A number is capicua if it is read equally from left to right as it is from right to the left. For example, $23432$ and $111111$ are capicua numbers. (a) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2022$ equal digits? (b) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2021$ equal digits?

Consider a semicircle with center $M$ and diameter $AB$. Let $P$ be a point in the semicircle, different from $A$ and $B$, and let $Q$ be the midpoint of the arc $AP$. The line parallel to $QP$ through $M$ intersects $PB$ at the point $S$. Prove that the triangle $PMS$ is isosceles.

You have a list of $2023$ numbers, where each one can be $-1$, $0$, $1$ or $2$. The sum of all numbers is $19$ and the sum of their squares is $99$. What are the minimum and maximum values of the sum of the cubes of those $2023$ numbers?

Find all positive integers $n$ such that: $$n = a^2 + b^2 + c^2 + d^2,$$where $a < b < c < d$ are the smallest divisors of $n$.

Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of $31$.

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$where $a$ and $b$ are positive integers with no prime factors in common.

$2023$ wise men are located in a circle. Each of them thinks either that the earth is the center of the universe, or that it is not. Once a minute, all the wise men express their opinion at the same time. Every wise man who is between two wise men with an opinion different from his will change his mind at that moment. The others don't change their minds. The others don't change their minds. Determine the smallest necessary time for all the wise men to have the same opinion, without regardless of initial opinions or your location.

Inside a quadrilateral $ABCD$ there exists a point $P$ such that $AP$ is perpendicular to $AD$ and the line $BP$ is perpendicular to $DC$. Besides, $AB = 7$, $AP = 3$, $BP = 6$, $AD = 5 $ and $CD = 10$. Calculate the area of the triangle $ABC$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

Let $I$ be the incenter of a triangle $ABC$ and let $D$ and $E$ be the touchpoints of the incircle with sides $BC$ and $AC$, respectively. The lines $DE$ and $BI$ intersect at point $P$. Prove that $AP$ is perpendicular to $BP$.

Let $p(x)$ be a polynomial of degree $2022$ such that: $$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$Find $p(2023)$.

A frog started from the origin of the coordinate plane and made $3$ jumps. Each time, the frog jumped a distance of $5$ units and landed on a point with integer coordinates. How many different position possibilities end of the frog there?