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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 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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Inspired by old results
sqing   1
N a few seconds ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+b^2=2$ . Prove that
$$ (a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2 $$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq 2 $$
1 reply
1 viewing
sqing
2 minutes ago
sqing
a few seconds ago
J
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prove that any quadrilateral satisfying this inequality is a trapezoid
mqoi_KOLA   2
N 13 minutes ago by mqoi_KOLA
Prove that any quadrilateral satisfying this inequality is a Trapezoid/trapzium $$ |r - p| < q + s < r + p $$where $p,r$ are lengths of parallel sides and $q,s$ are other two sides.
2 replies
mqoi_KOLA
Yesterday at 3:48 AM
mqoi_KOLA
13 minutes ago
J
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Inspired by KHOMNYO2
sqing   2
N 29 minutes ago by sqing
Source: Own
Let $ a,b>0 $ and $ a^2+b^2=\frac{5}{2}. $ Prove that $$ 2a + 2b + \frac{1}{a} + \frac{1}{b} +\frac{ab}{\sqrt 2}\geq 5\sqrt 2$$$$ a + b +\frac{2}{a} + \frac{2}{b} + ab\geq \frac{5}{4} + \frac{13}{\sqrt 5} $$$$ a + b +\frac{2}{a} + \frac{2}{b} + \frac{ab}{\sqrt 2}\geq \frac{5}{4\sqrt 2} + \frac{13}{\sqrt 5} $$
2 replies
sqing
Mar 28, 2025
sqing
29 minutes ago
J
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USAMO 2003 Problem 4
MithsApprentice   71
N 36 minutes ago by LeYohan
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
71 replies
MithsApprentice
Sep 27, 2005
LeYohan
36 minutes ago
J
No more topics!
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angles wanted, excircle related
parmenides51   1
N Sep 15, 2024 by SomeonecoolLovesMaths
Source: 2012 Brazil Cono Sur Training List 1.5
In the triangle $ABC$, let $J$ be the center of the excircle that is tangent to side $BC$ in $A_1$ and to the extensions of sides $AC$ and $AB$ in $B_1$ and $C_1$, respectively. Suppose lines $A_1B_1$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular of $C_1$ on the line $DJ$. Determine the angles $\angle BEA_1$ and $\angle AEB_1$.
1 reply
parmenides51
Sep 3, 2021
SomeonecoolLovesMaths
Sep 15, 2024
J
angles wanted, excircle related
G H J
Reveal topic tags
geometry
angles
excircle
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G H BBookmark kLocked kLocked NReply
Source: 2012 Brazil Cono Sur Training List 1.5
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parmenides51
30630 posts
parmenides51
#1 h  Sep 3, 2021, 7:58 AM
Y by
In the triangle $ABC$, let $J$ be the center of the excircle that is tangent to side $BC$ in $A_1$ and to the extensions of sides $AC$ and $AB$ in $B_1$ and $C_1$, respectively. Suppose lines $A_1B_1$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular of $C_1$ on the line $DJ$. Determine the angles $\angle BEA_1$ and $\angle AEB_1$.
This post has been edited 3 times. Last edited by parmenides51, Dec 10, 2022, 2:45 AM
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SomeonecoolLovesMaths
3186 posts
SomeonecoolLovesMaths
#2 h  Sep 15, 2024, 2:11 PM
Y by
bump....
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