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!

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Math

Appears 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}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears 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}`	$\boxed{123}$

Comments

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

Ahaan Rungta - 7 years, 5 months ago

Where will we post our answers?

Tierone James Santos - 7 years, 5 months ago

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

José Marín Guzmán - 7 years, 5 months ago

And the NT I believe is a USAMO problem.

Michael Tang - 7 years, 5 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.

Daniel Chiu - 7 years, 5 months ago

@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 - 7 years, 5 months ago

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

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

Zi Song Yeoh - 7 years, 5 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.

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

Are there any solutions to these questions???

Happy Melodies - 7 years, 5 months ago

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

José Marín Guzmán - 7 years, 4 months ago

Math	Appears 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}`	$\boxed{123}$