## Saturday, October 21, 2006

### 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 $n \ge m \ge 1$.

#### 2 comments:

1. What is the "integer closest to" a positive integer supposed to mean?

2. Sorry. Typo.