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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.
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[*]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
J
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Equal angles (a very old problem)
April   54
N a few seconds ago by joshualiu315
Source: ISL 2007, G3, VAIMO 2008, P5
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD = \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP = \angle DAQ$.

Author: Vyacheslav Yasinskiy, Ukraine
54 replies
April
Jul 13, 2008
joshualiu315
a few seconds ago
J
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Prove line tangent to circle
mofumofu   28
N a minute ago by joshualiu315
Source: China Mathematical Olympiad 2019 Q3
Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.
28 replies
mofumofu
Nov 14, 2018
joshualiu315
a minute ago
J
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IMO Shortlist 2012, Geometry 2
lyukhson   86
N a minute ago by joshualiu315
Source: IMO Shortlist 2012, Geometry 2
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
86 replies
lyukhson
Jul 29, 2013
joshualiu315
a minute ago
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C-B=60 <degrees>
Sasha   25
N 2 minutes ago by joshualiu315
Source: Moldova TST 2005, IMO Shortlist 2004 geometry problem 3
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.

Proposed by Hojoo Lee, Korea
25 replies
Sasha
Apr 10, 2005
joshualiu315
2 minutes ago
J
No more topics!
H
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Hard problem
Zuyong   4
N Yesterday at 12:13 PM by mpcnotnpc
Source: ?
Given that $P(x)$ is a cubic polynomial such that its discriminant is non-positive and two numbers $m,n \in \mathbb{R}$.
Prove that:
\begin{align*} \text{If} \ \begin{cases} P(m) >0 \\ P(n) >0 \end{cases} \text{then} \ P(x) \geqslant 0 \ \ \forall x \in (m,n) .\end{align*}
4 replies
Zuyong
Oct 24, 2024
mpcnotnpc
Yesterday at 12:13 PM
J
Hard problem
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Reveal topic tags
inequalities
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Source: ?
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Zuyong
669 posts
Zuyong
#1 h  Oct 24, 2024, 10:41 AM
Y by
Given that $P(x)$ is a cubic polynomial such that its discriminant is non-positive and two numbers $m,n \in \mathbb{R}$.
Prove that:
\begin{align*} \text{If} \ \begin{cases} P(m) >0 \\ P(n) >0 \end{cases} \text{then} \ P(x) \geqslant 0 \ \ \forall x \in (m,n) .\end{align*}
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tigerBoss101
4 posts
tigerBoss101
#2 h  Oct 24, 2024, 11:24 AM
Y by
Solution
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Zuyong
669 posts
Zuyong
#3 h  Oct 25, 2024, 11:57 PM
Y by
Sure??? Check it again
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Zuyong
669 posts
Zuyong
#4 h  Yesterday at 10:25 AM
Y by
Zuyong wrote:
Given that $P(x)$ is a cubic polynomial such that its discriminant is non-positive and two numbers $m,n \in \mathbb{R}$.
Prove that:
\begin{align*} \text{If} \ \begin{cases} P(m) >0 \\ P(n) >0 \end{cases} \text{then} \ P(x) \geqslant 0 \ \ \forall x \in (m,n) .\end{align*}

Any ideas?
Z K Y
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mpcnotnpc
23 posts
mpcnotnpc
#5 h  Yesterday at 12:13 PM
Y by
Essentially what this reduces to proving is that given any two positive values for the polynomial, there's no root in between them? Like you can think about it like a graph x^3 + 3 or smth.
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