Olympiad problems

IV Soros Olympiad 1997 - 98 (Russia), 11.10

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

2018 Singapore Junior Math Olympiad, 4

Tags: factor , sum , number theory , divisor

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

1978 Bundeswettbewerb Mathematik, 3

Tags: division , sum , power of 2 , number theory

For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$

2007 Junior Balkan Team Selection Tests - Moldova, 4

Tags: average , sum , combinatorics

The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?

1995 Poland - Second Round, 4

Tags: inequalities , algebra , n-variable inequality , sum

Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ . Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.

1989 Dutch Mathematical Olympiad, 3

Tags: sum , algebra

Calculate $$\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$

1992 Chile National Olympiad, 2

Tags: number theory , algebra , sum

For a finite set of naturals $(C)$, the product of its elements is going to be noted $P(C)$. We are going to define $P (\phi) = 1$. Calculate the value of the expression $$\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}$$

2012 Chile National Olympiad, 2

Tags: number theory , sum , 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 $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

2019 Durer Math Competition Finals, 15

Tags: diophantine , diophantine equation , number theory , sum

The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

2010 Singapore Junior Math Olympiad, 3

Tags: positive integer , sum , combinatorics

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , divisor , sum , divide , combinatorics

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.

2008 Indonesia TST, 2

Tags: algebra , inequalities , sequence , sum

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

1945 Moscow Mathematical Olympiad, 099

Tags: combinatorics , algebra , sum , number theory , 4-digit

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

2016 India Regional Mathematical Olympiad, 1

Tags: sum , number theory , algebra

Suppose in a given collection of $2016$ integer, the sum of any $1008$ integers is positive. Show that sum of all $2016$ integers is positive.

1937 Moscow Mathematical Olympiad, 033

Tags: sum , geometry , find point

* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.

1998 Tuymaada Olympiad, 1

Tags: algebra , sum , reciprocal sum

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

2006 Dutch Mathematical Olympiad, 3

Tags: sum , minimum , arithmetic progression , algebra

$1+2+3+4+5+6=6+7+8$. What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?

2020 Kyiv Mathematical Festival, 1.2

Tags: sum , combinatorial sum , algebra

Prove that (a) for each $n \ge 1$ $$\sum_{k=0}^n C_{n}^{k} \left(\frac{k}{n}-\frac{1}{2} \right)^2 \frac{1}{2^n}=\frac{1}{4n}$$ (b) for every n \ge m \ge 2 $$\sum_{\ell=0}^n \sum_{k_1+...+k_n=\ell,k_i=0,...,m} \frac{\ell!}{k_1!...k_n!} \frac{1}{(m+1)^n} \left(\frac{\ell}{n}-\frac{m}{2} \right)^2= \left(\frac{m^3-3m^2}{12(m+1)}+\frac{m}{2}-\frac{m}{3(m+1)}\right)n$$

1935 Moscow Mathematical Olympiad, 020

Tags: combinatorics , sum

How many ways are there of representing a positive integer $n$ as the sum of three positive integers? Representations which differ only in the order of the summands are considered to be distinct.

1967 Swedish Mathematical Competition, 4

Tags: algebra , sum , limit

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.

2018 Saudi Arabia BMO TST, 2

Tags: combinatorics , sum , product

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

2014 Greece JBMO TST, 4

Tags: set theory , subset , partition , sum , set , combinatorics

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

2009 Postal Coaching, 3

Tags: function , sum , recurrence relation , combination

Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions (i) $f(0, 0) = 1$, $f(0, 1) = 1$ (ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and (iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of $$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$

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. $17$

1976 Spain Mathematical Olympiad, 2

Tags: vector , sum , algebra

Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.