Let $a$, $b$ and $c$ be real numbers such that $\mid a \mid >2$ and $a^2+b^2+c^2=abc+4$. Prove that numbers $x$ and $y$ exist such that $a=x+\frac{1}{x}$, $b=y+\frac{1}{y}$ and $c=xy+\frac{1}{xy}$.

Given real numbers $x$ and $y$. Let $s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4$ and $t=xy$. [b]a)[/b] Prove, that number $t$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. [b]b)[/b] Prove, that number $s_{1}$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. [b]c)[/b] Can number $s_{1}$ be irrational, if $s_{2}$ and $s_{3}$ are rational numbers?

Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation $$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$ represent the vertices of an equilateral triangle in the complex plane.

For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of \[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]

Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.

Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$. Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$

Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.

In a row written in increasing order all the irreducible positive rational numbers, such that the product of the numerator and the denominator is less than 1988. Prove that any two adjacent fractions $\frac{a}{b}$ and $\frac{c}{d},$ $\frac{a}{b} < \frac{c}{d},$ satisfy the equation $b \cdot c - a \cdot d = 1.$

Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$ Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.

We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion $$ a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor $$ for all $n \ge 3$. Find the value of $a_{2022}$.

Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. [b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$. [b]p3.[/b] Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems? [b]p4.[/b] Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done? [b]p5.[/b] You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon? [b]p6.[/b] Let $a, b$, and $c$ be positive integers such that $gcd(a, b)$, $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$, but $gcd(a, b, c) = 1$. Find the minimum possible value of $a + b + c$. [b]p7.[/b] Let $ABC$ be a triangle inscribed in a circle with $AB = 7$, $AC = 9$, and $BC = 8$. Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$. Find the length of $\overline{BX}$. [b]p8.[/b] What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ? [b]p9.[/b] How many terms are in the simplified expansion of $(L + M + T)^{10}$ ? [b]p10.[/b] Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$. What are all possible slopes for a line tangent to both of the circles? PS. You had better use hide for answers.

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.

Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$ When does the equality take place?

Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients. Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients

The graph of a function $f: \mathbb{R}\to\mathbb{R}$ has two has at least two centres of symmetry. Prove that $f$ can be represented as sum of a linear and periodic funtion.

Find all triples ( $\alpha, \beta,\theta$) of acute angles such that the following inequalities are fulfilled at the same time $$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$ $$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$ $$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$

[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \] [b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)} \] Calculate the minimum value of $k.$

Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\, b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$ Prove that $$|a-b| <\frac{1}{99! 100!}$$ (G Galperin, Moscow)

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

Higher Secondary P2 Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.