# Santa Clara University

### Santa Clara University High School Mathematics Contest

Since 1958, the Department of Mathematics and Computer Science at Santa Clara University has sponsored, as a community outreach, an annual High School Mathematics Contest usually scheduled on the second Saturday in November. This is a three hour exam which tests problem solving skills and mathematical ingenuity rather than being directed toward the content of a specific course.

Calculators and computers may NOT be used. (Students should also bring enough pre-sharpened pencils since no pencil sharpeners are available.) Student must not make use of cell-phones or MP3 players (or similar devices) during the exam.

The 2012 Contest was held on Saturday, November 10, 2012.

Each year in October, letters are sent to high schools in the Bay Area inviting them to request tickets for those students interested in participating. Schools who have not received letters by November 1, should contact the Department at the number given below.

The following are examples of the type of questions asked:

• Order the following numbers from smallest to largest: 2600, 3500, 4400, 5300, 6200. (2012 exam, prob 3)
• A divisor of a positive integer N is a positive integer m such that N/m is also an integer. For example, the divisors of 12 are 1, 2, 3, 6 and 12. Of the integers in the set {1, 2, 3, ..., 2499, 2500}, which one or ones have exactly 27 divisors? (2011 exam, prob 5)
• How many different right triangles are there in the co-ordinate plane with all three vertices on the parabola y=x2, with all three vertices having integer x and y co-ordinates, and having an area no more than 2010 square units? (2010 exam, prob 8)
• Find all ordered pairs (x,y) which give solutions to the equation (2x + y - 4)2 + (3x + 2y + 1)2 = 0. (2009 exam, prob 3)
• When (x3 + x + 1)10 is multiplied out, what is the coefficient of x25? (2008 exam, prob 5)
• What is the largest number n < 2007 which can be written as the sum of five consecutive perfect squares? (2007 exam, prob 2)
• If x + y = 1 and x2 + y2 = 2, what is the value of (a) xy, (b) x3 + y3? (Write your answer without the use of operations [+, -, etc., ... or square roots].) (2006 exam, prob 3)
• How many distinct (that is, noncongruent) triangles are there with all sides of integer length, with the longest side (or sides) having length 2005? (2005 exam, prob 5)
• If a and b are numbers such that a - b = 4 and a2 - b2 = 44, what integer is a3 - b3? (2004 exam, prob 4)
• A positive integer n, when written in base b, has the form 211. When n is written in base b+2, it has the form 110. Determine n and b (in base 10). (2001 exam, prob 4)
• Find a positive integer a that satisfies the equation aa - (a - 1)a+1 = 13. (2000 exam, prob 1)
• In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged as a sequence u, v, w, x, y, z such that u + x = v + y = w + z? (1999 exam, prob 3)
• Find the largest positive integer e for which 3e is an exact divisor of 1998. (1998 exam, prob 1)

The first twenty-five exams were published in the book, Santa Clara Silver Anniversary Contest Book, by G.L. Alexanderson, A. Hillman, L.F. Klosinski, D. Logothetti (Palo Alto: Dale Seymour Publications, 1985). (This book is now out of print.)

• November 10, 2012 55th Contest, 233 contestants.
• November 12, 2011 54th Contest, 201 contestants.
• November 13, 2010 53rd Contest, 215 contestants.
• November 14, 2009, 52nd Contest, 332 contestants.
• November 15, 2008, 51st Contest, 231 contestants.
• November 10, 2007, 50th Contest, 277 contestants.
• November 11, 2006, 49th Contest, 255 contestants.
• November 12, 2005, 48th Contest, 275 contestants.
• November 13, 2004, 47th Contest, 258 contestants.
• November 8, 2003, 46th Contest, 230 contestants.
• November 9, 2002, 45th Contest, 238 contestants.
• November 10, 2001, 44th Contest, 143 contestants.
• November 11, 2000, 43rd Contest, 254 contestants.
• November 13, 1999, 42nd Contest, 232 contestants.
• November 14, 1998, 41st Contest, 363 contestants.
• November 15, 1997, 40th Contest, 384 contestants.
• November 16, 1996, 39th Contest, 358 contestants.
• November 11, 1995, 38th Contest, 296 contestants.
• November 12, 1994, 37th Contest, 298 contestants.

Information about the contest and forms requesting admission tickets are sent to schools in the San Francisco Bay Area in early October each year. Further information can be obtained by contacting Prof. Leonard F. Klosinski at 408-554-6897 or the Department Office at 408-554-4525 (to obtain a form for the admission tickets).