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# Weekly Math Olympiad Challenge! Problem Set #1

Number Theory

Given positive integers $$m$$ and $$n$$, prove that there exists a positive integer $$c$$ such that the numbers $$cm$$ and $$cn$$ have the same occurrences of each non-zero digit in base $$10$$.

Geometry

Let $$ABC$$ be a triangle with $$\angle A = 90^\circ$$. Points $$D$$ and $$E$$ lie on sides $$AC$$ and $$AB$$, respectively, such that $$\angle ABD = \angle DBC$$ and $$\angle ACE = \angle ECB$$. Segments $$ED$$ and $$CE$$ meet at $$I$$. Determine whether or not it is possible for segments $$AB, AC, BI, ID, CI, IE$$ to all have integer side lengths.

Algebra

Find all functions $$f : \mathbb{Z} \rightarrow \mathbb{Z}$$ such that for all integers $$a, b, c$$ that satisfy $$a+b+c = 0$$, the following equality holds:

$$(f(a))^2 + (f(b))^2 + (f(c))^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(a)f(c)$$.

Combinatorics

A circle is divided into $$432$$ congruent arcs by $$432$$ points. The points are colored with four colors such that some $$108$$ points are colored each of Red, Blue, Green, and Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

Extra Credit

Find all positive integers $$n$$ for which there exists non-negative integers $$a_1, a_2, \cdots, a_n$$ such that

$\large \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \ldots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac {n}{3^{a_n}}$

Want a category not included here? Problems are too hard? Too easy? Any feedback would be appreciated!

Note by Michael Tong
4 years ago

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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} $$\boxed{123}$$

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Calculus? Nice problems! Rules for the contest? I couldn't find out how to submit.

- 4 years ago

Where will we post our answers?

- 4 years ago

Wow, Extra Credit Problem is from IMO 2012 P6 and Algebra Problem is from IMO 2012 P4 ......

- 4 years ago

The Algebra and Extra Credit problems are problems 4 and 6 from IMO 2012, aren't they?

- 4 years ago

And the NT I believe is a USAMO problem.

- 4 years ago

The combinatorics is USAMO 2012 #2 and geometry is USAMO 2010 #4.

And the number theory is 2013 USAMO #5.

@Michael Tong: If you don't want people to have references I'll delete this.

- 4 years ago

I don't mind, I trust the community enough to not cheat

This isn't really a competition, it's just for fun and the "challenge." If I wanted to grade it, I would make up problems.

- 4 years ago

Well, it's pretty easy to just copy this onto AoPS and find out the sources if wanted. :P

- 4 years ago

Are there any solutions to these questions???

- 4 years ago

You can look the problems up in the Contests page of Art of problem solving

- 4 years ago

Will there be a second set? I had fun solving (or trying to solve) these problems. Nice selection!

- 4 years ago

Extra Credit

- 4 years ago

Combinatorics

- 4 years ago

Number Theory

- 4 years ago

Geometry

- 4 years ago

Algebra

- 4 years ago

Okay, thanks for the feedback. I think for answers, people can post their solution (full solution or just a general overview) to comments that I have posted here in the comment section, each corresponding to one question.

- 4 years ago