Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_{1}$ and $\ell_{2}$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_{1}$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_{2}$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_{1}$ and $m_{2}$ respectively. Prove that no matter how the square was placed, $m_{1}+m_{2}$ remains constant.
Let $a$, $b$, $c$ be the sides of a triangle, with $a + b + c = 1$, and let $n \geq 2$ be an integer. Show that
$$
\sqrt[n]{a^{n} + b^{n}} + \sqrt[n]{b^{n} + c^{n}} + \sqrt[n]{c^{n} + a^{n}} \lt 1 + \frac{\sqrt[n]{2}}{2}
$$
Let $k \geq 14$ be an integer, and let $p_{k}$ be the largest prime number which is strictly less than $k$. You may assume that $p_{k} \geq 3k/4$. Let $n$ be a composite integer. Prove:
a. if $n = 2p_{k}$, then $n$ does not divide $(n-k)!$;
Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.
Let $a, b, c, d, e, f$ be real numbers such that the polynomial
$$
p(x) = x^{8} - 4x^{7} + 7x^{6} + a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f
$$
factorises into eight linear factors $x - x_{i}$, with $x_{i} \gt 0$ for $i = 1, 2, \ldots, 8$. Determine all possible values of $f$.
On the computer screen there are initially two $1$ written. The *insert* program causes the sum of those numbers to be inserted between each pair of numbers by pressing the *Enter* key.
In the first step, a number is inserted and we obtain $1$-$2$-$1$; in the second step two numbers are inserted and we have $1$-$3$-$2$-$3$-$1$; In the third, four numbers are inserted and you have $1$-$4$-$3$-$5$-$2$-$5$-$3$-$4$-$1$; etc Find the sum of all the numbers that appear on the screen at the end of step number $25$.
Let $ABC$ be a triangle such that $A\hat{C}C = 2B\hat{A}A$; Furthermore, if $D$ denotes the point on the side $BC$ such that $AD$ is a bisector of the angle $C\hat{A}B$, we have $CD=AB$. Calculate the measures of the angles of the triangle $ABC$.
On a table there are $n \geq 2$ plastic bags, all of different colors. Each of them is in contact with the table or is inside one of the other bags. The allowed operation is to choose one of the bags that is in contact with the table and carry out the following exchange: all the bags inside are put in contact with the table, and all the other bags that are in contact with the table are placed inside the chosen bag (without modifying the content of any of the bags that change places). Determine how many different configurations can be obtained by repeatedly using this operation.
Clarification: Two configurations are considered the same if each bag contains the same bags in both configurations.
In a military school there are $200$ students. The principal decided that every day a group of $9$ students should patrol the school and prepared the list of patrols for the entire year, respecting the following rule: if a student is on patrol on a certain day, then he cannot be on patrol on the following $10$ days.
The deputy director believes that each patrol should have $10$ students.
Decide if the vice principal can add a student to each patrol while respecting the principal's list and the $10$ drop day rule for each student.
In a sum there are $1669$ fractions and the denominators rearrange the integers from $1$ to $1669$. If the denominator is a multiple of $5$, then the numerator is $4$ and the fraction is signed $-$; If the denominator is not a multiple of $5$, then the numerator is $1$ and the fraction is signed $+$:
$$
1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{4}{5} + \dots + \frac{1}{1661} + \frac{1}{1662} + \frac{1}{1663} + \frac{1}{1664} - \frac{4}{1665} + \frac{1}{1666} + \frac{1}{1667} + \frac{1}{1668} + \frac{1}{1669}
$$
If the sum is written as an irreducible (simplified) fraction, show that the numerator of this fraction is divisible by $2003$.
In a circle $\Gamma$, a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $\widehat{PQ}$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ be three circles tangent to the chord $PQ$ that are in the same half-plane as the center of $\Gamma$ with respect to the straight line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are interior tangents to $\Gamma$ and exterior tangents to $\Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are in different half-planes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.
Side a right angle $X \hat{A}Y$ of vertex $A$ and a semicircle $\Gamma$ interior to this angle with center on the side $AX$ and tangent to the side $AY$ in $A$, construct a tangent to $\Gamma$ such that the triangle that is cut from the angle $X \hat{A}Y$ is of minimum area.
Initially on the board, all the integers from $1$ to $2002$ inclusive, without repetitions, are written in one line and in some order.
In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step.
After performing $2001$ steps, only one number remains on the board.
Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).
Let $ABC$ be a triangle inscribed in a circle. Let $M$ be the midpoint of the arc $AB$ that does not contain $C$ and $N$ be the midpoint of the arc $AC$ that does not contain $B$.
Let $E$ and $F$ be the points where the line $MN$ cuts the sides $AB$ and $AC$ respectively.
Prove that if $ME = EF = FN$, then the triangle $ABC$ is equilateral.
In an acute triangle $ABC$, the points $H$, $G$, and $M$ lie on the side $BC$, so $AH$, $AG$, and $AM$ are height, bisector, and median of the triangle, respectively. It is known that $HG = GM$, $AB = 10$ and $AC = 14$. Determine the area of triangle $ABC$.
The quadrilateral $ABCD$ has its diagonals perpendicular and is inscribed in a circle $\Gamma$ with center $O$. A line parallel to $BD$ cuts the segments $AO$ and $AD$ at $P$ and $Q$, respectively. Show that the lines $AP$ and $CQ$ intersect at a point $\Gamma$.
At Gabriel's house they are very methodical. Every business day the mother leaves on her motorcycle at the same time, at the same speed and along the same path to pick up Gabriel from school. He arrives at school at exactly 12 o'clock and immediately returns home with Gabriel, along the same path and at the same speed. Of course, they arrive home at exactly the same time every day.
One day, Gabriel left school early, and at 11 hours and 15 minutes he began the walk home. On the way he met his mother, who was picking him up from school, like every day. As soon as they met, they immediately returned to the house, arriving 20 minutes earlier than usual.
Determine how many minutes earlier than usual they would have arrived home if Gabriel began the walk at 11 hours and 33 minutes.
Let $C$ and $D$ be two points of the semicircle of diameter $AB$ such that $B$ and $C$ are in different half-planes with respect to the straight line $AD$. Let $M$, $N$, and $P$ denote the midpoints of $AC$, $DB$, and $CD$, respectively. Let $O_A$ and $O_B$ be the circumcenters of the triangles $ACP$ and $BDP$. Show that the lines $O_A O_B$ and $MN$ are parallel.
Determine the smallest positive integer $k$ so that the equation
$$
2002x + 273y = 200201 + k
$$
have integer solutions, and for that value of $k$, find the number of solutions $(x, y)$ with $x, y$ positive integers that the equation has.