Download Worksheet 3 MATH 3283W Fall 2012

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.
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