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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
3 years, 1 month ago

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The Algebra and Extra Credit problems are problems 4 and 6 from IMO 2012, aren't they? José Marín Guzmán · 3 years, 1 month ago

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@José Marín Guzmán And the NT I believe is a USAMO problem. Michael Tang · 3 years, 1 month ago

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@Michael Tang 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. Daniel Chiu · 3 years, 1 month ago

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@Daniel Chiu 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. Michael Tong · 3 years, 1 month ago

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@Daniel Chiu Well, it's pretty easy to just copy this onto AoPS and find out the sources if wanted. :P Ahaan Rungta · 3 years, 1 month ago

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Calculus? Nice problems! Rules for the contest? I couldn't find out how to submit. Ahaan Rungta · 3 years, 1 month ago

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Where will we post our answers? Tierone James Santos · 3 years, 1 month ago

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Wow, Extra Credit Problem is from IMO 2012 P6 and Algebra Problem is from IMO 2012 P4 ...... Zi Song Yeoh · 3 years, 1 month ago

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Are there any solutions to these questions??? Happy Melodies · 3 years ago

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@Happy Melodies You can look the problems up in the Contests page of Art of problem solving José Marín Guzmán · 3 years ago

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

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Extra Credit Michael Tong · 3 years, 1 month ago

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Combinatorics Michael Tong · 3 years, 1 month ago

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Number Theory Michael Tong · 3 years, 1 month ago

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Geometry Michael Tong · 3 years, 1 month ago

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Algebra Michael Tong · 3 years, 1 month ago

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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. Michael Tong · 3 years, 1 month ago

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