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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   0
4 minutes ago
IMAGE
0 replies
sasu1ke
4 minutes ago
0 replies
J
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How to judge a number is prime or not?
mingzhehu   1
N 43 minutes 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
Today at 2:45 PM
scrabbler94
43 minutes ago
J
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Eazy equation clap
giangtruong13   0
Today 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
Today at 4:03 PM
0 replies
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inequality
revol_ufiaw   3
N Today 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
Today at 2:05 PM
MS_asdfgzxcvb
Today at 2:55 PM
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No more topics!
H
J
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equal areas from midpoints of ABCD (2007 Estonia Open Junior 1.2)
parmenides51   4
N Mar 6, 2024 by MattArg
The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.
4 replies
parmenides51
Mar 27, 2020
MattArg
Mar 6, 2024
J
equal areas from midpoints of ABCD (2007 Estonia Open Junior 1.2)
G H J
Reveal topic tags
equal areas
midpoints
geometry
parallelogram
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parmenides51
30629 posts
parmenides51
#1 h  Mar 27, 2020, 10:19 AM
Y by
The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.
This post has been edited 2 times. Last edited by parmenides51, Mar 29, 2020, 8:34 PM
Reason: typo mentioned in #2: replaced the word rectangle with the word parallelogram
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brainiacmaniac31
2170 posts
brainiacmaniac31
#2 h  Mar 29, 2020, 8:10 PM • 1 Y
Y by parmenides51
$EFGH$ isn't necessarily a rectangle, did you mean parallelogram?
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parmenides51
30629 posts
parmenides51
#3 h  Mar 29, 2020, 8:32 PM
Y by
yes, it means parallelogram, thanks for noticing
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Dumb_at_math_1729
14 posts
Dumb_at_math_1729
#4 h  Jan 23, 2024, 5:44 PM
Y by
Its easy to prove that EFGH is a parallelogram. Draw the diagonal AC & BD. Then its easy.........
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MattArg
54 posts
MattArg
#5 h  Mar 6, 2024, 12:27 PM
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It is sometimes called Varignon theorem
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