Found problems: 32
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 $1$.
1959 Putnam, B7
For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that
$\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$
$\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$
$\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$
Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$
2017 Bosnia And Herzegovina - Regional Olympiad, 4
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 $A_S$
1991 All Soviet Union Mathematical Olympiad, 536
$n$ 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 $\frac{1}{n}$.
1959 AMC 12/AHSME, 18
The arithmetic mean (average) of the first $n$ positive integers is:
$ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $
2014 JHMMC 7 Contest, 11
What number is exactly halfway between $\frac 1 6$ and $\frac 1 4$?
2017 India PRMO, 15
Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?