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A4

Part of 1950 Putnam

Problems(1)

Putnam 1950 A4

Source:

5/24/2022
Answer either (i) or (ii).
(i) In a right prism with triangular base, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base), show that these faces are of equal area and perpendicular to each other when the volume attains its maximum.
(ii) Show that x1+x31⋅3+x51⋅3⋅5+x71⋅3⋅5⋅7+⋯1+x22+x42⋅4+x62⋅4⋅6+⋯=∫0xe−t2dt. \frac{\frac x1 + \frac {x^3} {1 \cdot 3} + \frac {x^5} {1 \cdot 3 \cdot 5} + \frac {x^7} {1 \cdot 3 \cdot 5 \cdot 7} + \cdots }{1 + \frac {x^2} 2 + \frac {x^4}{2 \cdot 4} + \frac{x^6}{2 \cdot 4 \cdot 6} + \cdots} = \int_0^x e^{-t^2} \mathrm dt.
Putnam