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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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TST Junior Romania 2025
ant_   1
N a minute ago by wassupevery1
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
1 reply
ant_
Yesterday at 5:01 PM
wassupevery1
a minute ago
J
H
Very tight inequalities
KhuongTrang   2
N 30 minutes ago by SunnyEvan
Source: own
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{35a+12b+2}+\frac{1}{35b+12c+2}+\frac{1}{35c+12a+2}\ge \frac{4}{39}.}$$$$\color{black}{\frac{1}{4a+9b+6}+\frac{1}{4b+9c+6}+\frac{1}{4c+9a+6}\le \frac{2}{9}.}$$When does equality hold?
2 replies
KhuongTrang
May 17, 2024
SunnyEvan
30 minutes ago
J
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Sum of First, Second, and Third Powers
Brut3Forc3   47
N 43 minutes ago by cubres
Source: 1973 USAMO Problem 4
Determine all roots, real or complex, of the system of simultaneous equations
\begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}
47 replies
Brut3Forc3
Mar 7, 2010
cubres
43 minutes ago
J
H
Modified Sum of floors
prMoLeGend42   2
N an hour ago by cubres
Find the closed form of : $\sum _{k=0}^{n-1} \left\lfloor \frac{ak+b}{n}\right \rfloor$ where $\gcd(a,n)=1$
2 replies
+1 w
prMoLeGend42
Yesterday at 9:09 AM
cubres
an hour ago
J
No more topics!
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J
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Proving CB = MN
bluecarneal   1
N Apr 15, 2015 by tkhalid
Source: Spring 2006 Tournament of Towns Junior O-Level #1
Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$.

(3 points)
1 reply
bluecarneal
Apr 15, 2015
tkhalid
Apr 15, 2015
J
Proving CB = MN
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Reveal topic tags
geometry
perpendicular bisector
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G H BBookmark kLocked kLocked NReply
Source: Spring 2006 Tournament of Towns Junior O-Level #1
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bluecarneal
9294 posts
bluecarneal
#1 h  Apr 15, 2015, 4:06 AM • 2 Y
Y by Adventure10, Mango247
Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$.

(3 points)
Z K Y
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tkhalid
965 posts
tkhalid
#2 h  Apr 15, 2015, 4:25 AM • 1 Y
Y by Adventure10
Solution
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N Quick Reply
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