Maximizing

by steven_zhang123, Mar 23, 2025, 12:56 AM

Find the largest positive real number $c$ such that for any positive integer $n$ , satisfies $\{ \sqrt{7n} \} \geq \frac{c}{\sqrt{7n}}$ .

China TST

number theory

Math Olympiad Workshops

by kokcio, Mar 23, 2025, 12:11 AM

Hello Math Enthusiasts!

I'm excited to announce a series of free Math Olympiad Workshops designed to help you sharpen your problem-solving skills in preparation for competitions. Whether you're a beginner or a seasoned competitor, these workshops aim to provide a supportive, challenging, and collaborative environment to explore advanced math topics.

Workshop Overview

Duration: 6 months (with the possibility of extending based on participant interest)

Structure: Weekly cycles, each dedicated to one of the main areas of Math Olympiad:
Week 1: Number Theory
Week 2: Geometry
Week 3: Algebra
Week 4: Combinatorics

Weekly Format
Monday: Problem Set Release: Approximately 30 problems will be posted covering the week's topic, which you will have chance to discuss.
Throughout the Week:
Theory Notes: I will share helpful theory and insights relevant to the problem set, giving you the tools you need to approach the problems.
Submission Opportunity: You can work on the problems and submit your solutions. I’ll review your work and provide feedback.
End of the Week: Solutions Post: I’ll release detailed solutions to all problems from the problem set.
Leaderboard: For those interested, we can maintain a table tracking participants who solve the most problems during the week.

Cycle Finale – Mock Contest
At the end of each 4-week cycle, we’ll host a Mock Contest featuring 4 problems (one from each topic). This is a great chance to simulate the competition environment and test your skills in a timed setting. I will review and provide feedback on your contest submissions.

Starting date: June 2

How to participate? Just write /signup under this post.

I believe these workshops will provide a comprehensive, engaging, and collaborative way to tackle Math Olympiad problems. I'm looking forward to seeing your creativity and problem-solving prowess!
If you have any questions or suggestions, please leave a comment below.

number theory question?

by jag11, Mar 22, 2025, 10:41 PM

Find the smallest positive integer n such that n is a multiple of 11, n +1 is a multiple of 10, n + 2 is a
multiple of 9, n + 3 is a multiple of 8, n +4 is a multiple of 7, n +5 is a multiple of 6, n +6 is a multiple of
5, n + 7 is a multiple of 4, n + 8 is a multiple of 3, and n + 9 is a multiple of 2.

I tried doing the mods and simplifying it but I'm kinda confused.

This post has been edited 1 time. Last edited by jag11, 3 hours ago
Reason: edit

number theory

Whiteboard magic again

by navi_09220114, Mar 22, 2025, 1:01 PM

Fix positive integers $n$ and $k$ , and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$ , $b_1, \cdots, b_n$ . An equation is written on a whiteboard: $t=*\times*\times\cdots\times*$ where $t$ is a fixed positive real number, with exactly $k$ asterisks.

Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$ , while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$ , so that the equation on the whiteboard is correct. Suppose for every positive real number $t$ , the number of ways for Ebi and Rubi to do so are equal.

Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other.

(Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$ .)

Proposed by Wong Jer Ren

This post has been edited 1 time. Last edited by navi_09220114, Yesterday at 1:02 PM

algebra

Complex numbers should be easy

by RenheMiResembleRice, Mar 21, 2025, 8:32 AM

I cant do the last part.

Attachments:

complex numbers

algebra

Mathhhhh

by mathbetter, Mar 20, 2025, 11:21 AM

Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?

