Olympiad problems

2003 Junior Balkan Team Selection Tests - Romania, 3

A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.

1993 Czech And Slovak Olympiad IIIA, 4

Tags: Sum of powers , Digits , Sum , number theory , Sequence

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

1988 All Soviet Union Mathematical Olympiad, 484

Tags: trigonometry , system of equations , minimum , algebra , Sum

What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?

2003 Estonia National Olympiad, 4

Tags: number theory , Sum , reciprocal , Divisors

Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

2010 Argentina National Olympiad, 4

Tags: algebra , Sum

Find the sum of all products $a_1a_2...a_{50}$ , where $a_1,a_2,...,a_{50}$ are distinct positive integers, less than or equal to $101$, and such that no two of them add up to $101$.

2013 IMAC Arhimede, 6

Tags: inequalities , algebra , Sums and Products , Product , Sum

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

1988 Swedish Mathematical Competition, 3

Tags: inequalities , Sum , algebra

Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$. Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.

2015 Romania Team Selection Test, 5

Tags: maximization , algebra , inequalities , Sum

Given an integer $N \geq 4$, determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.

2000 Junior Balkan Team Selection Tests - Moldova, 2

Tags: number theory , least common multiple , Sum

The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$. Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.

1999 Tournament Of Towns, 1

Tags: Sum , algebra

In a row are written $1999$ numbers such that except the first and the last , each is equal to the sum of its neighbours. If the first number is $1$, find the last number. (V Senderov)

2011 Belarus Team Selection Test, 3

Tags: algebra , number theory , Sum

Any natural number $n, n\ge 3$ can be presented in different ways as a sum several summands (not necessarily different). Find the greatest possible value of these summands. Folklore

2014 Danube Mathematical Competition, 4

Tags: algebra , combinatorics , Sum

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

2003 Austrian-Polish Competition, 8

Tags: inequalities , Sum , algebra

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

2020 LIMIT Category 2, 18

Tags: trigonometry , Sum , limit , calculus

Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$

2013 Estonia Team Selection Test, 3

Tags: inequalities , Sum , algebra

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

1978 All Soviet Union Mathematical Olympiad, 267

Tags: algebra , Sum , inequalities , Average

Given $a_1, a_2, ... , a_n$. Define $$b_k = \frac{a_1 + a_2 + ... + a_k}{k}$$ for $1 \le k\le n.$ Let $$C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$$ Prove that $C \le D \le 2C$.

1956 Moscow Mathematical Olympiad, 335

Tags: algebra , combinatorics , Sum

a) $100$ numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive. Can the sum of all underlined numbers be (i) negative? (ii) equal to zero? b) $n$ numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.

2001 Estonia Team Selection Test, 4

Tags: Sum , Product , algebra , combinatorics

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

1974 Poland - Second Round, 2

Tags: algebra , Sum , inequalities

Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if $$ \sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.$$

2005 Singapore Senior Math Olympiad, 3

Tags: Sum , Subsets , combinatorics

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 $x+y=u+v$

2019 Tournament Of Towns, 4

Tags: combinatorics , Coloring , Sum , Product

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)

2019 Tournament Of Towns, 1

Tags: combinatorics , Sum , consecutive , Tournament of Towns , ToT

Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers? (Alexandr Shapovalov)

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , Digits , Sum , multiple

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

2020 Nordic, 3

Tags: trisector , Sum , area of a triangle , areas , geometry , diagonal

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

2018 Estonia Team Selection Test, 3

Tags: Sum , algebra , inequalities , max , min

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.