A good NT exercise

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

A good NT exercise

G H J

Reveal topic tags

relatively prime

L

G H BBookmark kLocked kLocked NReply

Source: Bulgaria JBMO TST 2022 Day 2 Problem 3

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

1188 posts

#1 h May 15, 2022, 2:05 PM

Y by

The integers $a$ , $b$ , $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$ . Determine the largest possible value of $d$ ,

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

194 posts

Iora

#2 h May 15, 2022, 2:39 PM

Y by

take $a= -2022, d=2022, b=c=0$ , so the answer is $d=2022$ , something feels wrong

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

1188 posts

#3 h May 15, 2022, 2:39 PM • 1 Y

Y by Iora

@above -2022 and 0 are not relatively prime.

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

825 posts

#4 h May 15, 2022, 2:52 PM • 1 Y

Y by Mango247

$b|ac\Rightarrow b|c$ ,ket $c=bk$ ,then $d=-ak$
so $a+b+bk-ak=0,(a-b)|(a+b)$
which means $|a-b|=1$ or $2$
(1) $|a-b|=1$ ,then $|k|=|a+b|,|d|=|a(a+b)|$ ,its maximum value is $32\cdot (32+31)=2016$
(2) $|a-b|=2,2\nmid a,b$ ,then $|k|=|\frac{a+b}{2}|,|d|=|a\frac{a+b}{2}|$
its maximum value is $45\cdot 44=1980$
so the maximum is $2016$

This post has been edited 2 times. Last edited by MathLover_ZJ, May 15, 2022, 2:56 PM

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a