(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 -> ∞ ( Σ 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.
(3a) Prove or disprove:
3⋅3 + 4⋅4 = 5⋅5
3⋅3⋅3 + 4⋅4⋅4 + 5⋅5⋅5 = 6⋅6⋅6
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.
(Names not etymologically-related, and interesting -- not just a color or occupation, say.)
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)
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