Olympiad problems

2013 Korea Junior Math Olympiad, 3

Tags: algebra , sequence , sum , positive integer , diophantine equation , recurrence relation

$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, define as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

2013 IFYM, Sozopol, 4

Tags: number theory , algebra , sum

Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find $\max S=\sum_{i\mid j}a_i a_j $.

1952 Moscow Mathematical Olympiad, 213

Tags: algebra , integer , sum , geometric sequence

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.

2008 Danube Mathematical Competition, 3

Tags: geometry , vector , geometric inequality , sum , semicircle , projection

On a semicircle centred at $O$ and with radius $1$ choose the respective points $A_1,A_2,...,A_{2n}$ , for $n \in N^*$. The lenght of the projection of the vector $\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}$ on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least $1$.

2001 Estonia National Olympiad, 5

Tags: combinatorics , magic square , difference , sum , square table

A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column. (a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table. (b) Show that there is such a table not all of whose entries are $0$.

2013 India PRMO, 2

Tags: sum , algebra

Let $S_n=\sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}$. What is the value of $\sum_{n=1}^{99}\frac{1}{S_n+S_{n-1}}$ ?

2014 IFYM, Sozopol, 4

Tags: combinatorics , sum , permutation

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 $d(a)$.

2020 Malaysia IMONST 1, 1

Tags: arithmetic , imonst , sum

Find the value of \[+1+2+3-4-5-6+7+8+9-10-11-12+\cdots -2020,\] where the sign alternates between $+$ and $-$ after every three numbers.

1984 All Soviet Union Mathematical Olympiad, 371

Tags: divisible , product , sum , number theory , combinatorics

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.

1989 Romania Team Selection Test, 1

Tags: number theory , sum , combinatorics

Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.

2006 BAMO, 2

Tags: number theory , algebra , consecutive , sum

Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers. (a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not. (b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.

1957 Moscow Mathematical Olympiad, 363

Tags: number theory , sum , sequence , fibonacci

Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.

1994 Poland - Second Round, 2

Tags: inequalities , sum , product , algebra

Let $a_1,...,a_n$ be positive real numbers such that $\sum_{i=1}^n a_i =\prod_{i=1}^n a_i $ , and let $b_1,...,b_n$ be positive real numbers such that $a_i \le b_i$ for all $i$. Prove that $\sum_{i=1}^n b_i \le\prod_{i=1}^n b_i $

2019 Saint Petersburg Mathematical Olympiad, 5

Tags: algebra , combinatorics , number theory , sum

Call the [i]improvement [/i] of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$. Prove that you can erase some of them, and [i]improve [/i] each of the other numbers so that the sum the resulting numbers were again $2^{100}$.

2003 Abels Math Contest (Norwegian MO), 2b

Tags: number theory , sum , inequalities

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

2009 Kyiv Mathematical Festival, 5

Tags: geometry , distance , sum , area of triangle

Assume that a triangle $ABC$ satisfies the following property: For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$. Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$

2017 Argentina National Olympiad, 2

Tags: number theory , sum , consecutive , algebra

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.

1988 Tournament Of Towns, (186) 3

Tags: number theory , sum , divide , divisible

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

1988 Mexico National Olympiad, 7

Tags: sum , subset , combinatorics

Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.

2020 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , inequalities , nonnegative , sum

In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number. Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.

1998 Belarus Team Selection Test, 1

Tags: number theory , inequalities , sum , divisor

Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$). Prove that $(S(n))^3 <n^4$.

2013 Tournament of Towns, 2

Tags: algebra , sum

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.

2011 Indonesia TST, 1

Tags: sum , inequalities , algebra

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

2012 IFYM, Sozopol, 6

Tags: algebra , binomial sum , sum , calculate

Calculate the sum $1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$

2004 Greece JBMO TST, 2

Tags: algebra , sum

Real numbers $x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}$ differ from $1$ and are such that $x_ky_k=1$ for every $k=1,2,...,2004$. Calculate the sum $$S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}$$