In this question we re-define the operations addition and multiplication as follows: $a + b$ is defined as the minimum of $a$ and $b$, while $a * b$ is defined to be the sum of $a$ and $b$. For example, $3+4 = 3$, $3*4 = 7$, and $$3*4^2+5*4+7 = \min(\text{3 plus 4 plus 4}, \text{5 plus 4}, 7) = \min(11, 9, 7) = 7.$$ Let $a, b, c$ be real numbers. Characterize, in terms of $a, b, c$, what the graph of $y = ax^2+bx+c$ looks like.

Let $n \ge 2$ be an integer. Show that there exist $n+1$ numbers $x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}$, so that $\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}$, where $\{ x \}$ is the fractionary part of $x$.

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.

A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[ a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}. \] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.

Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps. While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana. How many points in the plane, apart from $S$, are marked with a banana by the end of the $6$ minutes.

(1) Prove that $\sqrt[3]{2}$ is irrational. (2) Let $P(x)$ be a polynomoal with rational coefficients such that $P(\sqrt[3]{2})=0$. Prove that $P(x)$ is divisible by $x^3-2$. 35 points

Given three numbers $x, y, z$, and set $x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|$. From $x_1, y_1, z_1$, form in the same fashion the numbers $x_2, y_2, z_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z$ for some $n$. Find all possible values of $(x, y, z)$.

If $78$ is divided into three parts which are proportional to $1, \frac13, \frac16$, the middle part is: $ \textbf{(A)}\ 9\frac13 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 17\frac13 \qquad\textbf{(D)}\ 18\frac13\qquad\textbf{(E)}\ 26 $

Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation \[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\] has at least one real root.

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer. Note: $N^*$ denotes the set of positive integers.

Show that \[ a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1} \] for real $a,b,c \geq 0$ and $n$ a positive integer.

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$

Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct, [b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal. [b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational. [i]Marin Tolosi[/i]

Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Given that $a,b$, and $c$ are positive real numbers such that $ab + bc + ca \geq 1$, prove that \[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \geq \frac{\sqrt{3}}{abc} .\]

The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$: $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$, $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$?

Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$

Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

If the reals $a,b,c,d,e,f,g,h,i$ satisfy $$\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}$$show that $cf+fi+ic=3$ holds as well.