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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
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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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jlacosta
Apr 2, 2025
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trigonometric functions
VivaanKam   9
N 41 minutes ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Yesterday at 8:29 PM
aok
41 minutes ago
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Number Theory with set and subset and divisibility
SomeonecoolLovesMaths   3
N 3 hours ago by martianrunner
Let $S = \{ 1,2, \cdots, 100 \}$. Let $A$ be a subset of $S$ such that no sum of three distinct elements of $A$ is divisible by $5$. What is the maximum value of $\mid A \mid$.
3 replies
SomeonecoolLovesMaths
Apr 21, 2025
martianrunner
3 hours ago
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Basic geometry
AlexCenteno2007   7
N 3 hours ago by KAME06
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
7 replies
AlexCenteno2007
Feb 9, 2025
KAME06
3 hours ago
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Generating Functions
greenplanet2050   7
N 3 hours ago by rchokler
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
7 replies
greenplanet2050
Yesterday at 10:42 PM
rchokler
3 hours ago
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angle chasing, right and isosceles triangle wanted (2014 HOMC Junior Q8)
parmenides51   2
N Jul 19, 2019 by george_54
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.
2 replies
parmenides51
Jul 18, 2019
george_54
Jul 19, 2019
J
angle chasing, right and isosceles triangle wanted (2014 HOMC Junior Q8)
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Reveal topic tags
geometry
right triangle
isosceles
Isosceles Triangle
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parmenides51
30650 posts
parmenides51
#1 h  Jul 18, 2019, 9:47 PM • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.
This post has been edited 1 time. Last edited by parmenides51, Jul 18, 2019, 9:47 PM
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vanstraelen
9003 posts
vanstraelen
#2 h  Jul 19, 2019, 12:06 PM • 1 Y
Y by Adventure10
Let $A(a,b),B(0,0)C(c,0)$.
Then $D(a-b,a+b),E(a+b,b+c-a),F(\frac{c}{2},\frac{c}{2})$.
We have $DF \bot DE$ and $DF = DE$.
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george_54
1585 posts
george_54
#3 h  Jul 19, 2019, 3:47 PM • 2 Y
Y by Adventure10, Mango247
From Vecten's theorem , $DC=BE$ and $DC\bot BE.$ But $BHFC$ is cyclic, so red angles are congruent

and tirangles $BFE, CFD$ are also congruent. That means $\boxed{FE=FD}$ and blue angles are congruent.

Hence, $DHFE$ is cyclic and $\boxed{\angle EFD=\angle EHD=90^\circ}$
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