2021 Canadian Mathematical Olympiad Exam - CMS-SMC
The 2021 Canadian Mathematical Olympiad
A competition of the Canadian Mathematical Society and supported by the Actuarial Profession.
A full list of our competition sponsors and partners is available online at
Official Problem Set
1. Let ABCD be a trapezoid with AB parallel to CD, |AB| > |CD|, and equal edges |AD| = |BC|. Let I be the center of the circle tangent to lines AB, AC and BD, where A and I are on opposite sides of BD. Let J be the center of the circle tangent to lines CD, AC and BD, where D and J are on opposite sides of AC. Prove that |IC| = |JB|.
2. Let n 2 be some fixed positive integer and suppose that a1, a2, . . . , an are positive real numbers satisfying a1 + a2 + ? ? ? + an = 2n - 1.
Find the minimum possible value of
a1 + a2 +
a3
+???+
an
.
1 1 + a1 1 + a1 + a2
1 + a1 + a2 + ? ? ? + an-1
3. At a dinner party there are N hosts and N guests, seated around a circular table, where N 4. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the 2N people are seated at the dinner party, at least N pairs of guests will chat with one another.
4. A function f from the positive integers to the positive integers is called Canadian if it satisfies gcd f (f (x)), f (x + y) = gcd (x, y)
for all pairs of positive integers x and y.
Find all positive integers m such that f (m) = m for all Canadian functions f .
? 2021 Canadian Mathematical Society
p. 1/2
The 2021 Canadian Mathematical Olympiad
5. Nina and Tadashi play the following game. Initially, a triple (a, b, c) of nonnegative integers with a + b + c = 2021 is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer k and one of the three entries on the board; then the player increases the chosen entry by k and decreases the other two entries by k. A player loses if, on their turn, some entry on the board becomes negative. Find the number of initial triples (a, b, c) for which Tadashi has a winning strategy.
Important! Please do not discuss this problem set online for at least 24 hours.
? 2021 Canadian Mathematical Society
p. 2/2
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- grade 9 international junior math olympiad
- mathematics olympiad science olympiad foundation
- american mathematics olympiad 2017 answer entry sheet 奧冠教育中心
- national foundation phase maths olympiad grade3
- math 9 final practice test claremont secondary school
- sample questions mathematics category 1
- grade 9 november 2017 mathematics marking guideline maths 101
- math olympiad rules
- crest mathematics olympiad cmo previous year paper 2021 22 class 9
- 2020 australian mathematical olympiad
Related searches
- grade 11 mathematical literacy exam papers
- cms 2020 2021 flu vaccine
- cms 2021 final rule
- cms final rule 2021 changes
- cms 2021 ip only list
- cms 2021 e m code changes
- 2021 cms changes
- 2021 cms inpatient only list pdf
- 2021 cms documentation guidelines
- cms final rule 2021 e m
- cms e m guideline changes 2021 final rule
- cms 2021 changes