We are gratefull to William Cordwell for compiling this information.

Math Contests

Major National Math Contests

Math Olympiads for Elementary and Middle Schools http://www.moems.org/

One of the few contests for elementary students (upper grades).  Also can be useful for MS students as practice for MathCounts.  There is a fee for this competition.

MathCounts http://www.mathcounts.org/

The primary Middle School national-level competition.  Cost = approx. $80 / team + $20 per each of up to 4 individuals.  Top teams and individuals in the Chapter (local regional) Meet (usually February) progress to the State Meet.  The top four individuals at the State Meet (usually March) make up the State Team for the National Competition, usually in Washington, DC, in early May.  Travel expenses to the national competition for the individuals and the coach are usually paid for by a sponsor.  Restricted to 6th, 7th, and 8th graders.

American Math Competitions (AMC) http://www.unl.edu/amc/

The premier High School (and below) national-level competition.  The AMC->8 is for kids up through 8th grade, the AMC->10 and AMC->12 for kids up through 10th and 12th grades, respectively; good students should probably take the AMC->12, even if they are eligible to take the AMC->10.  The AMC->8 is given in November.  The AMC->10 and AMC->12 are given twice in February. Students are allowed to take tests on both days. 

For example, a 9th grader could take the AMC->10A in early February and the AMC->12B in late February.  About 250,000 students take these exams.  The top 25,000 or so students are invited to take the American Invitational Math Exam (AIME).  They may take either the AIME-A or the AIME-B, but not both.  The top 250 students (composite score = AMC score + 10*AIME score; some weighting in favor of younger students) are invited to write the USA Math Olympiad (USAMO). 

The top 25 scorers on the USAMO are invited to participate in the Math Olympiad Summer Program (MOP or MOsP) in June; in recent years, an extra 25 or so 9th graders have been invited, as well. The top 6 are chosen to represent the United States at the International Math Olympiad (IMO) in July.  There is a fee for the AMC->8, AMC->10, and AMC->12 exams; the invited exams are free, as is the summer math camp (including transportation).

USA Math Talent Search (USAMTS) http://www.usamts.org/

Now run by the ArtofProblemSolving, this is a contest that allows students to work for about a month on a problem set (four sets per year) and submit them.  Students are encouraged to learn LaTex, which is probably a good thing.  Students who are doing well after the 3rd round will get an invitation to the AIME, although, often, those students qualify via the AMC->12 or AMC->10, anyway.  The problems are challenging, but students can learn a lot by working on them.

American Regions Math League (ARML) http://www.arml.com/

Usually the first weekend in June, held concurrently at 3 (or more) sites; in 2006, at Penn State, UNLV, and the University of Iowa.  It consists of 2 team-like rounds, an individual round, and a relay round, where a team of 15 is broken into 5 groups of 3 each.  The groups of three solve a 3-part problem, where the answer to the first part is given to the 2nd person as an input, etc.  There are two divisions, A and B, and the highest-scoring teams in the B division must compete in the A Division for the next 3 years. 

Teams ("regions") may be from a particular school, part of a state, a state, or several states combined, but regions should not overlap.  Individual students may compete; they are usually put in a scratch team for the non-individual portions of the competition.  There is a nominal fee, plus transportation costs, but everyone stays at a dorm, which helps minimizes the expense.

The Mandelbrot Competition http://www.mandelbrot.org/

A national-level competition.  Teams of four compete, with team and individual exams.  Given four times a year, teams may enter at either the national or regional (east and west) levels.  Questions are reasonably difficult.  There is a fee.

Bay Area Math Olympiad (BAMO) http://mathcircle.berkeley.edu/BMC6/pages/BAMO/bamo.html

Although designed for the San Francisco Bay Area, they will allow individual students in other areas and states to participate.  Free, I think.

Additionally, several states offer state math competitions

New Mexico Specific

UNM/PNM State Math Contest http://www.math.unm.edu/math_contest/contest.html

One of the most important contests for NM students.  For students <= 12th grade, but geared towards 7th-12th grades.  Required for honors students at several high schools.  About 1500 students take the first round in the fall; about 200 qualify for the second round in February.  Free.

GoFigure Math Challenge http://www.sandia.gov/gofigure/challenge.html

Offered at several sites around the state in October.  For students <= 12th grade; geared towards 7th-12th grades.  Free.

Minor math contests that are worthwhile

American High School Internet Math Competitions (AHSIMC, or the iTest) http://www.ahsimc.com/

Free contest given in September.  Teams (up to 5 students) work over a period of a few days, submitting answers online.  Teams may be from different schools (even different states).

Purple Comet http://math.uww.edu/mathmeet/

MS or HS teams (of 6 or fewer) work for an hour or so in April.  Results are submitted online.  Registration is free.  Mixed teams (from different schools) are allowed.

ARML Power Contest http://www.arml.com/power.htm

Teams of students (students less than or equal to 12th grade) compete twice a year, November and February, in a contest like the ARML Power Round.  There is a fee for this contest.  Mixed teams are allowed.

Resources

The Art of Problem Solving http://www.artofproblemsolving.com/

Offers reasonably-priced online courses, books, and forums on a variety of math topics.  AoPS also runs the USAMTS contest.

Instruction

Educational Program for Gifted Youth (EPGY) http://epgy.stanford.edu/

Offers CD based courses, Kindergarten up through various college classes.  About $500 / quarter, but it is a well-designed program. 

Instruction, New Mexico Specific

University of New Mexico http://www.math.unm.edu/

Don't forget this!  Offers many courses in mathematics.  APS has a concurrent enrollment option. The UNM Math Dept has experience with having high-school (and younger) kids take their classes. About $500 a semester, for a 3-credit class

Summer Programs in Mathematics

The Ross Program http://www.artofproblemsolving.com/

Intensive, 8-week camp in Number Theory. Residential, at Ohio State University $2200 + transportation (2006)

PROMYS (Program in Mathematics for Young Scientists) http://math.bu.edu/INDIVIDUAL/promys/

Fairly Intense, 6-week camp in Number Theory Modeled after the Ross Program Residential, at Boston University $2350 + $100 (books) + transportation (2006)

Canada/USA Mathcamp http://www.mathcamp.org/

Five week course, offering a wide variety of classes over a wide range of skill levels.
Residential, at a college campus in the US or Canada $3195 + transportation (needs-based discount up to $100,000 AGI) (2006)

MathPath http://www.mathpath.org/index.htm

Four week course, it appears to offer a wide variety of classes. Geared towards rising 7th-9th graders (I have no personal experience with this camp) Residential, at a college campus in the US $4500 + transportation (needs/merit based scholarships available, up to $1000) (2006)

AwesomeMath Summer Program http://www.awesomemath.org/campfaq.html

Three week program; Focuses on problem-solving skills in different areas of math (I have no personal experience with this camp) Residential, this year at Univ. Texas at Dallas (this is the first year) $2995 + transportation (2006)

There are various other summer math camps, too.

Summer Programs in Mathematics, New Mexico Specific

UNM week-long summer program Offered by Prof. Nakamaye of UNM A variety of topics not usually offered in high school Residential (commuting optional) Free, including room/board