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Transcript
MATH 3283W Worksheet 3 Fall 2012 I am ready to swear that Masha [...] was a real beauty, but I don’t know how to prove it. Anton Chekhov, The Beauties 1. Let n be an integer. Prove that n3 − n is divisible by 6. 2. Let a1 , . . . , a22 be integers such that a1 · · · · · a22 = 1. Prove that a1 + · · · + a22 6= 0. 3. Prove that log2 3 is irrational. 4. Let b1 , b2 , . . . , b2012 be real numbers such that the sum of any five of them is positive. Prove that the sum of all of these numbers is positive. 5. Prove that 12 + 22 + · · · + 20122 is not divisible by 3. 6. Prove that for any real numbers x, y, z the inequality x2 + y 2 + z 2 ≥ xy + yz + xz holds. 7. There are thirty people at a party. Prove that (at least) two of them have the same number of friends there. 8. Prove that a triangle with all sides of different length cannot be cut into two equal triangles. 2