Olympiad problems

2010 Saudi Arabia BMO TST, 2

Tags: sum , algebra

Evaluate the sum $$1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 1 0 + . . . - 2010$$ , where each three consecutive signs $+$ are followed by two signs $-$.

1982 Austrian-Polish Competition, 9

Tags: algebra , min , sum , inequalities

Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$. Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$. (Note. The smaller $C$, the better the solution.)

2001 Estonia National Olympiad, 3

Tags: sum , angle , geometry

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]

2001 Estonia Team Selection Test, 4

Tags: algebra , sum , product , 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.

VMEO III 2006 Shortlist, N9

Tags: inequalities , number theory , sum

Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have $$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$

1987 Tournament Of Towns, (143) 4

Tags: geometry , algebra , chessboard , sum , distance

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)

2003 Junior Balkan Team Selection Tests - Moldova, 1

Tags: sum , perfect square , number theory

Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square. Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.

2018 Czech-Polish-Slovak Junior Match, 1

Tags: system of equations , algebra , sum

Are there four real numbers $a, b, c, d$ for every three positive real numbers $x, y, z$ with the property $ad + bc = x$, $ac + bd = y$, $ab + cd = z$ and one of the numbers $a, b, c, d$ is equal to the sum of the other three?

2015 Romania Team Selection Tests, 1

Tags: number theory , sum , divisibility , gcd

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 $y$ .

2016 Argentina National Olympiad, 2

Tags: algebra , sum , compare

For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$ .

2022 Indonesia TST, C

Tags: combinatorics , sum , equal , choose

Five numbers are chosen from $\{1, 2, \ldots, n\}$. Determine the largest $n$ such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).

Estonia Open Junior - geometry, 2009.2.1

Tags: rectangle , distance , sum , min , geometry

A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?

2021 Ukraine National Mathematical Olympiad, 5

Tags: number theory , sum

Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others. (Bogdan Rublev)

2019 India PRMO, 30

Tags: set , sum

Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.

1999 Estonia National Olympiad, 2

Tags: algebra , sum

Find the value of the expression $$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$ assuming $f(x) =\frac{x^2}{1 + x^2}$ .

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

2007 Singapore Junior Math Olympiad, 4

Tags: number theory , greatest common divisor , least common multiple , sum , difference , product

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.

2019 Flanders Math Olympiad, 2

Tags: sum , algebra

Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.

1999 Tournament Of Towns, 1

Tags: divisible , divide , sum , combinatorics

For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

2012 Chile National Olympiad, 2

Tags: sum , number theory , inequalities

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

1993 Austrian-Polish Competition, 3

Tags: number theory , power of prime , product , sum

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.

1985 Tournament Of Towns, (102) 6

Tags: sequence , sum , integer part , recurrence relation , algebra

The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

1995 Czech And Slovak Olympiad IIIA, 2

Tags: sum , integer , algebra

Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.

2019 SAFEST Olympiad, 4

Tags: minimum , inequalities , algebra , sum

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

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics , prime , sum , number theory

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 $ab \ne cd$.