HMMT November

Algebra Difficulty 4.8 AIME Find the answer

Find the number of ordered triples of integers $(a, b, c)$ with $1 \leq a, b, c \leq 100$ and $a^{2} b+b^{2} c+c^{2} a=a b^{2}+b c^{2}+c a^{2}$

Your answer

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

Solution

This factors as $(a-b)(b-c)(c-a)=0$ . By the inclusion-exclusion principle, we get $3 \cdot 100^{2}-3 \cdot 100+100=29800$ .

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