# High School Math 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 2016 Contest will be held on Saturday, November 5, 2016.**

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:

- For positive integers
*n,*if both*d*and*n/d*are positive integers, then we say*d*is a*divisor*of*n.*For*n*= 2^{8}3^{7}4^{6}5^{5}6^{4}7^{3}8^{2}, how many divisors of*n*are perfect cubes?*(2014 exam, prob 2)* - Let
*N*be the number 12345678910111213...201120122013 formed by concatenating the digits of the integers from 1 up to 2013.

a) What integer is the closest to log_{10}log_{10}*N*?

b) What is the remainder when*N*is divided by 75?*(2013 exam, prob 3)* - Order the following numbers from smallest to largest: 2
^{600}, 3^{500}, 4^{400}, 5^{300}, 6^{200}.*(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=x*, with all three vertices having integer^{2}*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 (2*x*+*y*- 4)^{2}+ (3*x*+ 2*y*+ 1)^{2}= 0.*(2009 exam, prob 3)* - When (
*x*^{3}+*x*+ 1)^{10}is multiplied out, what is the coefficient of*x*^{25}?*(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*x*= 2, what is the value of (a)^{2}+ y^{2}*xy*, (b)*x*? (Write your answer without the use of operations [+, -, etc., ... or square roots].)^{3}+ y^{3}*(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*a*= 44, what integer is^{2}- b^{2}*a*?^{3}- b^{3}*(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*a*= 13.^{a}- (a - 1)^{a+1}*(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 3^{e}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.)

Statistics about recent exams are:

Date | Contest | Contestants |
---|---|---|

November 15, 2014 | 57th Contest | 184 |

November 16, 2013 | 56th Contest | 223 |

November 10, 2012 | 55th Contest | 233 |

November 12, 2011 | 54th Contest | 201 |

November 13, 2010 | 53rd Contest | 215 |

November 14, 2009 | 52nd Contest | 332 |

November 15, 2008 | 51st Contest | 231 |

November 10, 2007 | 50th Contest | 277 |

November 11, 2006 | 49th Contest | 255 |

November 12, 2005 | 48th Contest | 275 |

November 13, 2004 | 47th Contest | 258 |

November 8, 2003 | 46th Contest | 230 |

November 9, 2002 | 45th Contest | 238 |

November 10, 2001 | 44th Contest | 143 |

November 11, 2000 | 43rd Contest | 254 |

November 13, 1999 | 42nd Contest | 232 |

November 14, 1998 | 41st Contest | 363 |

November 15, 1997 | 40th Contest | 384 |

November 16, 1996 | 39th Contest | 358 |

November 11, 1995 | 38th Contest | 296 |

November 12, 1994 | 37th Contest | 298 |

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).