Math Tournament

My college is hosting a math tournament and I need to come up with problems for the contestants. Since this is a highschool tournament, the problems have to be pre-cal level, so I'm having trouble thinking of anything intresting. Some ideas or examples of some really off the wall problems that could give some lulz would be appreciated.

integrate x^x

Get a problem solving book or look at previous problem solving competitions questions and adapt from that by making them easier or harder, whatever. For example, in the UK we have the Senior Maths Challenge, BMO 1, BMO 2 and from this the people are chosen for the IMO team.

>>2
Relative to y equals y*(x^x) + C, what's so hard about that?

>>4
Good, now do it with respect to x.

Go with discrete and combinatoric stuff, not only is it awesome, but its really easy to take down to a highschool, or even middle or elementary level.

>>2,4-5
We're not doing this again. Even the O.999... threads are more interesting.

Geometry problems can get tricky.

What difficulty are you looking for? Ah well. Here are some of varying difficulty:

Algebra:

Given f(x) = (x-(1/x))/(x+1), find [f(x^-1)][f(x)]^-1

Find all lattice points interior to the first quadrant, that is all points (m,n) where m and n are positive integers, that lie on the graph of x^2+y^2+2xy-4x-4y-5=0

Simplify [(a + a^-1)^-1]/[a(a^2 - 1)^-1], leaving an expression in which any exponents that appear are positive.

Find both solutions to (13x + 37)^(1/3)-(13x - 37)^(1/3) = 2^(1/3)

Given f(x) = [sqrt(x)+sqrt(2)]/[sqrt(x)-sqrt(2)], express f(8) + f(32) + f(128) as the quotient of two relatively prime integers.

Express [(x^3 + y^3) - (x + y)^3]/[x^2 - y^2] as the quotient of terms involving only the use of x and y to the first power.

Find all x and only those x that satisfy sqrt(x+2) = 2x+1

If x^4 + 2x^3 + ax^2 + bx + c is exactly divisible by x^3 + 3x^2 - 2x + 4, find a+b+c.


Geometry:

The sides of a right triangle ABC have lenghts a, b, and c. Each side is used as a chord of a circle having a radius equal to the length of the chord, Using Aa, Ab, and Ac to designate the areas of the circles with radii indicated by the subscript, write an equation relating Aa, Ab, and Ac.

Two chords of a circle having lengths of 7 and 8 intersect at right angles. The chord of length 7 is partitioned into lengths of 3 and 4, while the chord of length 8 is partitioned into lengths of 2 and 6. What is the radius of the circle?

Other:

The expansion of [(2/a) + (a^2/4)]^8 includes a term of the form ra where r is an integer. What is r?

It is well known that the sum 1 + 2 + 3 + ... + n can be expressed in the closed form [n(n+1)]/2. Find a similar closed form for the sums:
1^2 - 2^2 + 3^2 - 4^2 + ... - n^2 when n is even
1^2 - 2^2 + 3^2 - 4^2 + ... + n^2 when n is odd

For n greater than or equal to 4, find in terms of n three integers a, b, and c so that a>b>c and (n+1)! - 4n! + 2(n-1)! = abc.

G(1) = 3 and G(n+1) = [4G(n)+1]/4 for integers n>1/ Find G(2005).

Find the equation of the curve that contains those points that are equidistant from the origin and the point Q(4.2).

Find the equation of the curve that contains those points that are equidistant from A(2,0) and the y-axis.

Find the equation of the curve that contains those points that are equidistant from the origin and the line x + y = 4.

Give both coordinates of the highest point on the graph of 4x^2 + 8x + y^2 - 2y = 3

Have them find the dimension of the range of some linear transformation, and a base for it.

Have them find the Lebesgue measure of the Cantor set.

Have them find the binormal line, plane, and vector of Darboux to a helix.

Have them find the dimension of the solution space of a given system of linear equations.

Have them prove that multiplication is commutative.

Have them prove that the polynomials with rational coefficients form a countable set.

Have them prove that the set of all posets taken from the natural numbers form an uncountable set and has the same cardinality as the real numbers.

Have them find a set constructed from the natural numbers that has the ordering determined by the first uncountable ordinal.

Have them prove the consistency of the inconsistency of some large ordinal.

Have them give some connected and pathwise disconnected set.

Have them prove the Cauchy-Schwarz inequality.

Have them prove Hölder's inequality.

That's all I have so far. kthxbai

Let P(x) = (some random third-degree polynomial with two complex roots).  Find the sum of the cubes of the roots.

LET THEM SUCK SOME DICK

>>12
Let them eat cake.