Say we are given the odds of an upcoming horse race and we want to know how to bet in order to win no matter the outcome. Essentially, is there a way to bet on every horse and still turn a profit? Assume no scratches or ties, etc. Just a simple win money if horse you bet on wins, lose money if horse you bet on loses.
The (fixed) odds for the respective horses are the following:
$$a_1=\frac{9}{2}$$ $$a_2= 8$$ $$a_3=4$$ $$a_4=4$$ $$a_5=4$$ $$a_6=\frac{9}{2}$$ $$a_7=19$$ $$a_8=7$$
My thought was to do the following. Let $x_i$, where $i \in \{1,2,\dots,8\}$, be the amount bet on the horse $i$. If horse $1$ wins, then we want
$$a_1x_1-x_1-x_2-x_3-x_4-x_5-x_6-x_7-x_8 \gt 0$$
Similarly, if horse $2$ wins, we want
$$-x_1+a_2x_2-x_2-x_3-x_4-x_5-x_6-x_7-x_8 \gt 0$$
We continue in this manner for every horse. Hence, we have the following system of linear inequalities
$$\begin{cases} a_1x_1-x_1-x_2-x_3-x_4-x_5-x_6-x_7-x_8 \gt 0 \\ -x_1+a_2x_2-x_2-x_3-x_4-x_5-x_6-x_7-x_8 \gt 0 \\ -x_1-x_2+a_3x_3-x_3-x_4-x_5-x_6-x_7-x_8 \gt 0 \\ \vdots \\ -x_1-x_2-x_3-x_4-x_5-x_6-x_7+a_8x_8-x_8 \gt 0 \end{cases}$$
How I came up with theses inequalities is that I want the total amount I bet $$x_t=x_1+x_2+\cdots+x_8$$ to be less than my earnings $$a_ix_i \gt x_t.$$ I tried just solving the corresponding matrix but that does not work.
Lets define $f_i$ as the fraction of the total money bet on horse i: $$f_i \equiv \frac{x_i}{\sum x_i}$$ Then the conditions are $$a_if_i>1\ \ \forall i$$ And $$\sum f_i = 1$$ Assume for a moment that we want to get the same amount no matter which horse wins. Call the ratio of the winnings to the amount invested M. Then $$a_if_i -1 \equiv M \ \ \forall i$$ $$\Rightarrow f_i = \frac{M+1}{a_i}$$ And using the equation for the sum: $$\sum f_i = 1 = (M+1)\left(\sum\frac{1}{a_i}\right)$$ $$\Rightarrow M = \frac{1}{\sum \frac{1}{a_i}}-1$$ If we want the amount of money to be positive, we must have $$\frac{1}{\sum \frac{1}{a_i}}>1$$ $$\Rightarrow \sum \frac{1}{a_i} < 1$$
Now, to get this we had to assume that we won the same amount of money no matter what. But, if we change any of the $f_i$, then some $f_j$ must decrease, and we will make even less money in case horse j wins, so if we were losing money in every case before, we will be losing even more money now if horse j wins. So if we can't meet all the constraints when the outcomes are equal, we can't ever meet all the constraints. So this answer holds even without the assumption of equal outcomes.
In this case we have $$\sum \frac{1}{a_i} \approx 1.51$$ So there is no way to be guaranteed to win.