Ten words (Just for fun)

How well can we summarise proofs?

Come up with proofs for the following statements in ten words or less. Who can make the most rigorous, accurate proof in just ten words?

11. Among 5151 integers from 11 to 100100 inclusive, there exists two summing to 101101.


22. 8n5n8^{n}-5^{n} is a multiple of 33 where nn is a positive integer.


33. The point OO in equilateral triangle ABCABC such that OA+OB+OCOA+OB+OC is minimised is the center of ABCABC.

All operations, except for brackets, are words.

Note by Joel Tan
5 years 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](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 2×3 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}

Comments

Sort by:

Top Newest

1) There are 5050 pairs with sum 101101. PHP.

Mursalin Habib - 5 years ago

Log in to reply

2) 8n5n2n2n=0(mod3)8^n - 5^n \equiv 2^n - 2^n = 0 \pmod{3}

Sharky Kesa - 5 years ago

Log in to reply

Isn't it supposed to be 8n5n8^{n}-5^{n} in the very first step...??

Rohit Sachdeva - 5 years ago

Log in to reply

Yes, typoed it.

Sharky Kesa - 5 years ago

Log in to reply

Proofs:

1) Gauss' pairing trick paired with pigeonhole principle.

2) == GP with ends 5n1,8n15^{n-1}, 8^{n-1}, ratio 1.61.6, summed thrice.

3) By construction of Fermat point, O=X(13)O = X(13).

Comments:

1) Alliteration 'p's yay! No I really did not want to refer to any language stuff.

2) It is trivial to see that all terms of the GP are integers, and ends refer to initial and final terms. I would have preferred 3k=0n18n1k5k3\sum^{n-1}_{k=0}8^{n-1-k}5^k but I am not sure how many words would that be.

3) Sorry for the overkill :P But anyway I could have mentioned an equilateral triangle is acute for clarity.

By the way, nice choice of problems!

Yong See Foo - 5 years ago

Log in to reply

8n5n=8n(83)n=3A8^{n}-5^{n}=8^{n}-(8-3)^{n}=3A

Rohit Sachdeva - 5 years ago

Log in to reply

2.8^n-5^=(8-5)k

Bhaskar Immadisetty - 5 years ago

Log in to reply

Q.2 It has one factor(8-5) = 3 means it is divisible by 3 Q.3 O is crcumcentre

Ravibhole Bhole - 5 years ago

Log in to reply

Maximum is at the fermat point, not at the circumcentre, although it is true in this case since it is equilateral.

Joel Tan - 5 years ago

Log in to reply

There can be 49 pairs summing to 101

gunit Varshney - 5 years ago

Log in to reply

2) If N>=0, 8^N - 5^N = 0 (mod 3)

Ian Hoolihan - 4 years, 11 months ago

Log in to reply

  1. Prove that sum of two integers within 100 is 101.

uma shankar - 5 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...