Olympiad problems

1984 All Soviet Union Mathematical Olympiad, 386

Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.

1976 All Soviet Union Mathematical Olympiad, 224

Tags: 3-digit , cube , combinatorics

Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?

1940 Moscow Mathematical Olympiad, 062

Tags: 3-digit , factorial , number theory

Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

1945 Moscow Mathematical Olympiad, 096

Tags: number theory , power , 3-digit

Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

1998 Tournament Of Towns, 4

Tags: number theory , 3-digit , sum , product , digit

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

1954 Moscow Mathematical Olympiad, 263

Tags: ratio , maximum , 3-digit , sum of digits

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

1969 All Soviet Union Mathematical Olympiad, 122

Tags: number theory , digit , 3-digit

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

1974 All Soviet Union Mathematical Olympiad, 201

Tags: number theory , digit , 3-digit

Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.