Given 100 infinitely large boxes with markers in them, the following procedure is carried out. At step 1 one adds one marker in every box. At step 2 one marker is added in every box containing an even number of markers. At step 3 one marker is added in every box in which the number of markers is divisible by 3, and so on.
Before the process starts Bruno wants to distribute several markers in the boxes so that there is at least one marker in each box and the following holds: After any number of steps there exist two boxes containing different number of markers. Decide if this is possible to achieve.
In basketball the free-throw rate (FRT) of a player is the ratio of the number of his successful free throws to the number of all of his free throws. After the first half of a game Mateo's FRT was less than $75\%$, and at the end of the game it was greater than $75\%$. Can one claim with certainty that there was a moment when his FRT was exactly $75\%$? Answer the same question for $60\%$ instead of $75\%$?
Given 100 infinitely large boxes with markers in them, the following procedure is carried out. At step 1 one adds one marker in every box. At step 2 one marker is added in every box containing an even number of markers. At step 3 one marker is added in every box in which the number of markers is divisible by 3, and so on.
Before the process starts Bruno wants to distribute several markers in the boxes so that there is at least one marker in each box and the following holds: After any number of steps there exist two boxes containing different number of markers. Decide if this is possible to achieve.
Let $ABC$ be a right triangle with $\hat{C} = 90^\circ$. Points $D$ and $E$ on the hypotenuse $AB$ are such that $AD = AC$ and $BE = BC$. Points $P$ and $Q$ on $AC$ and $BC$ respectively are such that $AP = AE$ and $BQ = BD$. Let $M$ be the midpoint of segment $PQ$. Find $\hat{AM}\hat{B}$.
One of the numbers $1$, $2$, ..., $n$ is written in each cell of a $17 \times 17$ table for a certain $n \in \mathbb{N}$; all of these numbers are used. If a row contains two cells $C_1$ and $C_2$ with equal numbers $k$ and $C_1$ is to the left of $C_2$ then there are no numbers $k$ in the column of $C_1$ that are above $C_1$. Determine the minimal $n$ for which such a table exists.
Alex and Bibi play the following game. Alex chooses a natural number $k$ not exceeding $1000$. Then Bibi chooses a collection $B$ of $n$ integers in $\{0,1,...,1000\}$, not necessarily distinct, where $n \gt k$. Now Alex is allowed to apply repeatedly the following operation on $B$: choosing $k$ numbers $b_1, ..., b_k$ from $B$ and changing them as follows. For each $i = 1, ..., k$ the number $b_i$ is replaced by $b_i + 1$ if $b_i \lt 1000$ and by $0$ if $b_i = 1000$.
Alex wins if via several operations he succeeds in making all numbers in $B$ equal to $0$; if he fails then Bibi wins. Find all $k$ that guarantee Alex a win, regardless of the collection $B$ chosen by Bibi.
Point $D$ is chosen on side $BC$ of the acute triangle $ABC$ so that $AD = AC$. Let $P$ and $Q$ be respectively the feet of the perpendiculars from $C$ and $D$ to $\overline{AB}$. It is known that
$$AP^2 + 3BP^2 = AQ^2 + 3BQ^2.$$ Find $\hat{A}BC$.
Let $a_1, a_2, ..., a_{100}$ be a permutation of $1, 2, ..., 100$. For each triple $\{a_i, a_{i+1}, a_{i+2}\}$ of consecutive numbers, $1 \le i \le 98$, the middle number in the triple is marked. For instance if $a_i = 7, a_{i+1} = 99, a_{i+2} = 22$ then $a_{i+2} = 22$ is marked. Let $S$ be the sum of all marked numbers. Find the minimum value of $S$. (Each marked number enters the sum $S$ exactly once, although it may be marked more than once.)
Point $D$ is chosen on side $BC$ of the acute triangle $ABC$ so that $AD = AC$. Let $P$ and $Q$ be respectively the feet of the perpendiculars from $C$ and $D$ to side $AB$. It is known that
$$AP^2 + 3BP^2 = AQ^2 + 3BQ^2.$$ Find $\hat{A}BC$.
Let $n \ge 3$ an integer. Find the number of arrangements $a_1, a_2, ..., a_n$ of $1, 2, ..., n$ around a circle, in clockwise direction, such that $|a_1 - a_2| + |a_2 - a_3| + \dots + |a_{n-1} - a_n| + |a_n - a_1| = 2n - 2$.
Let $n \ge 2$ be a natural number. For each pair $a, b$ of relatively prime natural numbers let $d_{a,b}$ be the greatest common divisor of $na+b$ and $a+nb$. Find the maximum value of $d_{a,b}$.
A circle of length $999$ is divided into unit arcs by $999$ black points. Then $d$ arcs with lengths $1$, $2$, ..., $d$ are placed on the circle, with their end points black, so that none of these arcs contains another (otherwise the arcs may overlap). Find all $d$ for which such a configuration exists.
Decide if there is an arithmetic progression of $2016$ natural numbers that are not perfect powers but their product is a perfect power.
(A perfect power is a number of the form $n^k$ where $n$ and $k$ are natural numbers with $n \ge 2$, $k \ge 2$.)
For an integer $m \ge 3$ set $S(m) = 1 + \frac{1}{3} + \dots + \frac{1}{m}$ (the fraction $1/2$ does not participate in the sum). Let $n \ge 3$ and $k \ge 3$. Compare the numbers $S(nk)$ and $S(n) + S(k)$.
Agustin and Lucas take turns in marking one cell at a time in a $101 \times 101$ grid. Agustin starts the game. A cell cannot be marked if there are already two marked cells in its row or in its column. The one who cannot move loses. Decide which player has a winning strategy.
Find the angles of a convex quadrilateral $ABCD$ such that $\hat{ABD} = 29^\circ$, $\hat{ADB} = 41^\circ$, $\hat{ACB} = 82^\circ$ and $\hat{ACD} = 58^\circ$.
Let $a$ and $b$ be rational numbers such that $a+b = a^2 + b^2$. Suppose that the common value $s = a+b = a^2 + b^2$ is not an integer, and write it as an irreducible fraction: $s = \frac{m}{n}$. Let $p$ be the least prime divisor of $n$. Find the minimum value of $p$.
Let $AB$ a line segment of length $1$. Several elementary particles start moving simultaneously at constant speeds from $A$ to $B$. As soon as a particle reaches $B$, it turns around and heads to $A$; when reaching $A$, it starts moving to $B$ again, and so on indefinitely.
Find all rational numbers $r \gt 1$ with the following property: For each $n \ge 1$, if $n+1$ particles with speeds $1, r, r^2, \dots, r^n$ move as described, there is a moment when all particles are at the same interior point of segment $AB$. (Ignore the dimensions of the particles; assume that they can all gather at one point.)
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$.
Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.