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Problem 1022

AMC 12 late, AIME early

Algebra Difficulty 4.6 Find the answer HMMT February

Compute the value of $\sqrt{105^{3}-104^{3}}$ , given that it is a positive integer.

Your answer

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

Official solution

First compute $105^{3}-104^{3}=105^{2}+105 \cdot 104+104^{2}=3 \cdot 105 \cdot 104+1=32761$ . Note that $180^{2}=32400$ , so $181^{2}=180^{2}+2 \cdot 180+1=32761$ as desired.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.