This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 32

1999 AMC 12/AHSME, 3
Tags: arithmetic mean
The number halfway between $ \frac {1}{8}$ and $ \displaystyle \frac {1}{10}$ is $ \textbf{(A)}\ \frac {1}{80} \qquad \textbf{(B)}\ \frac {1}{40} \qquad \textbf{(C)}\ \frac {1}{18} \qquad \textbf{(D)}\ \frac {1}{9} \qquad \textbf{(E)}\ \frac {9}{80}$

2016 India PRMO, 16
Tags: algebra , geometric mean , arithmetic mean , perfect square , number theory
For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

2023 Brazil Team Selection Test, 3
Tags: inequalities , inequality , arithmetic mean , geometric mean , max
Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$

2017 India PRMO, 15
Tags: maximum , average , arithmetic mean , algebra
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?

2020 IMO Shortlist, N3
Tags: number theory , arithmetic mean , geometric mean
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]

2010 AMC 8, 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 AMC 10, 2
Tags: arithmetic mean
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? $\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$