Prove that for arbitrary fixed $a_1, a_2,.. , a_{31}$ the sum $\cos 32x + a_{31} \cos 31x +... + a_2 cos 2x + a_1 \cos x$ can take both positive and negative values as $x$ varies.

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

Let $p_n$ be the product of the $n$th roots of $1$. For integral $x > 4$, let $f(x) = p_1 - p_2 + p_3 - p_4 + ... + (-1)^{x+1}p_x$. What is $f(2010)$?

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

The given triangular number table is as follows: [img]https://cdn.artofproblemsolving.com/attachments/a/0/123b7511850047f3cc51494f107703f2757085.png[/img] Among them, the numbers in the first row are $1, 2, 3, ..., 98, 99, 100$. Starting from the second row, each number is equal to the sum of the left and right numbers in the row above it. Find the value of $M$.

$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.

Positive real numbers $ x$, $ y$ satisfy the equations $ x^2 \plus{} y^2 \equal{} 1$ and $ x^4 \plus{} y^4 \equal{} \frac {17}{18}$. Find $ xy$.

A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer. [i]($3$ points)[/i]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$. (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$. (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$. (c) Does (b) remain true if the word ”completely” is omitted?

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that: [b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$. [b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003. [b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.

Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ \[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

Find the polynomial $f$ with the following properties: $\bullet$ its leading coefficient is $1$, $\bullet$ its coefficients are nonnegative integers, $\bullet$ $72|f(x)$ if $x$ is an integer, $\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.

Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression: $$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$