Racing certainties: solution

If we discount the favourite (with odds of 1 to 1) and place none of our stake money on him then we are really betting on a three-horse race where Q is equal to $$ Q=\frac{1}{4}+\frac{1}{8}+\frac{2}{5}=\frac{31}{40}<1, $$ and by betting $\frac{1}{4}$ of our stake money on runner 1, $\frac{1}{8}$ on runner 2, and $\frac{2}{5}$ on runner 3 we are guaranteed a minimum return of $\frac{40}{31}-1=\ \frac{9}{31}$ of our total stake in addition to our original stake money!

In the case of $a_{i}=i(i+2)-1$ and an infinite number of runners ($N=\infty $) we have $$ Q=\sum_{i=1}^{N}\frac{1}{a_{i}+1}=\sum_{i=1}^{\infty }\frac{1}{i(i+2)}= \frac{3}{4}. $$