5
Part of 1996 All-Russian Olympiad
Problems(3)
arithmetic progression with infinitely many powers of 10.
Source: All-Russian Olympiad 1996, Grade 9, Second day, Problem 5
4/18/2013
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.L. Kuptsov
ratioEulerarithmetic sequencenumber theory proposednumber theory
cube with numbers
Source: All-Russian Olympiad 1996, Grade 10, Second Day, Problem 5
4/18/2013
At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges?A. Shapovalov
geometry3D geometrygreatest common divisornumber theory proposednumber theory
three natural numbers
Source: All-Russian Olympiad 1996, Grade 11, Second Day, Problem 5
4/19/2013
Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others?A. Golovanov
number theory proposednumber theory