Practice Makes Perfect. Topic: All. Level: AIME/Olympiad.

Some problems that I have done and not felt like writing solutions to...

Practice Problem #1: (2004 Putnam - A2) For $ i = 1, 2 $ let $ T_i $ be a triangle with side lengths $ a_i, b_i, c_i $, and area $ A_i $. Suppose that $ a_1 \le a_2 $, $ b_1 \le b_2 $, $ c_1 \le c_2 $ and that $ T_2 $ is an acute triangle. Does it follow that $ A_1 \le A_2 $?

Practice Problem #2: (2001 Putnam - B3) For any positive integer $ n $, let $ \langle n \rangle $ denote the integer closest to $ \sqrt{n} $. Evaluate

$ \displaystyle \sum_{n=1}^{\infty} \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n} $.

Practice Problem #3: (2000 Putnam - B2) Prove that the expression

$ \frac{gcd(m,n)}{n} (nCm) $

is an integer for all pairs of integers $ n \ge m \ge 1 $.