Is it possible to equally divide the set {$1, 2, 3, \cdots, 972$} into 12 non-intersect subsets so that each subset has exactly 81 elements, and the sums of those subsets are all equal?

Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

Suppose m and n are positive odd integers. Which of the following must also be an odd integer?

Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.

Prove: it is impossible to have two positive integers such that the product of their sum and their difference equals 1990.

17 people attend a party. Prove: it is impossible that everyone exactly shakes hands with 3 other attendees.

Let $n>1$ be a positive integer. Prove $1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}$ cannot be a whole integer.

Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?

Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2019$, inclusive. What is the probability that $(ad-bc)$ is even?

Let $a$, $b$, and $c$ be three odd integers. Prove the equation $ax^2 + bx + c=0$ does not have rational roots.

Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

Numbers $1,2,\cdots, 1974$ are written on a board. You are allowed to replace any two of these numbers by one number which is either the sum or the difference of these numbers. Show that after 1973 times performing this operation, the only number left on the board cannot be $0$.

On an $8\times 8$ chess board, there are $32$ white pieces and $32$ black pieces, one piece in each square. If a player can change all the white pieces to black and all the black pieces to white in any row or column in a single move, then is it possible that after finitely many movies, there will be exactly one black piece left on the board?

Four $x$'s and five $o$'s are written around the circle in an arbitrary order. If two consecutive symbols are the same, then a new $x$ is inserted in between. Otherwise, a new $o$ is inserted. After nine new symbols are inserted, the previous 9 old ones are erased. Is it possible to get nine $o$'s after having repeated this operation for a finite time?