MathDB

30

Part of 2019 India PRMO

Problems(2)

Set / Equal Sums

Source:

8/11/2019
Let EE denote the set of all natural numbers nn such that 3<n<1003 < n < 100 and the set {1,2,3,…,n}\{ 1, 2, 3, \ldots , n\} can be partitioned in to 33 subsets with equal sums. Find the number of elements of EE.
Setssums
Floors and Fracs

Source: PRMO Leg 2 P30

8/25/2019
For any real number xx, let ⌊x⌋\lfloor x \rfloor denote the integer part of xx; {x}\{ x \} be the fractional part of xx ({x}\{x\} == x−x- ⌊x⌋\lfloor x \rfloor). Let AA denote the set of all real numbers xx satisfying {x}=x+⌊x⌋+⌊x+(1/2)⌋20\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20} If SS is the sume of all numbers in AA, find ⌊S⌋\lfloor S \rfloor
number theoryfloor functionPRMO