MathDB

Number Theory identity

Source: 239 School Open MO, 2023, Senior league, Problem 3

April 1, 2023

number theoryresiduefloor function

Problem Statement

Let

n>1

be a natural number and

x_k{}

be the residue of

n^2

modulo

\lfloor n^2/k\rfloor+1

for all natural

k{}

. Compute the sum

\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.