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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
Sharygin 2025 CR P2
Gengar_in_Galar   4
N a few seconds ago by FKcosX
Source: Sharygin 2025
Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points.
Proposed by:A Kuznetsov
4 replies
Gengar_in_Galar
Mar 10, 2025
FKcosX
a few seconds ago
2 var inquality
sqing   5
N 2 minutes ago by ionbursuc
Source: Own
Let $ a ,  b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le   \frac{3}{2}. $ Show that$$ a+b+ab\geq1$$Let $ a ,  b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le   \frac{5}{6}. $ Show that$$ a+b+ab\geq2$$
5 replies
+1 w
sqing
Today at 4:06 AM
ionbursuc
2 minutes ago
100 Selected Problems Handout
Asjmaj   32
N 5 minutes ago by John_Mgr
Happy New Year to all AoPSers!
 :clap2:

Here’s my modest gift to you all. Although I haven’t been very active in the forums, the AoPS community contributed to an immense part of my preparation and left a huge impact on me as a person. Consider this my way of giving back. I also want to take this opportunity to thank Evan Chen—his work has consistently inspired me throughout my olympiad journey, and this handout is no exception.



With 2025 drawing near, my High School Olympiad career will soon be over, so I want to share a compilation of the problems that I liked the most over the years and their respective detailed write-ups. Originally, I intended it just as a personal record, but I decided to give it some “textbook value” by not repeating the topics so that the selection would span many different approaches, adding hints, and including my motivations and thought process.

While IMHO it turned out to be quite instructive, I cannot call it a textbook by any means. I recommend solving it if you are confident enough and want to test your skills on miscellaneous, unordered, challenging, high-quality problems. Hints will allow you to not be stuck for too long, and the fully motivated solutions (often with multiple approaches) should help broaden your perspective. 



This is my first experience of writing anything in this format, and I’m not a writer by any means, so please forgive any mistakes or nonsense that may be written here. If you spot any typos, inconsistencies, or flawed arguments whatsoever (no one is immune :blush: ), feel free to DM me. In fact, I welcome any feedback or suggestions.

I left some authors/sources blank simply because I don’t know them, so if you happen to recognize where and by whom a problem originated, please let me know. And quoting the legend: “The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me.” 



I’ll likely keep a separate file to track all the typos, and when there’s enough, I will update the main file. Some problems need polishing (at least aesthetically), and I also have more remarks to add.

This content is only for educational purposes and is not meant for commercial usage.



This is it! Good luck in 45^2, and I hope you enjoy working through these problems as much as I did!

Here's a link to Google Drive because of AoPS file size constraints: Selected Problems
32 replies
Asjmaj
Dec 31, 2024
John_Mgr
5 minutes ago
Where is the equality?
AndreiVila   2
N an hour ago by MihaiT
Source: Romanian District Olympiad 2025 9.3
Determine all positive real numbers $a,b,c,d$ such that $a+b+c+d=80$ and $$a+\frac{b}{1+a}+\frac{c}{1+a+b}+\frac{d}{1+a+b+c}=8.$$
2 replies
AndreiVila
Mar 8, 2025
MihaiT
an hour ago
Inequality with roots of cubic
Michael Niland   0
Today at 7:23 AM
The equation $x^3+px+q =0$ has real roots $ a_1 ,a_2 , a_3 $ where $a_1 \leq a_2 \leq a_3$.

Similarly the equation $x^3 +rx +s=0$ has real roots $ b_1, b_2, b_3 $ where $b_1 \leq b_2,\leq b_3$.

