Olympiad problems

2015 Ukraine Team Selection Test, 4

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

1990 Chile National Olympiad, 5

Tags: Sum , algebra

Determine a natural $n$ such that $$996 \le \sum_{k = 1}^{n}\frac{1}{k}$$

1952 Moscow Mathematical Olympiad, 213

Tags: geometric sequence , Integer , Sum , algebra

Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.

2004 VJIMC, Problem 2

Tags: Summation , Sum , real analysis

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

2000 Singapore Team Selection Test, 3

Tags: number theory , Sum

Let $n$ be any integer $\ge 2$. Prove that $\sum 1/pq = 1/2$, where the summation is over all integers$ p, q$ which satisfy $0 < p < q \le n$,$ p + q > n$, $(p, q) = 1$.

2000 Singapore Senior Math Olympiad, 3

Tags: number theory , inequalities , Sum , LCM

Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

2013 Tournament of Towns, 2

Tags: Sum , algebra

Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.

1986 All Soviet Union Mathematical Olympiad, 431

Tags: dodecagon , distance , Sum , geometry , combinatorial geometry

Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.

2018 Lusophon Mathematical Olympiad, 1

Tags: algebra , Sum

Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.

2018 Thailand Mathematical Olympiad, 8

Tags: Sum , algebra , maximum value , max

There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?

2019 Flanders Math Olympiad, 2

Tags: algebra , Sum

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

1990 Tournament Of Towns, (265) 3

Tags: number theory , divides , Sum

Find $10$ different positive integers such that each of them is a divisor of their sum (S Fomin, Leningrad)

1997 Abels Math Contest (Norwegian MO), 1

Tags: Perfect Square , Sum , number theory

We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that (a) $2t$ is [i]happy[/i], (b) $3t$ is not [i]happy[/i]

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , Sum , algebra , absolute value

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

2003 Junior Tuymaada Olympiad, 6

Tags: combinatorics , Sum , circle

On a circle, numbers from $1$ to $100$ are arranged in some order. We call a pair of numbers [i]good [/i] if these two numbers do not stand side by side, and at least on one of the two arcs into which they break a circle, all the numbers are less than each of them. What can be the total number of [i]good [/i] pairs?

1989 All Soviet Union Mathematical Olympiad, 499

Tags: irrational , Sum , Sum of powers , rational , algebra

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

2009 Greece JBMO TST, 1

Tags: number theory , Sum , algebra

One pupil has $7$ cards of paper. He takes a few of them and tears each in $7$ pieces. Then, he choses a few of the pieces of paper that he has and tears it again in $7$ pieces. He continues the same procedure many times with the pieces he has every time. Will he be able to have sometime $2009$ pieces of paper?

2012 Danube Mathematical Competition, 4

Tags: Sum , Subsets , set , combinatorics

Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$. Show that there are two distinct elements of $A$, having the same sum of their elements.

1992 Austrian-Polish Competition, 8

Tags: Product , Sum , algebra

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 $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

2018 Hanoi Open Mathematics Competitions, 4

Tags: number theory , remainder , Sum

A pyramid of non-negative integers is constructed as follows (a) The first row consists of only $0$, (b) The second row consists of $1$ and $1$, (c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure). Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$: A. $2$ B. $4$ C. $6$ D. $11$ E. $17$

1996 Estonia National Olympiad, 1

Tags: Sum , Product , Diophantine equation , diophantine , number theory

Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

2019 Tournament Of Towns, 5

Tags: Sequence , algebra , Sum , positive integers

Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence? (Alexandr Shapovalov)

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)

2014 IFYM, Sozopol, 4

Tags: combinatorics , permutations , Sum

Let $A$ be the set of permutations $a=(a_1,a_2,…,a_n)$ of $M=\{1,2,…n\}$ with the following property: There doesn’t exist a subset $S$ of $M$ such that $a(S)=S$. For $\forall$ such permutation $a$ let $d(a)=\sum_{k=1}^n (a_k-k)^2$ . Determine the smallest value of $d(a)$.

2005 Slovenia Team Selection Test, 4

Tags: combinatorics , Sum , Sequence , algebra

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.