Olympiad problems

1979 All Soviet Union Mathematical Olympiad, 272

Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain : a) $1/5$, b) an arbitrary rational number between $0$ and $1$.

2021 Balkan MO Shortlist, N1

Tags: number theory , shortlist , arithmetic mean

Let $n \geq 2$ be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\] be the set of the arithmetic means of the elements of all non-empty subsets of $\{1, 2, ..., n\}$. Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]

2010 Contests, 4

Tags: statistics , mean , median , mode , arithmetic mean

What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$? $ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $

2020 IMO, 5

Tags: arithmetic mean , geometric mean , number theory

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

1991 All Soviet Union Mathematical Olympiad, 536

Tags: combinatorics , game , arithmetic mean , game strategy

$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$.

1975 Bundeswettbewerb Mathematik, 3

Tags: number theory , arithmetic mean , geometric mean

For $n$positive integers $ x_1,x2,...,x_n$, $a_n$ is their arithmetic and $g_n$ the geometric mean. Consider the statement $S_n$: If $a_n/g_n$ is a positive integer, then $x_1 = x_2 = ··· = x_n$. Prove $S_2$ and disprove $S_n$ for all even $n > 2$.

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , combinatorics , arithmetic mean , set

Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$