The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $ Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers. (Here, $ [x] $ is the largest integer not exceeding $ x $.)

Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$ .

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

Let $a_n = 2^{n-1}$ for $n > 0$. Let \[ b_n = \sum\limits_{r+s \leq n} a_ra_s \] Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.

Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$

Let $a_1,...,a_n$ be positive real numbers such that $\sum_{i=1}^n a_i =\prod_{i=1}^n a_i $ , and let $b_1,...,b_n$ be positive real numbers such that $a_i \le b_i$ for all $i$. Prove that $\sum_{i=1}^n b_i \le\prod_{i=1}^n b_i $

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .

Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy: $\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$ What values can be taken by the product $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.

Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

Find integers \(a\) and \(b\) such that \[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\] [i]proposed by the ICMC Problem Committee[/i]

Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series $$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$ has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find the value of \[+1+2+3-4-5-6+7+8+9-10-11-12+\cdots -2020,\] where the sign alternates between $+$ and $-$ after every three numbers.

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

Let $a, b, c$ be positive real numbers with $abc = 1$. Find all possible values ​​of the expression $$\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}$$ can take.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.

Find all polynomials $P$ such that $P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$ $=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$ for all real numbers $x$.

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$. (A. Andj ans, Riga)

Given an arbitrary angle $\alpha$, compute $cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ . Generalize this result and justify your answer.

Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that $$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$