A list of $n$ positive integers $a_1, a_2, a_3, \dots, a_n$ is called *good* if both of the following conditions are satisfied:
* $a_1 \lt a_2 \lt a_3 \lt \dots \lt a_n$,
* $a_1 + a_2^2 + a_3^3 + \dots + a_n^n \le 2023$.
For each $n \ge 1$, find how many good lists of $n$ numbers are there.
There is a lamp on each cell of an infinite square grid. Initially, all lamps are *off*. A move consists of choosing either a $3 \times 3$, $4 \times 4$ or $5 \times 5$ square contained on the grid and switch all lamps inside that square from *on* to *off* or vice versa.
a. Prove that for any finite set $S$ of lamps it is possible to achieve, after a finite sequence of moves, that all lamps that are *on* are exactly the ones in $S$.
b. Prove that if a sequence of moves only involves two of the three sizes of squares available, then it is not possible to achieve that in the end all lamps that are *on* are exactly the ones inside a $2 \times 2$ square.
We say that an equilateral triangle on the plane is in *standard position* if one of its sides is horizontal and the opposite vertex lies above that side. We have $n$ equilateral triangles $S_1, S_2, \dots, S_n$, all of which are in standard position. For each triangle $S_i$, denote by $T_i$ its medial triangle. Let $S$ be the region of the plane covered by the triangles $S_1, S_2, \dots, S_n$, and let $T$ be the region of the plane covered by the triangles $T_1, T_2, \dots, T_n$. Prove that $\text{area}(S) \le 4 \cdot \text{area}(T)$.
A sequence of integers is defined as follows: $a_1 = 1$, $a_2 = 2$, and for each $n \ge 2$, $a_{n+1}$ is equal to the greatest prime divisor of $a_1 + a_2 + \dots + a_n$. Compute $a_{100}$.
Let $ABC$ be an acute triangle. Denote by $D, E, F$ the midpoints of sides $BC, CA, AB$ respectively. The circle with diameter $AD$ intersects lines $AB$ and $AC$ again at $P$ and $Q$ respectively. The line through $P$ parallel to $BC$ meets line $DE$ at $R$, the line through $Q$ parallel to $BC$ meets line $DF$ at $S$. The circumcircle of $DPR$ meets $AB$ again at $X$, the circumcircle of $DQS$ meets $AC$ again at $Y$, and those two circles meet again at $Z$. Prove that $Z$ is the midpoint of $XY$.
Let $x_1, x_2, \dots, x_n$ be positive real numbers. For each positive integer $k$, define
$$
S_k = x_1^k + x_2^k + \dots + x_n^k.
$$
(a) Prove that if $S_1 \lt S_2$ the sequence $S_1, S_2, S_3, \dots$ is strictly increasing.
(b) Prove that it is possible that $S_1 \gt S_2$ and yet the sequence $S_1, S_2, S_3, \dots$ is not strictly decreasing.
Let $ABCD$ be a convex quadrilateral that satisfies the following conditions:
$$
\angle BAC = 2\angle BCA, \quad \angle BCA + \angle CAD = 90^\circ \quad \text{and} \quad BC = BD.
$$
Find $\angle ADB$.
The 400-digit number $N = 20232023\ldots2023$, which is formed by 100 copies of $2023$, is written on a blackboard. Lionel has to erase some of the digits of $N$ in such a way that the resulting number is divisible by $84$, and the largest possible. Determine which digits Lionel should erase.
Initially we have a paper triangle $ABC$ such that $\angle BAC = 120^\circ$. In the first step, we draw the angle bisectors of the three angles of the triangle, which intersect in $I$, and then using a pair of scissors we cut along segments $AI$, $BI$ and $CI$, obtaining 3 triangles: $ABI$, $BCI$ and $CAI$. In the second step we repeat the same procedure with the three triangles, that is: each one of those is cut into three smaller triangles cutting along the angle bisectors. At the end of the second step we have 9 triangles in total. This procedure continues in the same way until we complete 10 steps.
How many of the triangles at the end of the process have an angle of $120^\circ$?
The vertices of a regular hexagon are marked on a blackboard. Ana draws some segments that are either sides or diagonals of the hexagon, in any way she wants to (she can even decide not to draw any segment at all, or to draw the 15 possible segments).
Afterwards, Beto writes a positive integer on each vertex, in such a way that the following condition is satisfied: if two vertices are connected by a segment drawn by Ana, then the corresponding numbers must have a common divisor greater than 1; otherwise, if they are not connected by a segment, the numbers must not have any common divisor greater than 1.
a. Show that Beto can always complete his task.
b. Once Beto completes his task, he must pay Ana $M$ pesos, where $M$ is the greatest of the 6 numbers that Beto wrote. Beto wants to pay as least as possible and Ana wants to get paid the greatest possible amount of pesos. Can Ana draw the segments in such a way that she is guaranteed to receive more than 2023 pesos?
