Strange Sequences

Two real sequences $a_1,a_2,\ldots a_{2014}$ and $b_1,b_2,\ldots b_{2014}$ both satisfy that for any two distinct positive integers $i,j$ , there exists a positive integer $k$ distinct from $i,j$ such that

$(a_i-a_j)(b_i-b_k)=(a_i-a_k)(b_i-b_j)$

where $1\le i,j,k \le 2014$ . Prove that

$(a_i-a_j)(b_i-b_k)=(a_i-a_k)(b_i-b_j)$

is true for all $i,j,k$ where $1\le i,j,k\le 2014$ .

Source: me

#Algebra #Contradiction #Sequences #Proofs #CleverInterpretation

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Sort by:

Top Newest

I have a one-line solution, given a 'well known fact'.

Hint: Take the obvious geometric interpretation.

Calvin Lin Staff - 6 years, 4 months ago

I think the solution you found is the intended one, although I wouldn't really call it a one-liner. Also, assume you need to prove that well known fact since that's basically most of the problem.

Daniel Liu - 6 years, 4 months ago

Haha, my definition of one-liner is basically that you just need that 1 idea, and things fall naturally after that.

This is a nice proof question. I really like the fact that is referenced, in part because it was one of the first applications of external combinatorics that I came across in Artur Engel, which showed me how beautiful such arguments could be.

Calvin Lin Staff - 6 years, 4 months ago

@Calvin Lin – Did you mean "proof question"?

Tan Li Xuan - 6 years, 4 months ago

@Tan Li Xuan – Yes, edited. Thanks.

Calvin Lin Staff - 6 years, 4 months ago

This is basically Sylvester–Gallai theorem: given a finite number of points in the plane, either all the points are collinear or there is a line containing exactly 2 points. I think best proof is using extreme principle.

Roger Lu - 6 years, 4 months ago

Yea, that's basically the solution. I also know of a proof by contradiction, but I'll try to find a proof using the extremal principle.

Daniel Liu - 6 years, 4 months ago

This is tough. I have the feeling PIE is a good place to start, but I'll work more on it later.

Finn Hulse - 6 years, 4 months ago

Good luck; this is a really tough problem to crack. I tried to turn it into a problem but I failed so this is the best I can do.

Daniel Liu - 6 years, 4 months ago

Okay. :D

Finn Hulse - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$