Let $\Gamma$ be the circumcircle of $\triangle A B C$. Let $D$ be a point on the side $B C$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $B A$ through $D$ at point $E$. The segment $C E$ intersects $\Gamma$ again at $F$. Suppose $B, D, F, E$ are concyclic. Prove that $A C, B F, D E$ are concurrent.
Show that $r=2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_{1}, a_{2}, \ldots$ of positive integers fulfills the inequalities
$$
a_{n} \leq a_{n+2} \leq \sqrt{a_{n}^{2}+r a_{n+1}}
$$
for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2}=a_{n}$ for every $n \geq M$.
Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n\gt m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_{1}, a_{2}, \ldots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i\lt j$ and an integer $k$ such that $a_{i}+a_{i+1}+\cdots+a_{j}=P(k)$.
Ignacio has a sheet of paper. He may cut it into either 6 or 8 pieces, as he chooses. At each subsequent stage he may choose an existing piece and cut it into 6 or 8 pieces.
a) Decide whether Ignacio can have exactly 24 pieces of paper after some stage.
b) Decide whether Ignacio can have exactly 32 pieces of paper after some stage.
If not, explain why; if so, specify how to make the cuts.
Fede must choose 50 distinct integers from $1$ to $100$ inclusive whose sum is $2900$. Determine the smallest possible number of even integers among his 50 choices.
For every positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Find, if one exists, a positive integer $n$ with 171 digits such that 7 divides both $S(n)$ and $S(n+1)$.
In the isosceles triangle $ABC$, let $D$ and $E$ lie on sides $AB$ and $AC$ respectively, with lines $BE$ and $CD$ meeting at $F$. Moreover, triangles $AEB$ and $ADC$ are congruent, with $AD=AE=10$ and $AB=AC=30$.
Calculate $\frac{\text{area}(ADFE)}{\text{area}(ABC)}$.
Let $n$ be a positive integer. There are $n$ colors, with $n \ge 1$. Each integer from $1$ to $1000$ is to be painted one of the $n$ colors so that any two distinct numbers have different colors whenever one divides the other. Find the smallest $n$ for which this is possible.
Let $k \gt 1$ be an integer. Determine the smallest positive integer $n$ such that some cells of an $n \times n$ board can be colored black with exactly $k$ black cells in every row and column, and with no two black cells sharing a side or a vertex.
The 18 integers from $1$ to $18$ are written on a board. Determine the smallest number that must be erased so that no two remaining numbers have a perfect-square sum.
Let $ABCD$ be a parallelogram with $\angle ABC = 105^\circ$. Inside it there is a point $E$ such that triangle $BEC$ is equilateral and $\angle CED = 135^\circ$. Let $K$ be the midpoint of side $AB$. Calculate angle $BKC$.
Let $ABC$ be an isosceles right triangle with right angle at $A$. Let $E$ and $F$ lie on $AB$ and $AC$ respectively, with $\angle ECB = 30°$ and $\angle FBC = 15°$. Lines $CE$ and $BF$ meet at $P$, and line $AP$ meets side $BC$ at $D$. Calculate angle $FDC$.
Ana and Beto play the following game. Ana writes four consecutive three-digit integers. Beto chooses three of them and adds them. If their sum can be written as a product of three positive integers greater than 1, Beto wins; otherwise Ana wins. Determine whether Ana can choose the four numbers to guarantee a win.
Juli has a deck of 54 cards and proposes the following game to Bruno. Juli puts the cards in a row, some face up and the others face down. Bruno may repeatedly choose a card and turn over that card and its two neighbors, changing face-up cards to face-down and vice versa. Bruno wins if this makes every card face up; otherwise Juli wins. Decide which player has a winning strategy and explain it.
An eight-sided polygon is drawn on a board. Mili writes an integer from $1$ to $16$ on each side and each vertex, without repetition. For each side she adds the numbers at its two endpoints and the number on the side, obtaining $8$ results. The goal is for all $8$ results to be equal. Let $S$ be the common value of the $8$ sums. Determine all possible values of $S$ and, for the smallest, give an arrangement of the $16$ numbers producing that value of $S$.
Twenty distinct positive integers are written around a circle. Alex divides each by its clockwise neighbor and records the remainders. Teo divides each by its counterclockwise neighbor and records the remainders. If Alex obtains only two distinct remainders among his 20 results, determine how many distinct remainders Teo obtains.
Determine the largest possible value of
$$
S = a_1 a_2 a_3 + a_4 a_5 a_6 + \dots + a_{2017} a_{2018} a_{2019} + a_{2020},
$$
where $(a_1, a_2, a_3, \dots, a_{2020})$ is a permutation of $(1, 2, 3, \dots, 2020)$.
a) Nico chooses 10 distinct positive integers, then Uriel chooses 6 and adds them. Uriel wins if the sum is a multiple of 6 and loses otherwise. Can Nico choose the 10 numbers so that Uriel cannot win?
b) Nico chooses 11 distinct positive integers, then Uriel chooses 6 and adds them. Uriel wins if the sum is a multiple of 6 and loses otherwise. Can Nico choose the 11 numbers so that Uriel cannot win?
In each case, give an example if yes and explain why if no.