JOM 2023

Part of JOM

Subcontests

(5)

1

Everything 0 mod 2023

m \times n

rectangle where

m,n\geq 2023

. The square in the

i

j

-th column is filled with the number

i+j

1\leq i \leq m, 1\leq j \leq n

. In each move, Alice can pick a

2023 \times 2023

subrectangle and add

1

to each number in it. Alice wins if all the numbers are multiples of

2023

after a finite number of moves. For which pairs

(m,n)

can Alice win?
Proposed by Boon Qing Hong

1

Minimum value of sum

n

positive real numbers

x_1,x_2,x_3,...,x_n

\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n

Determine the minimum value of

x_1+x_2+x_3+...+x_n

.
Proposed by Loh Kwong Weng

1

J, O, M colinear iff M is midpoint

Given an acute triangle

ABC

AB<AC

D

be the foot of altitude from

A

BC

M\neq D

be a point on segment

BC

\,J

K

AC

AB

respectively such that

K,J,M

lies on a common line perpendicular to

BC

. Let the circumcircles of

\triangle ABJ

\triangle ACK

O

J,O,M

are collinear if and only if

M

is the midpoint of

BC

.
Proposed by Wong Jer Ren

1

Product of digits

Ruby has a non-negative integer

n

. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or

0

86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6

)
Proposed by Wong Jer Ren

1

(x+2)^{2023}-x^{2023}

Does there exist a positive integer,

x

(x+2)^{2023}-x^{2023}

2023^{2023}

factors?
Proposed by Wong Jer Ren