct: first $A$ says a number, then $B$ says a number, then $A$ again, etc., in such a way that the sum of all the announced numbers is 20. Prove that there is a strategy by which, through a prior agreement, $A$ and $B$ can achieve the goal, no matter which number each one receives at the beginning of the puzzle.
What is the largest number of cells that can be coloured on a $7 \times 7$ board in such a way that every $2 \times 2$ sub-board has at most $2$ coloured cells?
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ with centre on the segment $AB$ is tangent to the equal sides $AC$ and $BC$. Consider a line tangent to $\Gamma$ that cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The lines perpendicular to $AC$ and $BC$ drawn through $D$ and $E$, respectively, meet at $P$. Let $Q$ be the foot of the perpendicular to the line $AB$ through $P$. Prove that
$$
\frac{PQ}{CP} = \frac{1}{2} \frac{AB}{AC}
$$