Olympiad problems

2013 Tournament of Towns, 4

Is it true that every integer is a sum of finite number of cubes of distinct integers?

2000 Junior Balkan Team Selection Tests - Moldova, 1

Tags: sum , algebra

Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.

2018 Austria Beginners' Competition, 3

Tags: combinatorics , sum

For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that $\bullet$ each row totals to the same sum $z$ and $\bullet$ each column totals to the same sum $s$. Prove: (a) If $n$ is even, such a table does not exist. (b) If $n = 5$, such a table does exist. (Gerhard J. Woeginger)

2021 Polish Junior MO First Round, 5

Tags: number theory , sum , product

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

1993 Bundeswettbewerb Mathematik, 1

Tags: sum , summand , integer , combinatorics

Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$

1977 All Soviet Union Mathematical Olympiad, 248

Tags: combinatorics , algebra , sum

Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $$(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)$$ Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.

2013 India PRMO, 3

Tags: algebra , sum , integer

It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?

2014 Denmark MO - Mohr Contest, 5

Tags: algebra , inequalities , sum

Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.

2007 Postal Coaching, 1

Tags: sum , geometry , constant , circumcircle , square

Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.

2020 Durer Math Competition Finals, 4

Tags: combinatorics , sum

Endre wrote $n$ (not necessarily distinct) integers on a paper. Then for each of the $2^n$ subsets, Kelemen wrote their sum on the blackboard. a) For which values of $n$ is it possible that two different $n$-tuples give the same numbers on the blackboard? b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.

2019 SAFEST Olympiad, 4

Tags: minimum , sum , algebra , inequalities

Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$. Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$ this minimum occurs

2004 German National Olympiad, 4

Tags: algebra , sum , integer , square root

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

2014 IMAC Arhimede, 4

Tags: polynomial , algebra , sum of powers , sum , rational

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

2016 Switzerland - Final Round, 7

Tags: combinatorics , sum

There are $2n$ distinct points on a circle. The numbers $1$ through $2n$ are randomly assigned to this one points distributed. Each point is connected to exactly one other point, so that no of the resulting connecting routes intersect. If a segment connects the numbers $a$ and $b$, so we assign the value $ |a - b|$ to the segment . Show that we can choose the routes such that the sum of these values results $n^2$.

2017 Argentina National Olympiad, 2

Tags: sum , algebra , number theory , consecutive

In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.

Oliforum Contest V 2017, 8

Tags: sum , product , algebra

Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$ for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$ show that at least one $a_i$ has to be equal to $\frac12$. (Paolo Leonetti)

2003 Singapore Senior Math Olympiad, 3

Tags: algebra , sum , product

(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)$

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: reciprocal sum , sum , irreducible , divisor , number theory

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: sum , combinatorics , number theory , prime

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

2016 Costa Rica - Final Round, F2

Tags: algebra , sum , radical

Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$ Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: sum , sequence , algebra

Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum $$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum

2015 Kyiv Math Festival, P3

Tags: number theory , sum , positive integer

Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

IV Soros Olympiad 1997 - 98 (Russia), 11.10

Tags: algebra , trigonometry , sum

Let $a_n = \frac{\pi}{2n}$, where $n$ is a natural number. Prove that for any $k = 1$,$2$,$...$, $n$ holds the equality $$\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn$$

2017 Purple Comet Problems, 28

Tags: sum , algebra

Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite 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 $m + n$.