[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

The product of the positive real numbers $x, y, z$ is 1. Show that if \[ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z \]then \[ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} \] for all positive integers $k$.

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals $\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

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.

Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

Find all function $f,g: \mathbb{Q} \to \mathbb{Q}$ such that \[\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}\] for all $x,y \in \mathbb{Q}$.

Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations \begin{align*}(a+b)(c+d)&=143,\\(a+c)(b+d)&=150,\\(a+d)(b+c)&=169.\end{align*} Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

If $z$ is a complex number such that \[ z + z^{-1} = \sqrt{3}, \] what is the value of \[ z^{2010} + z^{-2010} \, ? \]

Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations: \begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

During Christmas party Santa handed out to the children $47$ chocolates and $74$ marmalades. Each girl got $1$ more chocolate than each boy but each boy got $1$ more marmalade than each girl. What was the number of the children?

Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

Let $a,b,c>0$ be real numbers with sum 1. Prove that \[ \frac{a^2}b + \frac{b^2}c + \frac{c^2} a \geq 3(a^2+b^2+c^2) . \]

Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions : (a) $ x_{0} \equal{} x_{n} \equal{} 0$, and (b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$. Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.

Find a non-zero polynomial $f(x, y)$ such that $f(\lfloor 3t \rfloor, \lfloor 5t \rfloor) = 0$ for all real numbers $t$.