Two Floor Sums Adding Up

Problem

Let $a$ and $b$ be coprime positive integers with $a+b=2005$ . Let $[x]$ denote the integer part of $x$ , and set

A=\left[\frac{2005\times 1}a\right]+\left[\frac{2005\times 2}a\right]+\cdots+\left[\frac{2005\times a}a\right],

B=\left[\frac{2005\times 1}b\right]+\left[\frac{2005\times 2}b\right]+\cdots+\left[\frac{2005\times b}b\right].

Find $A+B$ .

Answer

Solution

Difficulty	8/10
Topics	number theory, Counting, Symmetry

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