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3

Part of 2003 Putnam

Problems(2)

Putnam 2003 A3

Source:

6/22/2011
Find the minimum value of ∣sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x∣|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}| for real numbers xx.
Putnamtrigonometryfunctionalgebradomaincalculusabsolute value
Putnam 2003 B3

Source:

6/23/2011
Show that for each positive integer n, n!=∏i=1n  lcm  {1,2,…,⌊ni⌋}n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\} (Here lcm denotes the least common multiple, and ⌊x⌋\lfloor x\rfloor denotes the greatest integer ≤x\le x.)
Putnamnumber theoryleast common multiplefloor functionprime factorizationcollege contests