max term

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

max term

G H J

Reveal topic tags

arithmetic sequence

algebra unsolved

L

G H BBookmark kLocked kLocked NReply

Source: CSEMO 2005-7

The post below has been deleted. Click to close.

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

417 posts

zhaoli

#1 h Jul 17, 2005, 3:21 PM • 2 Y

Y by Adventure10, Mango247

(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$ , where $x \in R$ and $a$ is parameter.

(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$ , and satisfy the condition
$\sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507.$
Find the maximum value of $n$ .

Z K Y

The post below has been deleted. Click to close.

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

8 posts

Micchi

#2 h Jul 21, 2005, 1:26 PM • 1 Y

Y by Adventure10

this question definitely had our team involved in a mathematical debate. nobody knew who was right. Alaban, our teammate told us there were no roots. Some answered 4, that's me and Jako... so need help here...

Z K Y

The post below has been deleted. Click to close.

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

803 posts

#3 h Jul 24, 2005, 9:46 AM • 2 Y

Y by Adventure10, Mango247

For (1), draw the graph of $|x + 1| + |x + 2| + |x + 3| = y$ , we find that there can be 0,1 or 2 roots.
For (2), I think case division should work (eg. the difference $d = 0$ , $1$ or more than $2$ )

Z K Y

The post below has been deleted. Click to close.

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

741 posts

enescu

#4 h Jul 24, 2005, 11:14 AM • 2 Y

Y by Adventure10, Mango247

For (2), $n=26$ , right?

Z K Y

The post below has been deleted. Click to close.

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

417 posts

zhaoli

#5 h Jul 24, 2005, 1:20 PM • 2 Y

Y by Adventure10, Mango247

enescu, you are right!

Z K Y

The post below has been deleted. Click to close.

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

1628 posts

#6 h Jul 24, 2005, 5:45 PM • 2 Y

Y by Adventure10, Mango247

Can someone post the solution to 2?

Z K Y

The post below has been deleted. Click to close.

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

741 posts

enescu

#7 h Jul 24, 2005, 6:08 PM • 3 Y

Y by Carl2015, Adventure10, Mango247

First, observe that if $a_{1}<a_{2}<\ldots<a_{n}$ are real numbers and $f:\mathbb{R}\rightarrow\mathbb{R}$ is defined as
$f\left( x\right) =\left| x-a_{1}\right| +\left| x-a_{2}\right| +\ldots+\left| x-a_{n}\right| ,$
then $f$ is first decreasing, and then increasing. More precisely, if $n$ is odd, say $n=2k+1,$ then $f$ is strictly decreasing on the interval $(-\infty,a_{k}]$ and strictly increasing on $[a_{k},+\infty)$ , while if $n$ is even, say $n=2k,$ then $f$ is strictly decreasing on the interval $(-\infty,a_{k}],$ constant on $[a_{k},a_{k+1}]$ and strictly increasing on $[a_{k+1},+\infty).$

Now, let WLOG $a_{1}\leq\ldots\leq a_{n}$ be our arithmetic sequence and $f$ defined as above. We see that $f(0)=f\left( -1\right) =f\left( 2\right)$ , hence three values of $f$ are equal. This is possible only if $n$ is even (say $n=2k$ ) and $0,-1,2\in\lbrack a_{k},a_{k+1}].$ We deduce that the common difference $d$ is at least $3,$ and $a_{k}\leq-1,a_{k+1}\geq2.$ A short computation shows that in this case
$507=\sum\left| a_{i}\right| \geq3k^{2},$
hence $k\leq13,$ that is, $n\leq26.$

Equality is obtained when $a_{1}=-37,a_{2}=-34,\ldots,a_{26}=38.$

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a