# Contests

## Mathematics and Statistics

## NCS Team Contest | Putnam Contest

## There are two mathematics contests that interested UW students can write each fall.

To be eligible for either, a student must be a regularly enrolled undergraduate who has not yet obtained a degree.

Below is a brief description of each contest followed by information about preparation and registration for the contests.

Any students who are interested or wish to receive more information should contact Professor Visentin (Office: 6L24, phone: 204-786-9374, email: t.visentin@uwinnipeg.ca).

## Problem Solving Sessions for Fall 2020

Stay tuned! Details coming Summer 2020.

## The 2019 NCS Team Contest took place on November 16.

The North Central Section of the Mathematical Association of America sponsors this Team Math Competition. Taking place in mid-November, this contest debuted in 1997 and we competed in it for the first time in 1998. A University may enter as many teams of up to 3 students as it likes. The students have 3 hours to solve 10 problems which range from straightforward to hard and they work together in teams to solve the problems. Our students always seem to enjoy this aspect of the competition and have performed well over the years. To prepare for the contest, students take time away from their normal course work to learn new solving techniques and work on practice problems.YEAR |
STUDENT |
RANK/# OF CONTESTANTS |

2019 |
Alex Stephens, Harsh Patel, Aalekh Patel |
6th of 56 |

2018 |
Kristaps Balodis, Joshua Wheelwright Aalekh Patel, Harsh Patel Colin Krawchuck, Alex Stephens |
19th of 60 26th of 60 27th of 60 |

2017 | Ian Kaye, Gaia Moravcik | 36th of 67 |

2015 |
Adam Borchert, Jeremy Nicholson, Alex Stephens |
17th of 82 |

2014 | Adam Borchert, Philip Lafrance, Trevor Thomson | 30th of 90 |

2013 | Philip Lafrance, Jehu Peters, Tim Pressey | 10th of 83 |

2012 | Philip Lafrance, Jehu Peters, Tim Pressey | 13th of 79 |

Our best performance came in 2005 when John-Paul Harris and Joel Peters-Fransen placed 7th out of 65 teams. This narrowly surpassed the 8th place performance of Erica Moodie, Jennifer Prokop and Go Suzuki in 1999.

#### Here are two sample problems from previous contests.

(1) A sequence begins with a_{1}, a_{2}, and for n>2 is defined by a_{n} = a_{n-1} - a_{n-2}. Find the sum of the first 2004 terms (in terms of a_{1} and a_{2}), and defend your answer.

(2) A card shuffling machine always rearranges cards in the same way relative to the order in which they are given to it. The thirteen spades arranged in the order

A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

are put into the machine, shuffled, and then the shuffled cards are put into the machine and shuffled again. If at this point the order of the cards is

3, K, 10, 2, Q, 9, 4, J, 8, 6, 7, A, 5,

What was the order of the cards after the first shuffle?

## The Annual Putnam Contest is held on the first Saturday in December.

The William Lowell Putnam Mathematical Competition is a traditional contest for individual students and has been held annually for the last 79 years. Written during the first week of December, it is a six hour exam (3 in the morning, 3 in the afternoon) in which students attempt to solve 12 fairly difficult problems. Each year, a list of the top 500 students is distributed to colleges and universities across the continent. A student who solves 3 or more problems would usually make this list.

Read the history behind the prestigious William Lowell Putnam competition.

YEAR |
STUDENT |
RANK/# OF CONTESTANTS |

2018 | Aalekh Patel | 1275th of 4623 |

2014 | Adam Borchert | 555th of 4320 |

2013 | Tim Pressey | 266th of 4113 |

2013 | Yifeng Zang | 597th of 4113 |

2012 | Tim Pressey | 415th of 4277 |

2012 | Yifeng Zang | 713th of 4277 |

2008 | Dylan Buhr | 473rd of 3627 |

2008 | Iain Crump | 619th of 3627 |

2005 | Joel Peters-Fransen | 219th of 3545 |

2004 | Joel Peters-Fransen | 596th of 3733 |

2001 | Sean Fitzpatrick | 555th of 2954 |

2000 | Kevin Doerksen | 422nd of 2919 |

1999 | Go Suzuki | 217th of 2900 |

Considering that at most universities only the best 5 or 10 math students enter this contest, placing highly in this elite group is a great accomplishment.

There is also a ranking for each school based on the total of the team members' scores, and in 2018 the University of Winnipeg placed 188th among the 568 universities across North America which competed.

Last year the team competition was won by Harvard.

#### Here are two of the easier(!) problems from a previous contest.

(1) Basketball star Shanille O'Keal's team statistician keeps track of the number, *S*(*N*), of successful free throws she has made in her first *N* attempts of the season.

Early in the season, *S*(*N*) was less than 80% of *N*, but by the end of the season, *S*(*N*) was more than 80% of *N*.

Was there necessarily a moment in between when *S*(*N*) was exactly 80% of *N*?

(2) Let *n* be a fixed positive integer. How many ways are there to write n as a sum of positive integers,

* n* = *a _{1} * +

*a*+ . . . +

_{2}*a*,

_{k}with *k* an arbitrary positive integer and *a _{1} * ≤

*a*≤ . . . ≤

_{2}*a*≤

_{k}*a*+

_{1}*1*?

For example, with

*n*= 4, there are four ways: 4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1.