MathDB

1

a and b coprime, prove if (a^s + b^s)|(a^r + b^r), then s|r

a, b, r,

and

s

be positive integers (

a \le 2

), where

a

and

b

have no common prime factor. Prove that if

a^r + b^r

is divisible by

a^s + b^s

, then

r

is divisible by

s

.

4

1

given convex polygon, if ABC = ADE and ADB = AEC, then BAC = DAE

Given a pentagon

ABCDE

with all its interior angles less than

180^\circ

. Prove that if

\angle ABC = \angle ADE

and

\angle ADB = \angle AEC

, then

\angle BAC = \angle DAE

.

2

3

Given n integers, sum = 0, product = n, find all n.

It is known that there are

n

integers

a_1, a_2, \cdots, a_n

such that

a_1 + a_2 + \cdots + a_n = 0 \qquad \text{and} \qquad a_1 \times a_2 \times \cdots \times a_n = n.

Determine all possible values of

n

.

[IMONST2 2022 Junior #2] Which is greater? 17^24 or 31^19

Without using a calculator, determine which number is greater:

17^{24}

or

31^{19}

[IMONST2 2022 Primary #2] Number sequence

The following list shows every number for which more than half of its digits are digits

2

, in increasing order:

2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots

If the

n

th term in the list is

2022

, what is

n

?

6

1

IMONST2 2022 Senior #6

A football league has

n

teams. Each team plays one game with every other team. Each win is awarded

2

points, each tie

1

point, and each loss

0

points. After the league is over, the following statement is true: for every subset

S

of teams in the league, there is a team (which may or may not be in

S

) such that the total points the team obtained by playing all the teams in

S

is odd. Prove that

n

is even.

1

3

IMONST2 Senior 2022 #1

Given a circle

\Gamma

with center

I

, and a triangle

\triangle ABC

with all its sides tangent to

\Gamma

. A line

\ell

is drawn such that it bisects both the area and the perimeter of

\triangle ABC

.
Prove that line

\ell

passes through

I

.

IMONST2 2022 Junior #1

Given a circle and a quadrilateral

ABCD

whose vertices all lie on the circle. Let

R

be the midpoint of arc

AB

. The line

RC

meets line

AB

at point

S

, and the line

RD

meets line

AB

at point

T

.
Prove that

CDTS

is a cyclic quadrilateral.

[IMONST2 2022 Primary #1] Number of diagonals

Given a polygon

ABCDEFGHIJ

. How many diagonals does the polygon have?

3

Prove multiple of 2^2022 that has 2022 digits can be written with 1 and 2 only

Prove that there is a multiple of

2^{2022}

that has

2022

digits, and can be written using digits

1

and

2

only.

[IMONST2 2022 Primary #3] sum of n(n+3)

Prove that

1\cdot 4 + 2\cdot 5 + 3\cdot 6 + \cdots + n(n+3) = \frac{n(n+1)(n+5)}{3}

for all positive integer

n

.

[IMONST2 2022 Junior #3] m+n=999999999

Given an integer

n

. We rearrange the digits of

n

to get another number

m

. Prove that it is impossible to get

m+n = 999999999

.

2022 Malaysia IMONST 2

Subcontests

a and b coprime, prove if (a^s + b^s)|(a^r + b^r), then s|r

given convex polygon, if ABC = ADE and ADB = AEC, then BAC = DAE

Given n integers, sum = 0, product = n, find all n.

[IMONST2 2022 Junior #2] Which is greater? 17^24 or 31^19

[IMONST2 2022 Primary #2] Number sequence

IMONST2 2022 Senior #6

IMONST2 Senior 2022 #1

IMONST2 2022 Junior #1

[IMONST2 2022 Primary #1] Number of diagonals

Prove multiple of 2^2022 that has 2022 digits can be written with 1 and 2 only

[IMONST2 2022 Primary #3] sum of n(n+3)

[IMONST2 2022 Junior #3] m+n=999999999