**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

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

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## Comments

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

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

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And the NT I believe is a USAMO problem.

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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.

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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.

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

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Where will we post our answers?

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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.

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Algebra

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Geometry

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Number Theory

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Combinatorics

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Extra Credit

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

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Are there any solutions to these questions???

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

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