An integer $n \ge 3$ is said to be a *polygonal pythagorean number* if there are $n$ positive integers, no two of them equal, which can be placed in the vertices of a regular $n$-gon in such a way that the sum of the squares of the numbers in any two consecutive vertices is a perfect square. For instance, 3 is a polygonal pythagorean number because placing 44, 117 and 240 in the vertices of a triangle, we have $44^2 + 117^2 = 125^2$, $117^2 + 240^2 = 267^2$, and $240^2 + 44^2 = 244^2$. Find all polygonal pythagorean numbers.
Let $ABCD$ be a convex quadrilateral such that $AB = CD$, $\angle BCD = 2\angle BAD$, $\angle ABC = 2\angle ADC$, and $\angle BAD \neq \angle ADC$. Find the angle between diagonals $AC$ and $BD$.
Let $n \gt d \gt 0$ be integers. Ana, Beto and Carlitos play *blind man's bluff* over an infinite grid. Initially, Ana and Carlitos are in cells at distance $n$, and there is a candy in a cell which is at distance $d$ from Carlitos. Carlitos is blindfolded and can only see his own cell, whereas Ana and Beto can see the whole grid. Two movements are performed alternately.
1. Carlitos moves to an adjacent cell. If he finds Ana there, Carlitos loses. If he finds the candy, but not Ana, Carlitos wins. If the cell was empty, Beto shouts "hot" or "cold", at his discretion.
2. Ana moves to an adjacent cell. If she finds Carlitos or the candy, Ana wins. Otherwise, the game continues.
Find, for each $d$, the least $n$ such that Beto and Carlitos can coordinate a strategy to ensure Carlitos' victory, regardless of the initial positions of Ana, Carlitos, and the candy.
*Remark:* Two cells are adjacent if they share a common side. The distance between cells $X$ and $Y$ is the least $p$ for which there is a sequence of cells $X = X_0, X_1, \dots, X_p = Y$ such that $X_i$ is adjacent to $X_{i-1}$ for all $i = 1, \dots, p$.
A set of points is called *antiparallelogram* if no four of them are the vertices of a parallelogram. Given a set $S$ of $2023$ points on the plane, no three of them on the same line, prove that there is a subset of $S$ containing $17$ points which is antiparallelogram.
We say that a positive integer $N$ is *rioplatense* if it satisfies the following two conditions:
* It is possible to find 34 consecutive integers such that their product is divisible by $N$ but none of them is divisible by $N$.
* It is *not* possible to find 30 consecutive integers such that their product is divisible by $N$ but none of them is divisible by $N$.
Find all rioplatense integers.
Let $\mathbb{Z}$ be the set of integer numbers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$
such that
$$
f(x + f(y + 1)) + f(xy) = f(x + 1)(f(y) + 1)
$$
for any integers $x, y$.
Let $ABCD$ be a convex quadrilateral with $AB \gt AD$ and $\angle B = \angle D = 90^\circ$. Let $P$ be the point on side $AB$ such that $AP = AD$. Lines $PD$ and $BC$ intersect at $Q$. The perpendicular line to $AC$ through $Q$ intersects $AB$ at $R$. Let $S$ be the foot of the perpendicular from $D$ to $AC$. Prove that $\angle PSQ = \angle RCP$.
The river city of Platense consists of several platforms and bridges between them. Each bridge connects two platforms and no two bridges are connecting the same two platforms. The mayor wants to change some bridges through a series of moves as follows: if there are three platforms $A$, $B$ and $C$, and bridges $AB$ and $AC$ but not $BC$, then $AB$ can be changed to $BC$.
A bridge configuration is *good* if you can go from any platform to any other using only the bridges. Starting from a good configuration, show that the mayor can reach any other good configuration, whose number of bridges is the same, through the movements described.
Mati is playing with some magic boxes and a machine. Each box has a value inside. When opening a box Mati sees its value, adds the value of the box to his score and it is destroyed (if the box's value is negative, Mati loses points). By placing a magic box with value $X$ in the machine, this box is destroyed and we obtain two magic boxes with values $X + 1$ and $X - 1$ (it is not known which is which, but the new boxes can be distinguished from the others). At the beginning of the game, Mati has 0 points and one magic box whose value he knows to be 0.
a. Show that Mati can ensure reaching 1000 points or more.
b. Can Mati ensure reaching 1000000 points or more, without having less than -42 points at any time?
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all non-negative real numbers $\alpha$ for which there exists a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$
f(x^{\alpha} + y) = (f(x + y))^{\alpha} + f(y)
$$
for any $x, y \in \mathbb{R}^+$.
