Research done by high-school students I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand.  To inspire them, I would like to give them examples of high-school students who have made original contributions in mathematics.  One example I have is the 11th-grader from Hawai'i named Kang Ying Liu who in 2010 "discover[ed] nine new geometric formulas for describing triangle inequalities."
Do you have any other examples of high-school students who have made original contributions in mathematics?
 A: You could look at the winners of the Intel Talent Search and projects at the Research Summer Insitute at MIT.
Some high schoolers have also written papers at Ken Ono's REU. You can find them by looking at the archive on his website. Here are two:
http://www.mathcs.emory.edu/~ono/REUs/archive/results/reu09FengKirschMcCallWage.pdf
http://www.mathcs.emory.edu/~ono/REUs/archive/results/reu09DummitGoldbergPerry.pdf
A: Gauss anecdotally solved the finite summation of linear sequences in primary school. Although to be fair this wasn't an original discovery as there are demonstrations of pairwise summation as early as 400 C.E. in Jewish religious works. 
A: In 1988 in the IMO, Australia decided to use the following question: 

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Prove $\frac{a^2+b^2}{ab+1}$ is a perfect square. 

The problem was proposed by Stephan Beck, West Germany. No one in  the committee was able to solve it. Two of it's members were George Szekeres (Erdős number of 1) and his wife, both famous problem solvers and problem creators. The problem was then sent to 4 prominent number theory researchers and they were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of 19th IMO marked with a double asterisk, which meant a super-hard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition. 
Eleven students gave perfect solutions. The solution to the question used a new technique in problem solving that had never been used before. However 11 high school students were able to surpass prominent number theorists in their own field by solving the question. The technique used for solving the problem is called Vieta Jumping.
A: The Siemens Competition is for high school students doing original research in math and science. They provide a list of the math paper abstracts of the winners and finalists.
As an example, here is the paper from 2009's winning project.
A: Sylvain Cappell's paper, "The Theory of Semi-cyclical Groups with Special Reference to Non-Aristotelian Logic," won him the 1963 Westinghouse Talent Search when he was 16 years old, as recollected in this Boy's Life article. This was the first of Sylvain's many important mathematical discoveries. Tangentially interesting is that the runner-up in this contest, Sylvain's rival, was a certain William Leonard Pickard.
A: I'm not sure this is really what you're looking for, but Britney Gallivan, then $16$, disproved the famous claim that it was impossible to fold a piece of paper in half ten times, by folding one twelve times.  She also came up with  a model that  correctly explained the limit, and predicted how big the original paper would have to be to be folded $n$ times.
Archive of page about Gallivan from the Pomona Historical Society
A: He may not have been in "high school" but he was certainly at that age when he "was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem."  I'm of course talking about Galois, whom I am amazed has not been mentioned yet.
A: The Nordstrom-Robinson $(16,2^8,6)$ nonlinear binary code was discovered by A. W. Nordstrom, then a high-school student in Illinois, after J. P. Robinson, a faculty member at U. Iowa, gave a talk at the high school about unsolved problems in coding theory. It is the simplest
example of nonlinear binary codes with more codewords than linear binary codes with the
same minimum distance $6$.  The generalization was discovered by F. P. Preparata.
A: The then 16-year old Sarah Flannery published an algorithm for public-key cryptography that she dubbed the "Cayley-Purser algorithm". There was some excitement when it was found to be a bit faster than RSA, but was subsequently found to be flawed. Nevertheless, it was still quite an accomplishment for a teenager.
Sarah's paper can be seen here. She has written a book with her father on her experiences.
A: Surprisingly (because it's so recent) missing is any mention of Jacob Lurie's research that earned him 1st place in the 1996 Westinghouse Science Talent Search. But perhaps not surprisingly, as others here may have felt it wasn't something the original poster could have used.
See the two articles about it in the July 1996 Notices of the American Mathematical Society.
In the 1st article John H. Conway wrote: His writing is already very much like that of a professional mathematician, and any professional mathematician would be justly proud if he could produce arguments as subtle and deep as Lurie's.
In the 2nd article Allyn Jackson wrote: Gasarch believes that Lurie's paper "could easily be half a Ph.D. thesis."
18 Sept. 2014 UPDATE: I just read that Jacob Lurie is among those selected this year (2014) for a MacArthur Fellowship.
A: In 2003 the Kemnitz Conjecture, a 20 year old open problem in combinatorial number theory, was independently proven by Christian Reiher and Carlos di Fiore.  Reiher at the time had just passed his Abitur (entrance exam for German universities), and di Fiore was still a high school student.  
A: The following link provides additional examples: http://isef.kpi.ua/index.php/tezy-proektiv/81-osnovna/pro-konkurs/tezy/96-tezy-isef-2011-ma
