Olympiad problems

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Found problems: 2

2017 USA TSTST, 2

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses. For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word. Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses? (The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.) [i]Proposed by Kevin Sun

2013 Princeton University Math Competition, 3

Tags: inequalities , evan orz , 1434 , xooks , i lost the game , otis

A graph consists of a set of vertices, some of which are connected by (undirected) edges. A [i]star[/i] of a graph is a set of edges with a common endpoint. A [i]matching[/i] of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph $G$ is larger than $2(k-1)^2$, then $G$ contains a matching of size $k$ or a star of size $k$.