**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!

## Comments

Sort by:

TopNewestThe Algebra and Extra Credit problems are problems 4 and 6 from IMO 2012, aren't they? – José Marín Guzmán · 3 years, 7 months ago

Log in to reply

– Michael Tang · 3 years, 7 months ago

And the NT I believe is a USAMO problem.Log in to reply

And the number theory is 2013 USAMO #5.

@Michael Tong: If you don't want people to have references I'll delete this. – Daniel Chiu · 3 years, 7 months ago

Log in to reply

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. – Michael Tong · 3 years, 7 months ago

Log in to reply

– Ahaan Rungta · 3 years, 7 months ago

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

Calculus? Nice problems! Rules for the contest? I couldn't find out how to submit. – Ahaan Rungta · 3 years, 7 months ago

Log in to reply

Where will we post our answers? – Tierone James Santos · 3 years, 7 months ago

Log in to reply

Wow, Extra Credit Problem is from IMO 2012 P6 and Algebra Problem is from IMO 2012 P4 ...... – Zi Song Yeoh · 3 years, 7 months ago

Log in to reply

Are there any solutions to these questions??? – Happy Melodies · 3 years, 6 months ago

Log in to reply

– José Marín Guzmán · 3 years, 6 months ago

You can look the problems up in the Contests page of Art of problem solvingLog in to reply

Will there be a second set? I had fun solving (or trying to solve) these problems. Nice selection! – José Marín Guzmán · 3 years, 6 months ago

Log in to reply

Extra Credit – Michael Tong · 3 years, 7 months ago

Log in to reply

Combinatorics – Michael Tong · 3 years, 7 months ago

Log in to reply

Number Theory – Michael Tong · 3 years, 7 months ago

Log in to reply

Geometry – Michael Tong · 3 years, 7 months ago

Log in to reply

Algebra – Michael Tong · 3 years, 7 months ago

Log in to reply

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. – Michael Tong · 3 years, 7 months ago

Log in to reply