Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$ It is known that: [list] [*] If $f, g\in A$ then $f (g (x)) \in A.$ [*] For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$ [/list] Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

Find all real numbers $r$, such that the inequality \[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\] holds for any real $a,b,c>0$.

Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$. Find the remainder when $N$ is divided by $1000$.

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

The function $f(x) = ax^2 + bx + c$ satisfies the following conditions: $f(\sqrt2)=3$ and $ |f(x)| \le 1$ for all $x \in [-1, 1]$. Evaluate the value of $f(\sqrt{2013})$

Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$. Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$. [i]($6$ points)[/i]

Let $\{ a_n \}_{n=0}^{11}$ and $\{ b_n \}_{n=0}^{11}$ be sequences of real numbers. Suppose $a_0 = b_0 = -1$, $a_1 = b_1$, and for all integers $n \in \{2, 3, \ldots, 11\}$, \begin{align*} a_n & = a_{n-1} - (11 - n)^2(1 - (11 - (n - 1))^2)a_{n-2} \quad \text{and} \\ b_n & = b_{n-1} - (12 - n)^2(1 - (12 - (n - 1))^2)b_{n-2}. \end{align*} If $b_{11} = 2a_{11}$, then determine the value of $a_1$.

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

For each positive integer $n$ we consider the sequence of $2004$ integers$$\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]$$Determine the smallest integer $n$ such that the $2004$ numbers in the sequence are $2004$ consecutive integers. Clarification: The brackets indicate the integer part.

Let $$P(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n$$ where $a_0,\ldots,a_n$ are integers. Show that if $P$ takes the value $2020$ for four distinct integral values of $x$, then $P$ cannot take the value $2001$ for any integral value of $x$. [i]Proposed by Ángel Plaza[/i]

Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$ holds for infinitely many real numbers $x$.

(a) Prove that $$3-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3$$ (b) Find some positive integers $a$, $b$ and $c$ such that for any $n > 2$, $$b-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b$$ (V Senderov, NB Vassiliev)

Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

A train is going up a hill with vertical velocity given as a function of $t$ by $\frac{1}{1 - t^4}$ , where $t$ is between $[0, 1)$. Determine its height as a function of $t$.