# 2015-8 all lines

Does there exist a subset $$A$$ of $$\mathbb{R}^2$$ such that $$\lvert A\cap L\rvert=2$$ for every straight line $$L$$?

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# 2014-11 Subsets of a countably infinite set

Prove or disprove that every uncountable collection of subsets of a countably infinite set must have two members whose intersection has at least 2014 elements.

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# 2011-16 Odd Sets with Even Intersection

Let A1, A2, A3, …, An be finite sets such that |Ai| is odd for all 1≤i≤n and |Ai∩Aj| is even for all 1≤i<j≤n. Prove that it is possible to pick one element ai in each set Ai so that a1, a2, …,an are distinct.

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Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying $$\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1$$. Prove that there is a one-to-one function from the set of all real numbers to S.