2. Problem: Is it possible to pair up the numbers $0,1,2,3, \ldots, 61$ in such a way that when we sum each pair, the product of the 31 numbers we get is a perfect fifth power?
Solution: (Jamie Craik)
Claim: It can be achieved. One way to achieve it is:
- to pair 0 with 1 , and
- to pair $k$ with $(63-k)$ for $k=2,3,4, \ldots, 31$.

This would result in a product equal to:
$$
(0+1) \prod_{k=2}^{31}(k+(63-k))=1 \times 63^{30}=\left(63^{6}\right)^{5}
$$
which is a perfect $5^{\text {th }}$ power of $63^{6}$.
Note that there are many ways to solve this problem, for example:
$$
\begin{aligned}
(0+1) & \times(2+11)(3+10)(4+9)(5+8)(6+7) \\
& \times(12+21)(13+20)(14+19)(15+18)(16+17) \\
& \times(22+31)(23+30)(24+29)(25+28)(26+27) \\
& \times(32+41)(33+40)(34+39)(35+38)(36+37) \\
& \times(42+51)(43+50)(44+49)(45+48)(46+47) \\
& \times(52+61)(53+60)(54+59)(55+58)(56+57) \\
& =1 \times 13^{5} \times 33^{5} \times 53^{5} \times 73^{5} \times 93^{5} \times 113^{5} .
\end{aligned}
$$