A positive integer is *happy* if
* all of its digits are different and non-zero,
* one of its digits is equal to the sum of the others.
For example, $253$ is a happy number. How many happy numbers are there?
Prove that for every positive integer $n$, there exists a positive integer $k$ such that each of the numbers $k, k^2, \dots, k^n$ has at least one 2022 block in its decimal representation.
(For example, the numbers $4202213$ and $544202212022$ have at least one 2022 block in their decimal representation.)
Ana and Beto play the following game on a $2022 \times 2022$ grid. First Ana colors some unit segments on the grid red, so that no cell has two red sides who share a vertex. Next, Beto must draw a blue path that joins two of the four corners of the grid, following the sides of the grid and without using any red segments. If Beto succeeds, he is the winner; otherwise, Ana wins. Who has a winning strategy?
A positive integer $n \gt 1$, whose positive divisors are
$$
1 = d_1 \lt d_2 \lt \dots \lt d_k = n,
$$
is called *sureño* if all of the numbers $d_2 - d_1, d_3 - d_2, \dots, d_k - d_{k-1}$ are divisors of $n$.
a. Find a positive integer that is not sureño and has exactly 2022 positive divisors that are sureño.
b. Prove that there are infinitely many positive integers that are not sureño and have exactly 2022 positive divisors that are sureño.
The numbers $1, 2, 3, \dots, 170$ are written on a board. We want to color each number with one of the $k$ colors $C_1, C_2, \dots, C_k$, so that the following condition is fulfilled: for each $i$ with $1 \le i \lt k$, the sum of all the numbers colored $C_i$ divides the sum of all the numbers colored $C_{i+1}$. Determine the maximum value of $k$ for which it is possible to perform this coloring.
Four teams $A$, $B$, $C$, $D$ play a football tournament in which every team faces each of the other teams exactly twice (for a total of 12 matches). In each match, if the teams draw against each other, both of them earn 1 point; otherwise, the winner earns 3 points and the loser earns no points.
Given that at the end of the tournament teams $A$, $B$, $C$ scored 8 points each, find all possible scores of team $D$.
Let $L$ be the 2022-digit number made only by ones, that is $L = \underbrace{111\ldots11}_{2022 \text{ digits}}$. Find the sum of the digits of $9L^2 + 2L$.
Let $ABCDEFGHI$ be a regular 9-gon with its vertices labelled anticlockwise, and let $ABJKLM$ be a regular hexagon with its vertices also labelled anticlockwise. Prove that the angles $\angle HMG$ and $\angle KEL$ are equal.
In Eventown all authentic coins weigh an even amount of grams and all fake coins weigh an odd amount of grams.
There are $2022$ coins and it is given that exactly $2$ of them are fake. We have an electronic scale which only shows if the total weight of the objects put on it is even or odd.
Find the least value of $k$ such that there is a strategy that allows us to identify the two fake coins using the scale at most $k$ times.
Determine the number of permutations $a_1, a_2, \ldots, a_{2021}$ of the numbers $2, 3, \ldots, 2022$ such that $a_k$ is divisible by $k$, for all $k = 1, 2, \ldots, 2021$.
Let $m, n \ge 2$. You want to completely cover an $m \times n$ board without any gaps or overlaps, using only pieces of the following two types:
Type A
Type B
Each type A piece must cover exactly 4 squares on the board, and each type B piece must cover exactly 5 squares on the board. Rotating the pieces is allowed. Determine all pairs $(m, n)$ for which this can be done.
Let $ABC$ be a triangle with $AB \lt AC$. On the angle bisector of $\angle BAC$, two points $X$ and $Y$ are marked such that $X$ is between $A$ and $Y$, and $BX$ is parallel to $CY$. Let $Z$ be the reflection of $X$ with respect to $BC$. Let $P$ be the point of intersection of lines $YZ$ and $BC$. If lines $BY$ and $CX$ intersect at $K$, prove that $KA = KP$.
Let $ABCD$ be a parallelogram whose diagonals meet at $M$. Let $N$ be an interior point of triangle $AMB$ such that $\angle AND = \angle BNC$. Prove that $\angle MNC = \angle NDA$ and $\angle MND = \angle NCB$.
Let $n$ be a positive integer. Using the integers from $1$ to $4n$ inclusive, pairs are to be formed such that the product of the numbers in each pair is a perfect square. Each number can be part of at most one pair, and the two numbers in each pair must be different. Determine, for each $n$, the maximum number of pairs that can be formed.
Let $N(a, b)$ be the number of ways to cover an $a \times b$ board using domino tiles. Additionally, let $N^*(a, 2b + 1)$ be the number of ways to cover an $a \times (2b+1)$ board using domino tiles, without having vertical dominoes in the central column. Prove that $N^*(2m, 2n + 1) = 2^m N(2m, n)N(2m, n - 1)$.
Prove that there are infinitely many positive integers $n$ such that the equation
$$
x^2 + y^{11} - z^{2022!} = n
$$
has no solution $(x, y, z)$ over the integers.
Let $ABC$ be an acute triangle with $AB \lt AC$. Let $D$, $E$, $F$ be the feet of the altitudes from $A$, $B$, $C$ respectively. The circumcircles of $AEF$ and $ABC$ meet again at $M$. Suppose the line $BM$ is tangent to the circumcircle of $AEF$. Prove that $M$, $F$, $D$ are collinear.
Let $I$ be the incenter of triangle $ABC$. The incircle of $ABC$ is tangent to side $BC$ at $D$. Let $P$ and $Q$ be points on rays $IB$ and $IC$ respectively such that $\angle IAP = \angle CAD$ and $\angle IAQ = \angle BAD$. Prove that $AP = AQ$.
Let $n \ge 4$ and $k$ be positive integers. We consider $n$ lines on the plane such that no two of them are parallel and no three of them intersect in a single point. On each of the $\frac{n(n-1)}{2}$ intersection points of these lines there are $k$ coins. Ana and Beto play the following game: each player, in their turn, chooses a point that does not lie on the same line as the point chosen in the previous turn by the other player, and discards one coin from that point. Ana makes the first move and she can choose any point. The player who cannot make a move loses the game.
Determine, for each value of $n$ and $k$, which player has a winning strategy.
