# 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
5 years, 8 months ago

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

- 5 years, 8 months ago

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

- 5 years, 8 months ago

And the NT I believe is a USAMO problem.

- 5 years, 8 months 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.

- 5 years, 8 months 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.

- 5 years, 8 months ago

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

- 5 years, 8 months ago

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

- 5 years, 8 months ago

Where will we post our answers?

- 5 years, 8 months 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.

- 5 years, 8 months ago

Algebra

- 5 years, 8 months ago

Geometry

- 5 years, 8 months ago

Number Theory

- 5 years, 8 months ago

Combinatorics

- 5 years, 8 months ago

Extra Credit

- 5 years, 8 months ago

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

- 5 years, 7 months ago

Are there any solutions to these questions???

- 5 years, 7 months ago

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

- 5 years, 7 months ago