19. Darts (from 9th grade. 5 points). According to one implausible legend, Cauchy ${ }^{5}$ and Bunyakovsky ${ }^{6}$ loved to play darts in the evenings. But their target was unusual - the sectors on it were unequal, so the probabilities of hitting different sectors were not the same. One day, Cauchy threw a dart and hit the target. Next, Bunyakovsky throws. What is more likely: that Bunyakovsky will hit the same sector that Cauchy hit, or that he will hit the next sector clockwise?
Solution. Let the probabilities of hitting the sectors be $p_{1}, p_{2}, \ldots, p_{n}$ (indices indicate the sector numbers). The probability that Bunyakovsky will hit the same sector as Cauchy is

$$
p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+\ldots+p_{n}^{2}
$$

and the probability that Bunyakovsky will hit the next sector clockwise is

$$
p_{1} p_{2}+p_{2} p_{3}+\ldots+p_{n-1} p_{n}+p_{n} p_{1}
$$

It remains to use the Cauchy-Bunyakovsky inequality:

$$
\begin{aligned}
p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+\ldots+p_{n}^{2}= & \sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+\ldots+p_{n-1}^{2}+p_{n}^{2}} \cdot \sqrt{p_{2}^{2}+p_{3}^{2}+p_{4}^{2}+\ldots+p_{n}^{2}+p_{1}^{2}} \geq \\
& \geq p_{1} p_{2}+p_{2} p_{3}+\ldots+p_{n-1} p_{n}+p_{n} p_{1}
\end{aligned}
$$

with equality only if $p_{1}=k p_{2}, p_{2}=k p_{3}, \ldots, p_{n-1}=k p_{n}, p_{n}=k p_{1}$ for some $k$. Equality is possible only if $k=1$ (then all probabilities are the same) or $k=0$ (all probabilities are zero). Both these options contradict the condition. Therefore,

$$
p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+\ldots+p_{n}^{2}>p_{1} p_{2}+p_{2} p_{3}+\ldots+p_{n-1} p_{n}+p_{n} p_{1}
$$

Answer: it is more likely that in the same.[^4]


[^0]:    ${ }^{1}$ The plot of the problem was suggested by Boris Frenkin

[^1]:    ${ }^{2}$ This and other voting paradoxes, as well as the famous Arrow's theorem, are described in detail in the book "Mathematics of Elections" (R.E. Klima, J.K. Hodg. Moscow, MCCME, 2007, translated by N.A. Shikhova.

[^2]:    ${ }^{3}$ Author Evgeny Smirnov

[^3]:    ${ }^{4}$ James Stirling - Scottish mathematician of the 18th century.

[^4]:    ${ }^{5}$ Augustin-Louis Cauchy - French mathematician of the 19th century.

    ${ }^{6}$ Viktor Yakovlevich Bunyakovsky - Russian mathematician of the 19th century.