MathDB

2021 EGMO

Part of EGMO

Subcontests

(6)
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2021 EGMO P6: floor(m/1) + ... + floor(m/m) = n^2 + a

Does there exist a nonnegative integer aa for which the equation ⌊m1⌋+⌊m2⌋+⌊m3⌋+⋯+⌊mm⌋=n2+a\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a has more than one million different solutions (m,n)(m, n) where mm and nn are positive integers?
The expression ⌊x⌋\lfloor x\rfloor denotes the integer part (or floor) of the real number xx. Thus ⌊2⌋=1,⌊π⌋=⌊22/7⌋=3,⌊42⌋=42,\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42, and ⌊0⌋=0\lfloor 0 \rfloor = 0.
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