HMMT November, 2013

Algebra Difficulty 4.2 AIME Find the answer United States

Problem:
Find the remainder when $1^{2} + 3^{2} + 5^{2} + \cdots + 99^{2}$ is divided by $1000$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
We have $S = \sum_{i=0}^{49} (2i+1)^{2} = \sum_{i=0}^{49} \left(4i^{2} + 4i + 1\right) = 4 \cdot \sum_{i=0}^{49} i^{2} + 4 \cdot \sum_{i=0}^{49} i + 50$ .

Now,
$\sum_{i=0}^{49} i = \frac{49 \cdot 50}{2} = 1225$ ,
$\sum_{i=0}^{49} i^{2} = \frac{49 \cdot 50 \cdot 99}{6} = 40425$ .

So,
$S = 4 \cdot 40425 + 4 \cdot 1225 + 50 = 161700 + 4900 + 50 = 166650$ .

Now, $166650 \bmod 1000 = 650$ .

Therefore, the remainder is $\boxed{650}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.