From here you can browse amongst examples of student work. These not only include "correct" solutions, and there are plenty of those, but also examples of partial work. Each one is accompanied with some commentary as well as the written response that the student received during the course.

Sum of series: Uses an investigation to determine an infinite series

Sum and difference of numbers: A good example of the pigeonhole principle

Are these friendships possible: A problem from graph theory

Locked lockers: A disguised problem from number theoy

Sum of the odd term Fibonacci numbers: A classic induction problem

Covering the chessboard: Example of a solution by the principle of pairing.

Number of variables: A good example of application of the invariance principle to show that the number of variables in a homogeneous polynomial must be divisible by 4.

Existence of lattice point (example one, example two): Application of pigeonhole principle to show that amongst a set of lattice points some pair has another lattice point on the interior of their line.

Number of regions of the plane made by n lines: A classical problem about what is the greatest number of regions n lines.

Path of bug on surface of a cube: Using an alternative representation to find the shortest path on the surface of a cube that takes a bug from one vertex to an opposite vertex.

Representing numbers as sum of distinct Fibonaccis: In this problem the student gives a novel representation to the problem of showing that every natural number can be written as a product of distinct Fibonacci numbers.

Weighted cube: The invariance principle is used to show that a cube which starts with a weight of one at a single vertex which is repeatedly transformed by adding equal weights to vertices on a shared edge can never all be made equal.

Summing digits: An example of using investigation to conjecture what number is obtained if one starts with 2^100 and adds its digits and adds the digits of that number and so on.

Proof by contradiction: A Putnam competition problem about subsets of the real line satisfying some multiplicative properties.

Monochromatic rectangle (example one, example two): The pigeonhole principle is used to show that if all points in the plane are labeled blue and red then there is some rectangle all of whose vertices are the same color.

Binomials and Fibonacci Numbers: The sum of several binomial coefficients are investigated and it is conjectured that the result is a Fibonacci number.

Non-divisibility of polynomial: In this problem it is shown that a certain quadratic polynomial is never divisible by 15.

Number of paths (example one, example two): Here are two examples of different treatments of determining the number of paths between lattice points when the path is restricted to go up one unit or to the right by one unit on each move.

Unfriendly subsets: A good example of a partial solution obtained by doing an investigation and recognizing a patttern. In this case one is trying to count the number of subsets of 1,2,...,n which do not contain consecutive integers.

Midpoints: A very elegant example of the application of the extremal princple. In this problem it is shown that a subset of the plane with the property that every one of its points is the midpoint of a pair of points from the set is an infinite set.

Factor of polynomial: In this problem, a representation of a polynomial f(x) is used to show that if a certain inversion of f(x) is equal to -f(x) then x-1 divides f(x).

Existence of prime: The pigeonhole principle and prime factorization are used to show that among 15 pairwise relatively prime numbers all less than 2000 at least one must be a prime.

Only one pair: After an investigation, elementary number theory is used to show that the only pair (p, p^2+2) of primes is (3,11).

Relatively prime: Here the pigeonhole principle is used to show that amongst 10 consecutive integers at least one is relatively prime to the other nine.

Product of four consecutive: After an investigation it is shown by straightforward algebra that the product of four consecutive natural numbers is always one less than a perfect square (and therefore never equal to a square).

Intersecting chords: Combinatorics is used after an investigation to show that the maximum number of intersections produced by all the chords joining n points on the circumference of a circle is n choose 4.

Triangle from lengths: The extremal principle and the triangle inequality is applied to show that if 10 line segments are given with lengths between 1 and 55 then three can be picked which will make a triangle.

Number of reseatings: A recursion is uncovered and applied with mathematical induction to show that the number of ways a row of n child can be reseated if each occupies their original seat or an adjacent seat.

Number of even words: Combinatorics and the binomial expansion are used to determine the number of words from a four letter alphabet have an even number of occurances of a given lettter.

Number of even words: Another treatment in which the number of words from a four letter alphabet have an even number of occurances of a given lettter.

Divisible by power of ten: The pigeonhole principle and simple ideas from number theory are used to prove that for any number relatively prime to ten has a power whose difference from 1 is divisible by a large power of 10.

Crossing segments, two examples. Example one is a good illustration of the heuristic "try a simpler case" is applied to show that it is not possible to join points successively 1999 points successively and then the last to the first and find a line which intersects each of the line segments in an interior point. Example two gives a direct proof.

Sum of product of reciprocals (one, two, three): Here you will find three attempts to prove that if one takes the non-empty subsets of 1,2,...,n and for each one takes the product of the reciprocals and adds all these numbers up then the surprising result is n.

Infinite solutions: In this problem it is shown that the equation x^2 + y^2 + x^2 = 3xyz has infinitely many solutions in integers.

The return of infinite solutions: In example one the method of the immediately preceeding problem solution is used to find infinitely many solutions in integers to the equation x^2 + y^2 + z^2 = xyz. In example two the problem is solved by getting a one-to-one correspondence between the solutions of the first equation and the second.

Rows of evens in Pascal: An investigation to determine for which rows of Pascal's triangle are all the interior terms even.