An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Four friends, Anna, Bob, Celia, and David, exchanged some money. For any two of these friends, exactly one gave money to the other. For example, Celia could have given some money to David but then David would not have given money to Celia. In the end, each person broke even (meaning that no one made or lost any money). (a) Is it possible that the amounts of money given were $10$, $20$, $30$, $40$, $50$, $60$? (b) Is it possible that the amounts of money given were $20$, $30$, $40$, $50$, $60$, $70$? For each part, if your answer is yes, show that the situation is possible by describing who could have given what amounts to whom. If your answer is no, prove that the situation is not possible.

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$

Point $P$ lies inside a triangle $ABC$. Let $D,E$ and $F$ be reflections of the point $P$ in the lines $BC,CA$ and $AB$, respectively. Prove that if the triangle $DEF$ is equilateral, then the lines $AD,BE$ and $CF$ intersect in a common point.

A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions: (A) Gives a candy to the student sitting on his left or to the student sitting on his right. (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right. At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions. (Two distributions are different if there is a student who has a different number of candy in each of these distributions.) （Forgive my poor English）

Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that: $$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$

Find all integer sequences $a_1 , a_2 , \ldots , a_{2024}$ such that $1\le a_i \le 2024$ for $1\le i\le 2024$ and $$i+j|ia_i-ja_j$$ for each pair $1\le i,j \le 2024$.

$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1. Show that $f$ preserves distances.

The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$, $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$. Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$

Let $ABCD$ be a cyclic quadrilateral whose center of the circumscribed circle is inside this quadrilateral, and its diagonals intersect in point $S{}$. Let $P{}$ and $Q{}$ be the centers of the curcimuscribed circles of triangles $ABS$ and $BCS$. The lines through the points $P{}$ and $Q{}$, which are parallel to the sides $AD$ and $CD$, respectively, intersect at the point $R$. Prove that the point $R$ lies on the line $BD$.

Suppose $a,b,c>0$ and $abc=64$, show that $$\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}$$

For each positive real number $\alpha$, define $$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$ Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$, $$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$ Determine, with proof, the smallest positive size of $S$.

Let $ABCD$ be a rectangle with $AB=9$ and $BC=3$. Suppose that $D$ is reflected across $AC$ to a point $E$. Compute the area of trapezoid $AEBC$.

Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game. For each positive integer $n$, find a winning strategy for one of the players.

Let $ABCD$ be a cyclic quadrilateral such that $AB=AD$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $BE+DF=EF$. Prove that $\angle BAD = 2 \angle EAF$.

A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying \[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\] Find all smooth numbers.

$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

Benedek scripted a program which calculated the following sum: $1^1+2^2+3^3+. . .+2021^{2021}$. What is the remainder when the sum is divided by $35$?

Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases . a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$. b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$. c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.

Paca-Vaca decided to note every day a single quadratic polynomial of the form $x^2+ax+b$, where $a$ and $b$ are positive integers, less or equal than $100$. He follows the rule that the polynomial he writes must not have any common roots with the polynomials previously noted. What is the maximum amount of days Paca-Vaca can follow this plan?

Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying $$P(x + Q(y)) = Q(x + P(y))$$ for all real numbers $x$ and $y$.