Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.

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?

Given $a_1, a_2, ... , a_n$. Define $$b_k = \frac{a_1 + a_2 + ... + a_k}{k}$$ for $1 \le k\le n.$ Let $$C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$$ Prove that $C \le D \le 2C$.

Let $S$ be a set of $n$ distinct real numbers. Let $A_{S}$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_{S}$?

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?

The positive integers from $ 1$ to $n$ inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining $n - 1$ numbers is $22$. Knowing that $n$ is odd, determine $n$ and the number that was erased. Explain your reasoning.

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.

A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.

A group of $100$ kids has a deck of $101$ cards numbered by $0, 1, 2,\dots, 100$. The first kid takes the deck, shuffles it, and then takes the cards one by one; when he takes a card (not the last one in the deck), he computes the average of the numbers on the cards he took up to that moment, and writes down this average on the blackboard. Thus, he writes down $100$ numbers, the first of which is the number on the first taken card. Then he passes the deck to the second kid which shuffles the deck and then performs the same procedure, and so on. This way, each of $100$ kids writes down $100$ numbers. Prove that there are two equal numbers among the $10000$ numbers on the blackboard.

A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour? (NN Konstantinov, Moscow)

The intelligence quotient (IQ) of a country is defined as the average IQ of its entire population. It is assumed that the total population and individual IQs remain constant throughout. (a) (i) A group of people from country $A$ has emigrated to country $B$ . Show that it can happen that as a result , the IQs of both countries have increased. (ii) After this, a group of people from $B$, which may include immigrants from $A$, emigrates to $A$. Can it happen that the IQs of both countries will increase again? (b) A group of people from country $A$ has emigrated to country $B$, and a group of people from $B$ has emigrated to country $C$ . It is known that a s a result , the IQs o f all three countries have increased. After this, a group of people from $C$ emigrates to $B$ and a group of people from $B$ emigrates to $A$. Can it happen that the IQs of all three countries will increase again? (A Kanel, B Begun)

Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.

On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where $x = \frac14 (a + b + c + d)$.

What number is exactly halfway between $\frac 1 6$ and $\frac 1 4$?

Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.

(a) On a blackboard are written $100$ different numbers. Prove that you can choose $8$ of them so that their average value is not equal to that of any $9$ of the numbers on the blackboard. (b) On a blackboard are written $100$ integers. For any $8$ of them, you can find $9$ numbers on the blackboard so that the average value of the $8$ numbers is equal to that of the $9$. Prove that all the numbers on the blackboard are equal. (A Shapovalov)

A list of integers with average $89$ is split into two disjoint groups. The average of the integers in the first group is $73$ while the average of the integers in the second group is $111$. What is the smallest possible number of integers in the original list? [i] Proposed by David Altizio [/i]