Low Key Lyesmith

If an army X miles long advances Y miles while a dispatch rider gallops from the rear to the front, delivers a dispatch to the commanding general, and returns to the rear, how far has he to travel?

(Adapted from a problem by H.E. Dudeney. I am looking for elegant solutions that require a minimum of calculation.)

Let ABC be a triangle such that the degree measure of each angle is a rational number. Prove that ABC can be dissected into a finite number of isosceles triangles.

Let AB be a horizontal line segment.

Let P be a point on the segment AB.

Let C be a circle of radius r which lies above AB and which is tangent to AB at P.

Locate a point X so that AX and BX are tangent to C.

Thus C is an inscribed circle for the triangle ABX.

Now, the location of X depends on the radius r. Can you described the path along which X travels, as r varies?

Is it true that for all positive integers m, n, p > 1, there exists a nonabelian group G and elements x, y in G such that the order of x is m, the order of y is n, and the order of xy is p? If not, what restrictions exist between m, n, and p?

Find all pairs of positive integers such that m > n and 2^m - 2^n divides 3^m - 3^n.

A pentagon with perimeter p is formed by placing an isosceles triangle on top of a rectangle. Find the maximum area of the pentagon, as a function of p.

Show that for every positive integer n there exists a polynomial function f of degree n such that (f(x))^2 - 2 = f(x^2-2).

He said that he was 894th out of 899 students. According to my math, this makes him sixth from the bottom.

What is the maximum possible height of a stack of 100 bowls? Assume that the bowls are perfect hemispheres with negligible thickness. The bowls can have different sizes, but the bowl at the base of the stack has radius 1.

I know the answer already. This is based on an exercise in Stewart's Calculus.

64 players take part in a single-elimination tournament. 32 of the players are beginners, and 32 of the players are experts. The players are randomly matched in every round. If a beginner plays an expert, then his probability of winning is p, where 0 < p < 1/2. What is the probability that a beginner will win the tournament?

What do you do when you find an error in an answer that you posted to Yahoo! Answers? Do you just edit the response, or do you include a note saying that the answer was changed? Does it depend on how long ago the previous response was posted?

Here's a little challenge, inspired by a recent Y!A question.

Show that the complex numbers x, y, and z form an equilateral triangle if and only if x^2 + y^2 + z^2 = x*y + y*z + z*x.