Parity - "A Lethal Weapon"

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

PARITY'PARITY’ means that odd+odd=even,odd+even=odd,even+even=even,odd(odd)=odd,odd(even)=even,even(even)=evenodd+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 RMO2016RMO 2016

QUESTION : f(x)=x3(k3)x211x+(4k8)f(x)=x^{3}-(k-3)x^{2}-11x+(4k-8) Find all integers kk such that roots of f(x)f(x) are integers.

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

This implies a+b+c=(k3),ab+bc+ac=11,abc=4(2k)a+b+c=(k-3),ab+bc+ac=-11,abc=4(2-k)

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

Therefore only one of them is even .

Let aa is even .

Also a+b+c=even;k3=even;k=odda+b+c=even;k-3=even;k=odd

f(x)=(xa)(xb)(xc)f(x)=(x-a)(x-b)(x-c) , this implies for xx be an integer f(x)f(x) is also an integer . Now put x=2x=2(because at x=2x=2 the kk term diappeas in f(x)f(x)) in f(x)f(x) this gives f(2)=(2a)(2b)(2c)=10f(2)=(2-a)(2-b)(2-c)=-10 . Now since aa is even therefore (2a)=2,2(2-a)={-2,2} where 2a=22-a=-2 is the essential case as 2a=22-a=2 makes k=evenk=even.

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

Note by Shivam Jadhav
4 years, 8 months ago

No vote yet
1 vote

  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.

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]( 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

Awesome use of parity!

Harsh Shrivastava - 4 years, 8 months ago

Log in to reply

Great solution!

A Former Brilliant Member - 4 years, 8 months ago

Log in to reply

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

A Former Brilliant Member - 4 years, 8 months ago

Log in to reply

ab+bc+ca=-11 not satisfied

Shivam Jadhav - 4 years, 8 months 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 - 4 years, 1 month ago

Log in to reply

Log in to reply

Log in to reply


Problem Loading...

Note Loading...

Set Loading...