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Part of 2009 APMO

Problems(1)

Sum of 5 fractions

Source: APMO 2009 Q.2

3/13/2009
Let a1 a_1, a2 a_2, a3 a_3, a4 a_4, a5 a_5 be real numbers satisfying the following equations: \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2} for k \equal{} 1, 2, 3, 4, 5 Find the value of \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41} (Express the value in a single fraction.)
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