JOM 2024

Part of JOM

Subcontests

(5)

1

3 perfect squares

Do there exist infinitely many triplets of positive integers

(a, b, c)

such that the following two conditions hold: 1.

\gcd(a, b, c) = 1

a+b+c, a^2+b^2+c^2

abc

are all perfect squares?
(Proposed by Ivan Chan Guan Yu)

1

Appending 1 or 3

Minivan chooses a prime number. Then every second, he adds either the digit

1

3

to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely?
(Proposed by Wong Jer Ren)

1

f(x)/y^2 - f(y)/x^2 <= (1/x - 1/y)^2

Find all functions

f:\mathbb{R}^+\rightarrow\mathbb{R}^+

such that for all

x, y\in\mathbb{R}^+

\frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2

\mathbb{R}^+

denotes the set of positive real numbers.)
(Proposed by Ivan Chan Guan Yu)

1

Changing 4 consecutive numbers

1, 2, \dots, 2023, 2024

is written on a whiteboard. Every second, Megavan chooses two integers

a

b

, and four consecutive numbers on the whiteboard. Then counting from the left, he adds

a

to the 1st and 3rd of those numbers, and adds

b

to the 2nd and 4th of those numbers. Can he achieve the sequence

2024, 2023, \dots, 2, 1

in a finite number of moves?
(Proposed by Avan Lim Zenn Ee)

1

JBM is isosceles

\triangle MAB

with a right angle at

A

and circumcircle

\omega

. Take any chord

CD

perpendicular to

AB

A, C, B, D, M

\omega

in this order. Let

AC

MD

intersect at point

E

O

be the circumcenter of

\triangle EMC

J

is the intersection of

BC

OM

JB = JM

.
(Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)