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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
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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
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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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altitude = half bisector ( 2018 I European Math Tournament - Geometry p4)
parmenides51   2
N Jul 10, 2020 by sunken rock
Calculate the angles of an isosceles triangle if its height drawn to the base is half the bisector of the angle at the base.
2 replies
parmenides51
Jul 9, 2020
sunken rock
Jul 10, 2020
J
altitude = half bisector ( 2018 I European Math Tournament - Geometry p4)
G H J
Reveal topic tags
geometry
isosceles
angles
altitude
angle bisector
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parmenides51
30629 posts
parmenides51
#1 h  Jul 9, 2020, 5:47 AM
Y by
Calculate the angles of an isosceles triangle if its height drawn to the base is half the bisector of the angle at the base.
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Vin0997
107 posts
Vin0997
#2 h  Jul 9, 2020, 12:11 PM
Y by
108°,36°,36°
By the very nature of golden ratio, it comes out to this
This post has been edited 1 time. Last edited by Vin0997, Jul 9, 2020, 12:12 PM
Reason: No
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sunken rock
4379 posts
sunken rock
#3 h  Jul 10, 2020, 2:13 PM
Y by
Take reflexion $D$ of the apex $A$ about the base $BC$, get a rhombus; draw the line bisector $BE$, notice that $ABDE$ is an isosceles trapezoid. If $m(\widehat{ABC})=2\alpha$, clearly $m(\widehat{EBD})=3\alpha=...=m(\widehat{BAD})$. Thus from $m(\widehat{ABC})+m(\widehat{DAB})=90^\circ\implies 5\alpha=90^\circ$, so the answer is as above.

Best regards,
sunken rock
This post has been edited 1 time. Last edited by sunken rock, Jul 10, 2020, 2:14 PM
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