MathDB

P(a_i)=i?

Source: Own. IMO 2022 Malaysian Training Camp 1

1/29/2022

It is known that a polynomial

P

with integer coefficients has degree

2022

. What is the maximum

n

such that there exist integers

a_1, a_2, \cdots a_n

with

P(a_i)=i

for all

1\le i\le n

?
[Extra: What happens if

P \in \mathbb{Q}[X]

and

a_i\in \mathbb{Q}

instead?]

algebra

All 2k x 2k are balanced

Source: Own. IMO 2022 Malaysian Training Camp 2

3/13/2022

Let

n

,

k

be fixed integers. On a

n \times n

board, label each square

0

or

1

such that in each

2k \times 2k

sub-square of the board, the number of

0

's and

1

's written are the same. What is the largest possible sum of numbers written on the

n\times n

board?

combinatorics

(EXF) tangent to gamma

Source: Own. IMO 2022 Malaysian Training Camp 1

2/27/2022

Let

ABCD

be a circumscribed quadrilateral with incircle

\gamma

. Let

AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L

. Let a circle with diameter

KL

intersect

\gamma

at one of the points

X

. Prove that

(EXF)

is tangent to

\gamma

.

geometry

2^n contains abcd?

Source: Own. IMO 2022 Malaysian Training Camp 2

3/14/2022

Given a four digit string

k=\overline{abcd}

,

a, b, c, d\in \{0, 1, \cdots, 9\}

, prove that there exist a

n<20000

such that

2^n

contains

k

as a substring when written in base

10

.
[Extra: Can you give a better bound? Mine is

12517

]

number theory

Furthest and nearest point have the same color as P

Source: Own. Malaysian IMO TST 2022 P2

5/7/2022

Let

\mathcal{S}

be a set of

2023

points in a plane, and it is known that the distances of any two different points in

S

are all distinct. Ivan colors the points with

k

colors such that for every point

P \in \mathcal{S}

, the closest and the furthest point from

P

in

\mathcal{S}

also have the same color as

P

.
What is the maximum possible value of

k

?
Proposed by Ivan Chan Kai Chin

combinatorial geometrycombinatorics

2

Problems(5)