MathDB

2

Part of 2000 IMO Shortlist

Problems(3)

3 numbers have their fractional parts lying in the interval

Source: IMO Shortlist 2000, A2

8/10/2008
Let a,b,c a, b, c be positive integers satisfying the conditions b>2a b > 2a and c>2b. c > 2b. Show that there exists a real number λ \lambda with the property that all the three numbers λa,λb,λc \lambda a, \lambda b, \lambda c have their fractional parts lying in the interval (13,23]. \left(\frac {1}{3}, \frac {2}{3} \right].
floor functionalgebrafractional partIMO Shortlist
Staircase-brick with 3 steps of width 2 using 12 unit cubes

Source: IMO Shortlist 2000, C2

8/10/2008
A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers n n for which it is possible to build a cube of side n n using such bricks.
3D geometrycombinatoricspackingColoringIMO Shortlist
number of divisors

Source: IMO Shortlist 2000, Problem N2

9/6/2003
For a positive integer nn, let d(n)d(n) be the number of all positive divisors of nn. Find all positive integers nn such that d(n)3=4nd(n)^3=4n.
number theoryDivisorsIMO Shortlist