5

Part of 2012 Saint Petersburg Mathematical Olympiad

Problems(3)

Number of Divisors

Source: St Petersburg Olympiad 2012, Grade 11, P5

n \geq k

-two natural numbers.

S

-such natural, that have not less than

n

divisors. All divisors of

S

are written in descending order. What minimal number of divisors can have number from

k

Divisors of divisor

Source: St Petersburg Olympiad 2012, Grade 10, P5

S

is natural, and

S=d_{1}>d_2>...>d_{1000000}=1

are all divisors of

S

. What minimal number of divisors can have

d_{250}

Numbers in square

Source: St Petersburg Olympiad 2012, Grade 9, P5

100 \times 100

table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?

combinatoricsnumber theory