2024 Canadian Open Math Challenge

Part of Canadian Open Math Challenge

Subcontests

(12)

1

2024 COMC C4

Call a polynomial

f(x)

excellent if its coefficients are all in [0, 1) and

f(x)

is an integer for all integers

x

. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most

n

n

. c) Find the minimum

n\ge3

for which there exists an excellent polynomial of the form

\frac{1}{n!}x^n+g(x)

g(x)

is a polynomial of degree at most

n-3

1

2024 COMC C3

ABC

be a triangle for which the tangent from

A

to the circumcircle intersects line

BC

D

O

be the circumcenter. Construct the line

l

that passes through

A

and is perpendicular to

OD

l

OD

E

BC

F

. Let the circle passing through

ADO

BC

H

. It is given that

AD=AO=1

OE

b) Suppose for this part only that

FH=\frac{1}{\sqrt{12}}

: determine the area of triangle

OEF

. c) Suppose for this part only that

BC=\sqrt3

: determine the area of triangle

OEF

. d) Suppose that

B

lies on the angle bisector of

DEF

. Find the area of the triangle

OEF

1

2024 COMC C2

a) How many ways are there to pair up the elements of

\{1,2,\dots,14\}

into seven pairs so that each pair has sum at least

15

? b) How many ways are there to pair up the elements of

\{1,2,\dots,14\}

into seven pairs so that each pair has sum at least

13

? c) How many ways are there to pair up the elements of

\{1,2,\dots,2024\}

1012

pairs so that each pair has sum at least

2001

1

2024 COMC C1

Let the function

f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}

for all real values

x,y

t\not=\pm1

f(2,0,3)

f(0,2,3)

f(x,y,0)=f(y,x,0)

for any values of

(x,y)

f(x,y,t)=f(y,x,t)

for any values of

(x,y)

t\not=\pm1

g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}

for all real values

x,y

s\not=\frac{\pi}{2}+2\pi k

k

is an integer number, show that

g(x,y,s)=g(y,x,s)

for any values of

(x,y)

s

in the domain of

g(x,y,s)

1

2024 COMC B4

Initially, the integer

80

is written on a blackboard. At each step, the integer

x

on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless

x=0

, in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let

P(a,b)

be the probability that after

a

steps, the integer on the board is

b

\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}

(that is, the value that the function

\frac{P(a,80)}{P(a,2024)}

a

goes to infinity).

1

2024 COMC B3

a,b,c,d

be four distinct integers such that:

\text{min}(a,b)=2

\text{min}(b,c)=0

\text{max}(a,c)=2

\text{max}(c,d)=4

\text{min}(a,b)

\text{max}(a,b)

denote respectively the minimum and the maximum of two numbers

a

b

. Determine the fifth smallest possible value for

a+b+c+d

1

2024 COMC B2

David wanted to calculate the volume of a prism with an equilateral triangular base. He was given the height of the prism

H=15

and the height of the base

h=6

. He accidentally swapped the values of

H

h

in his calculations. By what number should he multiply his result to get the correct volume?

1

2024 COMC B1

For any positive integer number

k

, the factorial

k!

is defined as a product of all integers between

1

k

k!=k\times{(k-1)}\times\dots\times{1}

s(n)

denote the sum of the first

n

factorials, i.e.

s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}

Find the remainder when

s(2024)

8

1

2024 COMC A4

Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row.
The beginning of this arrangement is shown below:

0

2\; 4

6\;\underline{8}\;10

12\: 14\: 16\: 18

20\: 22 \: 24 \: 26\: 28

The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?

1

2024 COMC A3

Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey.
How many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?

1

2024 COMC A2

Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200 meter side and run in the same direction, Alice running twice as fast as Bob.
After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the building?

1

2024 COMC A1

Two locations A and B are connected by a 5-mile trail which features a lookout C. A group of 15 hikers started at A and walked along the trail to C. Another group of 10 hikers started at B and walked along the trail to C. The total distance travelled to C by all hikers from the group that started in A was equal to the total distance travelled to C by all hikers from the group that started in B.
Find the distance (in miles) from A to C along the trail.