Olympiad problems

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2017 Miklós Schweitzer, 10

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$. Let $Y_1$, $Y_2$, $Y_3$, and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$. Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.

2023 Olympic Revenge, 6

Tags: group theory , algebra , random

We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that $$|\bigcap_{H'\in S} H'|>1$$

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