MathDB

1

Maximum number of rational roots in a family of quadratics

a, b, c

be three distinct positive integers. Define

S(a, b, c)

as the set of all rational roots of

px^2 + qx + r = 0

for every permutation

(p, q, r)

of

(a, b, c)

. For example,

S(1, 2, 3) = \{ -1, -2, -1/2 \}

because the equation

x^2+3x+2

has roots

-1

and

-2

, the equation

2x^2+3x+1=0

has roots

-1

and

-1/2

, and for all the other permutations of

(1, 2, 3)

, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in

S(a, b, c)

.

7

1

Another circle tangent to circle problem

Given a triangle

ABC

with

\angle ACB = 90^{\circ}

. Let

\omega

be the circumcircle of triangle

ABC

. The tangents of

\omega

at

B

and

C

intersect at

P

. Let

M

be the midpoint of

PB

. Line

CM

intersects

\omega

at

N

and line

PN

intersects

AB

at

E

. Point

D

is on

CM

such that

ED \parallel BM

. Show that the circumcircle of

CDE

is tangent to

\omega

.

6

1

Number of permutation satisfying a summation relation

Determine the number of permutations

a_1, a_2, \dots, a_n

of

1, 2, \dots, n

such that for every positive integer

k

with

1 \le k \le n

, there exists an integer

r

with

0 \le r \le n - k

which satisfies

1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}.

5

1

Another NT with GCD and LCM

Let

a

and

b

be positive integers such that

\text{gcd}(a, b) + \text{lcm}(a, b)

is a multiple of

a+1

. If

b \le a

, show that

b

is a perfect square.

2

1

Simple FE on National Contest

Determine all functions

f : \mathbb{R} \to \mathbb{R}

such that the following equation holds for every real

x,y

:

f(f(x) + y) = \lfloor x + f(f(y)) \rfloor.

Note:

\lfloor x \rfloor

denotes the greatest integer not greater than

x

.

4

1

Existence of some Large Number

Determine whether or not there exists a natural number

N

which satisfies the following three criteria: 1.

N

is divisible by

2^{2023}

, but not by

2^{2024}

, 2.

N

only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of

N

are odd.

1

Equal Circumradii

An acute triangle

ABC

has

BC

as its longest side. Points

D,E

respectively lie on

AC,AB

such that

BA = BD

and

CA = CE

. The point

A'

is the reflection of

A

against line

BC

. Prove that the circumcircles of

ABC

and

A'DE

have the same radii.

3

1

A Game with Limited Steps

A natural number

n

is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is

X

. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor

d

of

X

, and replace

X

with

X+d

if Neneng chose the sign "up" or

X-d

if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero.
Prove that if

n \geq 14

, Asep can win in at most

(n-5)/4

steps.

2023 Indonesia MO

Subcontests

Maximum number of rational roots in a family of quadratics

Another circle tangent to circle problem

Number of permutation satisfying a summation relation

Another NT with GCD and LCM

Simple FE on National Contest

Existence of some Large Number

Equal Circumradii

A Game with Limited Steps