Waste less time on Facebook — follow Brilliant.
×

RMO Part -9

1)Let \(1<a_1<a_2<a_3<.....a_{51}<142\) for positive integers \(a_1,a_2,a_3,.....a_{51}\).

Prove that among the 50 consecutive differences some value must occur at least 12 times.

2)Prove that in any perfect square the three digits immediately to the left of the unit digit cannot be 101.

Try to solve these 2 problems in 1 hour.

Also try my set RMO.

Note by Naitik Sanghavi
2 years, 3 months 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

Hints 1) Try Pigeonhole

2) Mods.

Alan Yan - 2 years, 3 months ago

Log in to reply

Log in to reply

@Rajdeep Dhingra ,I want to ask you something can u please give me your email or whatsapp no.

My email - nkscool21@gmail.com. Reply asap.

Naitik Sanghavi - 1 month, 1 week ago

Log in to reply

For question 2

The last four digits will be :

1010 - Not possible because perfect squares can't end in odd number of zeroes.

1011 - Not possible because perfect squares aren't of the form 4k + 3.

1012 - Not possible because perfect squares aren't of the form 4k + 2.

1013 - Not possible because perfect squares aren't of the form 8k + 5.

1014 - Not possible because perfect squares aren't of the form 4k + 2.

1015 - Not possible because perfect squares aren't of the form 4k + 3.

1016 - Not possible because perfect squares aren't of the form 16k + 8.

1017 - Not possible because perfect squares aren't of the form 5k + 2.

1018 - Not possible because perfect squares aren't of the form 4k + 2.

1019 - Not possible because perfect squares aren't of the form 4k + 3.

@naitik sanghavi Hope this works!

Ankit Kumar Jain - 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...