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 3 of 5: Determine the polynomial identity

P(x)-x^2Q(x)=-\frac1{120}\prod_{k=1}^{5}(x^2-k^2)

Detailed analysis

Both sides have degree at most $10$ , so $P(x)-x^2Q(x)=A\prod_{k=1}^5(x^2-k^2)$ . Setting $x=0$ gives $A=P(0)/[(-1)(-4)(-9)(-16)(-25)]=-1/120$ .