Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$. For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount. For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.

What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?

(i) Find a formula for $S_n = -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^n n^2 \times (n + 1)$ in terms of the positive integer $n$. Justify your answer. (As an example, one has $1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$) (ii) Using your formula in (i), find the value of $ -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^{100} 100^2 \times (100 + 1)$

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$

Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.

There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to: a) $25$, b) $50$?

The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges. Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

Find the number of ways to write the number $2013$ as the sum of two integers greater than or equal to zero so that when adding there is no carry over. Clarification: In the sum $2008+5=2013$ there is carry over from the units to the tens

For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]

A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers $b_1=a_1$ $b_2= a_1 + a_2$ $b_3=a_1 + a_2 + a_3$ ... $b_{100}=a_1 + a_2 + ...+a_{100}$ Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different . ( L . D . Kurlyandchik , Leningrad)

Let $S$ be a subset of $\{1,2,3,...,24\}$ with $n(S)=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.

Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.

$n$ natural numbers ($n>3$) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number. Prove that the sum of those relations is a) not less than $2n$ b) less than $3n$

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

Let n be a positive integer and $x_1, x_2, ..., x_n > 0$ be real numbers so that $x_1 + x_2 +... + x_n =\frac{1}{x_1^2}+\frac{1}{x_2^2}+...+\frac{1}{x_n^2}$ Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, ..., x_n $ whose sum is at least $k$.

For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $. a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $. b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $

Evaluate the sum $$\sum_{n=0}^\infty\operatorname{arctan}\left(\frac1{1+n+n^2}\right).$$