Parity - "A Lethal Weapon"

This note is to show how \(‘PARITY’\) can be used very effectively to solve olympaid problems .

\('PARITY’\) means that \(odd+odd=even,odd+even=odd,even+even=even,odd(odd)=odd,odd(even)=even,even(even)=even\)

To explain its significance a example of question of \(RMO 2016\)

QUESTION : \(f(x)=x^{3}-(k-3)x^{2}-11x+(4k-8)\) Find all integers \(k\) such that roots of \(f(x)\) are integers.

Proof : let \(a,b,c\) are integral roots of \(f(x)\)

This implies \(a+b+c=(k-3),ab+bc+ac=-11,abc=4(2-k)\)

Since \(a,b,c\) are integers and \(abc=4(2-k)\) therefore atleast one of them is even or \(k=2\) but \(k=2\) doesn’t give integral roots but \(ab+bc+ac=-11\) therefore at a time 2 or 3 of \({a,b,c}\) can’t be even .

Therefore only one of them is even .

Let \(a\) is even .

Also \(a+b+c=even;k-3=even;k=odd\)

\(f(x)=(x-a)(x-b)(x-c)\) , this implies for \(x\) be an integer \(f(x)\) is also an integer . Now put \(x=2\)(because at \(x=2\) the \(k\) term diappeas in \(f(x)\)) in \(f(x)\) this gives \(f(2)=(2-a)(2-b)(2-c)=-10\) . Now since \(a\) is even therefore \((2-a)={-2,2}\) where \(2-a=-2\) is the essential case as \(2-a=2\) makes \(k=even\).

So \(a=4\) is a root of \(f(x)\) . after putting values of \(x=4\) in \(f(x)\) we get \(k=5,b=1,c=-3\) which matches our condition that \({k,b,c}=odd\) .

Note by Shivam Jadhav
2 years ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears 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

Awesome use of parity!

Harsh Shrivastava - 2 years ago

Log in to reply

Yes. Parity is of great help in many cases. It needs a little practice before we can make great use of it. I had used it in silution of quadratic equations. Thank you for the notes.

Niranjan Khanderia - 1 year, 5 months ago

Log in to reply

Why didn't you consider the case 2-a=|10|?

Brilliant Member - 2 years ago

Log in to reply

ab+bc+ca=-11 not satisfied

Shivam Jadhav - 2 years ago

Log in to reply

Great solution!

Brilliant Member - 2 years ago

Log in to reply

Log in to reply

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...