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# Olympiad Proof Problem - Day 2

This is the second proof problem I am posting. Try these problems and post your working. I will eventually post my solution to all of these notes, named "Olympiad Proof Problems", three days after I post them. So here is the problem.

Consider any ten distinct points $$P_{1}$$, $$P_{2}$$, $$P_{3}$$ $$\ldots$$ $$P_{10}$$ in the square $$\mathcal{S}$$ of unit side length. Prove that one can find a pair of points among those ten such that their distance is atmost $$\dfrac{\sqrt{2}}{3}$$ units.

Try more proof problems from this set Olympiad Proof Problems

Note by Surya Prakash
1 year, 10 months ago

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Divide the unit square into $$9$$ small squares of side length $$\frac {1}{3}$$ . $$10$$ points are to be chosen within the unit square which means that there will be at least one small square with more than $$1$$ point. The maximum distance between any two points in such a square is when they are at diagonally opposite points of that square. By the Pythagorean theorem this distance is $$\frac{\sqrt2}{3}$$ . · 1 year, 10 months ago

Awesome solution! · 1 year, 10 months ago

Nice solution!! (+1) · 1 year, 10 months ago

It's just PHP followed by Pythagoras Theorem. Great solution BTW. +1 · 1 year, 10 months ago

ni shuo sha? · 1 year, 10 months ago

Nándào nǐ bù zhīdào yīngyǔ · 1 year, 10 months ago

What? English, please. · 1 year, 10 months ago

Here is a variant of the above question:

Consider a equilateral triangular dartboard of side 2ft. If you throw five darts with no misses, then show that at least two will be within one feet of each other.

###### Pakistan First Round 2015
· 1 year, 10 months ago

Extremal Principle!! · 1 year, 10 months ago

Actually I solved it in the paper with PHP.. · 1 year, 10 months ago

I solved it without paper using Penguin-hole Principle. · 1 year, 10 months ago

WOW Sharky(-the genius) you win! · 1 year, 10 months ago

Once again, PHP followed by some simple maths.

Connect the medians of the triangle to create for equilateral triangles of side length 1ft. Now, at least 2 darts go into on of these triangles by PHP. The furthest distance 2 points can be separated in an equilateral triangle is the 2 vertices (pick any other 2 points and they are closer). This length is the side length, which is 1ft. Therefore, at least 2 darts are within 1ft of each other.

(Note: PHP stands for Pigeonhole Principle (or as I like to call it 'Penguin-hole Principle') · 1 year, 10 months ago

I like to call it Pig-on-hole Principle. · 1 year, 10 months ago