Prove that if $ \frac{a_1}{b_1} \leq \frac{a_2}{b_2} \leq \frac{a_3}{b_3}$ (s non zero), then $(\frac{p}{r})^3 =(\frac{q}{s})^2$
0 replies
Michael Niland
Today at 7:23 AM
0 replies
Inequalities
nhathhuyyp5c   3
N Today at 6:36 AM by giangtruong13
Let $a,b,c$ be positive reals such that $a+b+c+2=abc$. Find the maximum value of $$\frac{a+1}{a^2+2}+\frac{b+1}{b^2+2}+\frac{c+1}{c^2+2}$$Given $n\ge 2$ non-zero reals $x_1,x_2,\cdots x_n$ such that their sum is $100$. Prove that there exists two numbers $x_i,x_j$ such that $\frac{1}{2}\le \left|\frac{x_i}{x_j}\right|\le 2$
3 replies
nhathhuyyp5c
Jan 10, 2025
giangtruong13
Today at 6:36 AM
Algebra-1
JetFire008   3
N Today at 5:11 AM by BackToSchool
Find real numbers $p$,$q$ if $1+i$ is a root of $x^3+px^2+qx+6=0$ and solve the equation.
3 replies
JetFire008
Yesterday at 3:19 PM
BackToSchool
Today at 5:11 AM
Cool problem
jkim0656   3
N Today at 4:09 AM by RedFireTruck
Hey AoPS!
I came across a problem recently and it goes like this:
Which is greater:
$$2 ^ {100!} $$or $$2^{100}!$$Soo... can u guys help? thx!
:yoda: and may the force be with u!
Notes: I don't exactly know where this problem came from, but if u find that u are the orig maker of this problem feel free to drop me a PM and ill add u to my post :)
3 replies
jkim0656
Today at 3:04 AM
RedFireTruck
Today at 4:09 AM
help me at this pls i'm so confused
myname17042005   12
N Today at 3:07 AM by sqing
if a, b, c are real numbers and a, b, c >0, prove that
pls help meee
12 replies
myname17042005
Jun 12, 2020
sqing
Today at 3:07 AM
Trig Multiplication
szhang7   1
N Today at 2:24 AM by joeym2011
Find the exact value of $(2-\sin^2(\frac{\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))$.
1 reply
szhang7
Yesterday at 3:50 PM
joeym2011
Today at 2:24 AM
Algebra-2
JetFire008   1
N Today at 2:11 AM by joeym2011
Prove that if $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$, then $f(x^n)$ is divisible by $x^n-1$.
1 reply
JetFire008
Yesterday at 3:23 PM
joeym2011
Today at 2:11 AM
a/b + b/a never integer ?
MTA_2024   3
N Yesterday at 10:35 PM by ohiorizzler1434
Let $a$ and $b$ be 2 distinct positive integers.
Can $\frac a b +\frac b a $ be in an integer. Prove why ?
3 replies
MTA_2024
Yesterday at 3:08 PM
ohiorizzler1434
Yesterday at 10:35 PM
2014 Community AIME / Marathon ... Algebra Medium #1 quartic
parmenides51   5
N Yesterday at 7:17 PM by CubeAlgo15
Let there be a quartic function $f(x)$ with maximums $(4,5)$ and $(5,5)$. If $f(0) = -195$, and $f(10)$ can be expressed as $-n$ where $n$ is a positive integer, find $n$.

proposed by joshualee2000
5 replies
parmenides51
Jan 21, 2024
CubeAlgo15
Yesterday at 7:17 PM
interesting problem
sausagebun   1
N Yesterday at 4:05 PM by mathprodigy2011
Six points, labeled A, B, C, D, E, and F, are positioned consecutively on a straight line. Let G be a point not located on this line. The following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. Determine the area of triangle BGE.
I brute forced this with trig, was wondering if theres a more elegant way of doing this
1 reply
sausagebun
Yesterday at 3:21 PM
mathprodigy2011
Yesterday at 4:05 PM
IMO 2012 P5
mathmdmb   122
N Today at 2:51 AM by KevinYang2.71
Source: IMO 2012 P5
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Proposed by Josef Tkadlec, Czech Republic
122 replies
mathmdmb
Jul 11, 2012
KevinYang2.71
Today at 2:51 AM
IMO 2012 P5
G H J
Source: IMO 2012 P5
G
H
=
a