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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.

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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
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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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inequality
revol_ufiaw   3
N an hour ago by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
2 hours ago
MS_asdfgzxcvb
an hour ago
J
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How to judge a number is prime or not?
mingzhehu   0
an hour ago
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
0 replies
1 viewing
mingzhehu
an hour ago
0 replies
J
H
What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N an hour ago by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
an hour ago
J
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Any nice way to do this?
NamelyOrange   3
N 2 hours ago by pooh123
Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$?
3 replies
NamelyOrange
Apr 2, 2025
pooh123
2 hours ago
J
No more topics!
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J
H
k octagon area wanted inside square (2014 Euler Teachers' MO I p2)
parmenides51   5
N Jul 24, 2020 by Ezmaths
The side of the square is a. The midpoints of the sides of the square are connected by segments with its opposite vertices (see figure). Find the area of the resulting octagon.
IMAGE
5 replies
parmenides51
Jul 24, 2020
Ezmaths
Jul 24, 2020
J
octagon area wanted inside square (2014 Euler Teachers' MO I p2)
G H J
Reveal topic tags
geometry
area
square
octagon
L
G H BBookmark kLocked kLocked NReply
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parmenides51
30629 posts
parmenides51
#1 h  Jul 24, 2020, 7:46 AM
Y by
The side of the square is a. The midpoints of the sides of the square are connected by segments with its opposite vertices (see figure). Find the area of the resulting octagon.
https://cdn.artofproblemsolving.com/attachments/7/e/ca21c59f41e03fd7af157217be5226ddaeae23.png
This post has been edited 1 time. Last edited by parmenides51, Jul 24, 2020, 7:46 AM
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parmenides51
30629 posts
parmenides51
#2 h  Jul 24, 2020, 7:46 AM
Y by
posted for the image link
Attachments:
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Ezmaths
87 posts
Ezmaths
#3 h  Jul 24, 2020, 7:51 AM
Y by
very famous question and there is also a video solution to the problem by mindyourdecisions here is the link https://www.youtube.com/watch?v=rwDgfxTulpI and a very general case of it(parallelogram).
This post has been edited 3 times. Last edited by Ezmaths, Jul 24, 2020, 6:41 PM
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Ezmaths
87 posts
Ezmaths
#4 h  Jul 24, 2020, 7:52 AM
Y by
btw the answer is $\frac{a^2}{6}$
This post has been edited 1 time. Last edited by Ezmaths, Jul 24, 2020, 7:52 AM
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Possible
6504 posts
Possible
#5 h  Jul 24, 2020, 4:43 PM • 1 Y
Y by Mango247
Ezmaths wrote:
very famous question and there is also a video solution to the problem by mindyourdecisions here is the link https://www.youtube.com/watch?v=rwDgfxTulpI and a very general case of it(parallelogram).

FTFY
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Ezmaths
87 posts
Ezmaths
#6 h  Jul 24, 2020, 6:36 PM
Y by
@above thanks
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