This post has been edited 1 time. Last edited by mathbetter, Mar 20, 2025, 11:21 AM

combinatorics

FE on Stems

by mathscrazy, Dec 29, 2024, 12:35 PM

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ , $xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)$ Proposed by Aritra Mondal

functional equation

Quality FE

by pablock, Nov 17, 2020, 9:11 PM

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(xf(x-y))+yf(x)=x+y+f(x^2),$ for all real numbers $x$ and $y.$

algebra

functional equation

Iberoamerican

three "old" circles and four concurrent lines

by pohoatza, Jun 28, 2007, 6:41 PM

Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$ . Let $AB$ be the diameter of $w$ perpendicular to $t$ , so that $A, E, O_{1}$ are on the same side of $t$ . Prove that lines $AO_{1}$ , $BO_{2}$ , $EF$ and $t$ are concurrent.

Problem 4 (second day)

by darij grinberg, Jul 13, 2004, 2:46 PM

Let $n \geq 3$ be an integer. Let $t_1$ , $t_2$ , ..., $t_n$ be positive real numbers such that $n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).$ Show that $t_i$ , $t_j$ , $t_k$ are side lengths of a triangle for all $i$ , $j$ , $k$ with $1 \leq i < j < k \leq n$ .

Attachments:

This post has been edited 2 times. Last edited by djmathman, Jun 27, 2015, 11:54 PM

n-variable inequality

IMO

Submit

!!!!!!!!
by stroller, Mar 10, 2020, 8:15 PM
cooooooool
nice blog
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Victor is one of the GOATs.
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This blog is truly amazing.
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what a gem.
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by ahaanomegas, Dec 15, 2013, 7:21 PM
yo dawg i heard you imo
by Mewto55555, Aug 1, 2013, 10:25 PM
Good job on 5 problems on USAMO. I know that is a pretty bad score for someone as awesome as you on a normal USAMO, but no one got all 6 this year so it's GREAT!

VICTOR WANG 42 ON IMO 2013 LET'S GO.

See you at MOP this year (probably ).
by yugrey, May 3, 2013, 10:32 PM
you can just write "Solution
" instead of "Click to reveal hidden text
"

by math154, Feb 6, 2013, 6:33 PM
Hello. May I ask a stupid question: how to turn "Hidden Text" into hidden "Solution" tag?
by ysymyth, Feb 1, 2013, 12:59 PM
This is a good blog.I like it.
by Lingqiao, Jul 17, 2012, 4:21 PM
hi.i like the blog.
by Dranzer, Feb 9, 2012, 12:25 PM
sorry, i've decided this blog is just for the random stuff i do. don't take it personally... you can always post things on your own blog
by math154, Nov 23, 2011, 10:26 PM
what i got uncontribbed for posting a nice geo problem?
by yugrey, Nov 23, 2011, 1:32 AM
Can I be a contributor?
by Binomial-theorem, Oct 5, 2011, 12:39 AM
by gauss1181, Mar 8, 2009, 5:09 PM
i hate you
by alsyy, Mar 8, 2009, 4:37 PM
Fine. I don't care.
by math154, Mar 8, 2009, 1:34 AM
Math154: It was West Somethign Middle School from Columbia that got 2nd.
by Mewto55555, Mar 8, 2009, 1:20 AM
Oh dang, I already feel like deleting this blog. Maybe I should not do that for a while, though.
by math154, Mar 6, 2009, 2:50 AM
math154 mew and tinytim tied for first at chapter...
by AIME15, Mar 6, 2009, 1:06 AM
Thanks! Good luck to you too.
by math154, Mar 5, 2009, 1:00 AM
good luck on saturday
by gauss1181, Mar 5, 2009, 12:51 AM
Yeah gauss, our chapter sends the top 50 individuals and the top 10 teams.
by math154, Mar 4, 2009, 10:49 PM
Me wants be contrib.
by AIME15, Mar 4, 2009, 5:07 PM
I'll contrib in the future when I feel like it. I'm not gonna forget you, though, so don't worry.
by math154, Mar 2, 2009, 11:55 PM
third! contrib?
by nikeballa96, Mar 2, 2009, 10:40 PM
2nd Shout!
by gauss1181, Feb 28, 2009, 5:25 PM
1st Shout!
by mathemagician1729, Feb 28, 2009, 4:02 AM