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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)!

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]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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circle w_B' is symmetric to w_B wrt the midpoint of AC
orl   17
N a few seconds ago by Nari_Tom
Source: ARO 2005 - problem 10.4
$w_B$ and $w_C$ are excircles of a triangle $ABC$. The circle $w_B'$ is symmetric to $w_B$ with respect to the midpoint of $AC$, the circle $w_C'$ is symmetric to $w_C$ with respect to the midpoint of $AB$. Prove that the radical axis of $w_B'$ and $w_C'$ halves the perimeter of $ABC$.
17 replies
orl
Apr 30, 2005
Nari_Tom
a few seconds ago
J
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collinear wanted, 2 circles passing through circumcenter related
parmenides51   4
N 20 minutes ago by AshAuktober
Source: Estonia IMO TST 2016 p5 / Slovenia IMO TST 2015
Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear
4 replies
parmenides51
Jul 22, 2020
AshAuktober
20 minutes ago
J
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V \le RS/2 in tetrahderon with equil base
Miquel-point   1
N an hour ago by kiyoras_2001
Source: Romanian IMO TST 1981, Day 4 P2
Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron.

Stere Ianuș
1 reply
Miquel-point
Sunday at 7:54 PM
kiyoras_2001
an hour ago
J
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a+b+c+abc=4 inequality
KhuongTrang   0
an hour ago
Source: own
Problem. Let $a,b,c$ be three non-negative real numbers with $a+b+c+abc=4.$ Prove that $$\color{black}{\sqrt{a^{2}+b^{2}+2c}+\sqrt{b^{2}+c^{2}+2a}+\sqrt{c^{2}+a^{2}+2b}\ge 6\sqrt{a+b+c-2}.}$$When does equality hold?
0 replies
KhuongTrang
an hour ago
0 replies
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No more topics!
H
J
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Prove parallelism
José   3
N May 28, 2024 by parmenides51
Source: Cono Sur 1991-problem 1
Let $A, B$ and $C$ be three non-collinear points and $E$ ($\ne B$) an arbitrary point not in the straight line $AC$. Construct the parallelograms $ABCD$ and $AECF$. Prove that $BE \parallel DF$.
3 replies
José
May 29, 2006
parmenides51
May 28, 2024
J
Prove parallelism
G H J
Reveal topic tags
geometry
parallelogram
geometry unsolved
L
G H BBookmark kLocked kLocked NReply
Source: Cono Sur 1991-problem 1
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José
1828 posts
José
#1 h  May 29, 2006, 4:13 PM • 2 Y
Y by Adventure10, Mango247
Let $A, B$ and $C$ be three non-collinear points and $E$ ($\ne B$) an arbitrary point not in the straight line $AC$. Construct the parallelograms $ABCD$ and $AECF$. Prove that $BE \parallel DF$.
Z K Y
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Andreas
578 posts
Andreas
#2 h  May 29, 2006, 5:12 PM • 2 Y
Y by Adventure10, Mango247
By simple angle chasing $\Delta ADF \equiv \Delta CBE$, $\Delta ABF \equiv \Delta CDE$, and $BEDF$ is a parallelogram.
Z K Y
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crazyfehmy
1345 posts
crazyfehmy
#3 h  Jun 8, 2008, 8:45 PM • 2 Y
Y by Adventure10, Mango247
too easy? :wink:
Z K Y
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parmenides51
30630 posts
parmenides51
#4 h  May 28, 2024, 1:42 PM
Y by
ABCD # => midpoint M of AC = midpoint of BD (1)
AECF # => midpoint M of AC = midpoint of EF (2)
(1), (2) = > midpoint of BD = midpoint of EF = > FBED # => BE// DF
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