Problem 2

Let

a_1,a_2,a_3,a_4,a_5

be real numbers satisfying

\frac{a_1}{k^2+1}+\frac{a_2}{k^2+2}+\frac{a_3}{k^2+3}+\frac{a_4}{k^2+4}+\frac{a_5}{k^2+5}=\frac1{k^2}

for

k=1,2,3,4,5

. Find the value of

\frac{a_1}{37}+\frac{a_2}{38}+\frac{a_3}{39}+\frac{a_4}{40}+\frac{a_5}{41}

(express the value in a single fraction).

Step 1 of 5: Encode the equations

R(x)=\sum_{j=1}^{5}\frac{a_j}{x^2+j},\quad R(\pm k)=\frac1{k^2}\ (1\le k\le5)

Detailed analysis

Define R(x)=a_1/(x^2+1)+⋯+a_5/(x^2+5). The given equations say R(±k)=1/k^2 for k=1,…,5 because R depends only on x^2.