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Unformatted text preview: every subset of 3 attendees spoke a common language. Prove that some language was spoken by at least 200 of the people at the meeting. 3. Prove that there exist integers a, b, c , not all zero and each of absolute value less than one million, such that  a + b √ 2 + c √ 3  < 1011 . 4. Given two relatively prime positive integers a, b and an integer m ≥ ( a1)( b1), prove that there exist nonnegative integers x, y such that m = ax + by . 5. A set S of integers is called “sumfree” if there do not exist x, y, z ∈ S with x + y = z . What is the maximum size of a sumfree subset of { 1 , 2 , 3 , . . . , 2 n + 1 } ? 6. Let B be a set of more than 2 n +1 n distinct points with coordinates of the form ( ± 1 , ± 1 , . . . , ± 1) in R n , with n ≥ 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle. 2...
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 Spring '08
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 Math, Natural number, Prime number

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