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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.

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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
J
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Olympiad
sasu1ke   3
N 4 minutes ago by sasu1ke
IMAGE
3 replies
sasu1ke
an hour ago
sasu1ke
4 minutes ago
J
H
How to judge a number is prime or not?
mingzhehu   1
N 2 hours ago by scrabbler94
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
1 reply
mingzhehu
Yesterday at 2:45 PM
scrabbler94
2 hours ago
J
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Eazy equation clap
giangtruong13   0
Yesterday at 4:03 PM
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$$
0 replies
giangtruong13
Yesterday at 4:03 PM
0 replies
J
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inequality
revol_ufiaw   3
N Yesterday at 2:55 PM 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
Yesterday at 2:05 PM
MS_asdfgzxcvb
Yesterday at 2:55 PM
J
No more topics!
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J
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equilaterals, 1 wanted, 5 given (2005-06 Savin Competition 6-8 p13)
parmenides51   4
N Jun 11, 2021 by manlio
Triangles $ABC$, $ADE$, $BFD$, $CEG$, $BGH$ are equilateral and equally oriented. Prove that triangle $EFH$ is also equilateral.
IMAGE
4 replies
parmenides51
Jun 10, 2021
manlio
Jun 11, 2021
J
equilaterals, 1 wanted, 5 given (2005-06 Savin Competition 6-8 p13)
G H J
Reveal topic tags
geometry
Equilateral
equilaterals
L
G H BBookmark kLocked kLocked NReply
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parmenides51
30629 posts
parmenides51
#1 h  Jun 10, 2021, 10:19 PM • 3 Y
Y by samrocksnature, centslordm, Mango247
Triangles $ABC$, $ADE$, $BFD$, $CEG$, $BGH$ are equilateral and equally oriented. Prove that triangle $EFH$ is also equilateral.
https://cdn.artofproblemsolving.com/attachments/e/9/333ac68b704a8e9f90d4fd7703449b6f8b1e0e.png
This post has been edited 1 time. Last edited by parmenides51, Jun 10, 2021, 10:20 PM
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parmenides51
30629 posts
parmenides51
#2 h  Jun 10, 2021, 10:20 PM • 2 Y
Y by samrocksnature, Mango247
posted for the image link
Attachments:
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franzliszt
23531 posts
franzliszt
#3 h  Jun 10, 2021, 10:37 PM • 1 Y
Y by samrocksnature
da hek? gliding principal or something
Z K Y
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greenturtle3141
3541 posts
greenturtle3141
#4 h  Jun 10, 2021, 11:00 PM • 2 Y
Y by samrocksnature, centslordm
Oh this is cute.

Solution
This post has been edited 1 time. Last edited by greenturtle3141, Jun 10, 2021, 11:43 PM
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manlio
3251 posts
manlio
#5 h  Jun 11, 2021, 5:39 PM • 1 Y
Y by centslordm
Nice problem.
This is my reasoning.

Triangle AEC=ABD=BGC=BHA=CFB by SAS criterium as all angles are shifted in the same manner , so AE=CF=BH and AH=CE=BF
finally again by SAS triangle AEH=CFE=BHF so EF=FH=HE
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