**The Rules. **These are way too many problems to consider.
Just pick a few problems you

like and play around with them. You are not allowed to try a problem that you
already

know how to solve. Otherwise, work on the problems you want to work on.

**The Hints.** Work in groups. Try small cases. Do examples. Look for patterns. Use
lots

of paper. Talk it over. Choose effective notation. Try the problem with
different numbers.

Work backwards. Argue by contradiction. Eat pizza. Modify the problem.
Generalize.

Don’t give up after five minutes. Don’t be afraid of a little algebra. Sleep on
it if need be.

Ask.

**Things to remember.** Sums of squares are non-negative. The arithmetic
mean-geometric

mean inequality (AM-GM): (Qai)1/n if the
are non-negative. The triangle

inequality (the shortest distance between two points is a straight line).
Cauchy’s inequality

. Lagrange multipliers.

1. (a) Find, without using calculus, the minimum surface area of a rectangular
box which

holds volume V. (Mark Lucianovic)

(b) Find, without using calculus, the minimum area of an open box — with no top
—

which holds volume V.

2. Determine themaximumvalue of (sinA_{1})(sinA_{2}) · · · (sinA_{n}) given that
(tanA_{1})(tanA_{2}) · · · (tanA_{n}) =

1. (Hint: don’t use calculus without thinking!) (Mark Lucianovic) [We discovered
that this

is hard for a # 2!]

3. Given a, b, c nonnegative real numbers such that (a + 1)(b + 1)(c + 1) ≤ 8.
Prove that

abc ≤ 1. (Mark Lucianovic)

4. If
are positive real numberswhose product is 1, show that
.

(Soren Galatius)

5. The polynomial 4x^{4} − ax^{3} + bx^{2} − cx + 5 has four positive (real) roots such
that

. Find them. (Mark Lucianovic)

6. Suppose a, b, c are positive real numbers. Prove that

(Soren Galatius)

7. Suppose are positive real numbers. Prove that

Hint: Don’t be afraid to introduce square roots. (Soren
Galatius)

8. (A number-theoretic inequality) Given integers 0 < a < b < c < d < e, prove
that

1/ lcm(a, b) + 1/ lcm(b, c) + 1/ lcm(c, d) + 1/ lcm(d, e) ≤15/16. Generalize to
integers

. (Mark Lucianovic)

9. (The Power Mean inequality) Suppose are positive real numbers.
Define the

kth power mean as

if k ≠ 0, and . Show that if a > b,
then , with equality if

and only if all the are equal. (This can be very handy! If k = 1, we get
the arithmetic

mean; if k = 0 we get the geometric mean; if k = −1 we get the harmonic mean; if
k = 2 we

get the quadratic mean.)

10. Prove the “logarithmic mean” inequality for a > b > 0:

(Mark Lucianovic)

11. Suppose f(x) is a continuous function [a, b] →
(a < b). Figure out what
the

integral version of the arithmetic mean should be (see the Power Mean statement
above).

State the integral version of the quadratic mean-arithmetic mean inequality.
Prove it!