MathDB

1

Scanning rectangles for treasure

Treasure was buried in a single cell of an

M\times N

(

2\le M

,

N

) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid

[a,b]\times[c,d]

with

1\le a\le b\le M

and

1\le c\le d\le N

. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up

Q

detectors, which may only be run simultaneously after all

Q

detectors are ready.
In terms of

M

and

N

, what is the minimum

Q

required to gaurantee to determine the location of the treasure?

5

1

Constructing orthocenter using ruler with width

Initially, three non-collinear points,

A

,

B

, and

C

, are marked on the plane. You have a pencil and a double-edged ruler of width

1

. Using them, you may perform the following operations:
[*]Mark an arbitrary point in the plane. [*]Mark an arbitrary point on an already drawn line. [*]If two points

P_1

and

P_2

are marked, draw the line connecting

P_1

and

P_2

. [*]If two non-parallel lines

l_1

and

l_2

are drawn, mark the intersection of

l_1

and

l_2

. [*]If a line

l

is drawn, draw a line parallel to

l

that is at distance

1

away from

l

(note that two such lines may be drawn).
Prove that it is possible to mark the orthocenter of

ABC

using these operations.

3

1

Even number of irreducible polynomials

Let

N{}

be the number of positive integers with

10

digits

\overline{d_9d_8\cdots d_0}

in base

10

(where

0\le d_i\le9

for all

i

and

d_9>0

) such that the polynomial

d_9x^9+d_8x^8+\cdots+d_1x+d_0

is irreducible in

\Bbb Q

. Prove that

N

is even.
(A polynomial is irreducible in

\Bbb Q

if it cannot be factored into two non-constant polynomials with rational coefficients.)

2

1

2024 numbers in a circle

Jane writes down

2024

natural numbers around the perimeter of a circle. She wants the

2024

products of adjacent pairs of numbers to be exactly the set

\{ 1!, 2!, \ldots, 2024! \}.

Can she accomplish this?

1

Reflecting triangle sides across angle bisector

Let

ABC

be a triangle with incenter

I

. Suppose the reflection of

AB

across

CI

and the reflection of

AC

across

BI

intersect at a point

X

. Prove that

XI

is perpendicular to

BC

.

2024 Canada National Olympiad

Subcontests

Scanning rectangles for treasure

Constructing orthocenter using ruler with width

Even number of irreducible polynomials

2024 numbers in a circle

Reflecting triangle sides across angle bisector