MathDB

1

Fake NT \gcd(a_1, a_2, \dots, a_n)

n > 2

, or show that no such natural numbers

n

exists, that satisfy the following condition: There exists natural numbers

a_1, a_2, \dots, a_n

such that

\gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}}

6

1

Famous Theorem of R...

Given a cyclic quadrilateral

ABCD

. Let

X

be a point on segment

BC

(

X \not= BC

) such that line

AX

is perpendicular to the angle bisector of

\angle CBD

, and

Y

be a point on segment

AD

(

Y \not= D)

such that

BY

is perpendicular to the angle bisector of

\angle CAD

. Prove that

XY

is parallel to

CD

.

7

1

Polynomial equations with floor

Determine all real-coefficient polynomials

P(x)

such that

P(\lfloor x \rfloor) = \lfloor P(x) \rfloor

for every real numbers

x

.

5

1

Classic combinatorics

A set

A

contains exactly

n

integers, each of which is greater than

1

and every of their prime factors is less than

10

. Determine the smallest

n

such that

A

must contain at least two distinct elements

a

and

b

such that

ab

is the square of an integer.

4

1

Colouring 2n tiles and rectangles, combinatorics

Problem 4. A chessboard with

2n \times 2n

tiles is coloured such that every tile is coloured with one out of

n

colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.

3

1

Functional Equation and Divisibility

The wording is just ever so slightly different, however the problem is identical.
Problem 3. Determine all functions

f: \mathbb{N} \to \mathbb{N}

such that

n^2 + f(n)f(m)

is a multiple of

f(n) + m

for all natural numbers

m, n

.

2

1

System of Quadratic Polynomials: sum equal to zero or not all are distinct

Problem 2. Let

P(x) = ax^2 + bx + c

where

a, b, c

are real numbers. If

P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab

then prove that

(a - b)(b - c)(c - a)(a + b + c) = 0.

1

Circles centred on sides leading to a bisection

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here.
Problem 1. Given an acute triangle

ABC

and the point

D

on segment

BC

. Circle

c_1

passes through

A, D

and its centre lies on

AC

. Whereas circle

c_2

passes through

A, D

and its centre lies on

AB

. Let

P \neq A

be the intersection of

c_1

with

AB

and

Q \neq A

be the intersection of

c_2

with

AC

. Prove that

AD

bisects

\angle{PDQ}

.

2020 Indonesia MO

Subcontests

Fake NT \gcd(a_1, a_2, \dots, a_n)

Famous Theorem of R...

Polynomial equations with floor

Classic combinatorics

Colouring 2n tiles and rectangles, combinatorics

Functional Equation and Divisibility

System of Quadratic Polynomials: sum equal to zero or not all are distinct

Circles centred on sides leading to a bisection