(1) Prove or disprove:

The product of any pair of twin primes, when increased by 1, always gives a perfect square.

(2) Let γ be the Euler Gamma --

γ = lim _{n -> ∞} ( Sum _{k = 1...n} 1/k – log n ) = 0.5772156649...

Prove or disprove:

π^{ log γ} = γ^{ log π}

(3) Prove or disprove:

1 + 2 = 3

1*2 + 2*3 + 3*4 = 4*5

1*2*3 + 2*3*4 + 3*4*5 + 4*5*6 = 5*6*7

etc.

(4) 2 2 2 3 2 2 4 2 3 ?

(5) Compute 1/49 to 12 digits, *in your head.*

(6) Consider the altitudes of a non-degenerate triangle as vectors (say, vertex-to-side). Divide each by the square of its length. Prove or disprove: the resulting vectors add up to 0.

(7) Name two famous mathematicians whose names are synonymous.

Bonus true story:

Grading papers in a graph-theory course. Problem: prove that if G is not connected, then the complement of G is connected.

"We use a contrapositive proof..."

Bonus joke, not by me, and most likely unintentional:

"We beat [the tin can] out flat; we beat it back square; we battered it into every form known to geometry -- but we could not make a hole in it."

(Jerome K. Jerome, *Three Men in a Boat*)

© 1985-2014, S.E.

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