Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

Compute the value of 1053−1043\sqrt{105^{3}-104^{3}}, given that it is a positive integer.

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

Solution

First compute 1053−1043=1052+105⋅104+1042=3⋅105⋅104+1=32761105^{3}-104^{3}=105^{2}+105 \cdot 104+104^{2}=3 \cdot 105 \cdot 104+1=32761. Note that 1802=32400180^{2}=32400, so 1812=1802+2⋅180+1=32761181^{2}=180^{2}+2 \cdot 180+1=32761 as desired.

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