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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)!

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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   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
J
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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!
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J
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computational, perpendiculars, angle bisectors (2010 HSO Высшая проба 11.7)
parmenides51   2
N Oct 18, 2020 by jasperE3
Point $D$ is taken on the side $AB$ of the triangle $ABC$. In the triangle $ADC$, bisectors $AP$ and $CQ$ are drawn. A point $R$ is taken on the side $AC$ of triangle $ADC$ so that $PR \perp CQ$. It is known that the bisector of angle $D$ of triangle $BCD$ is perpendicular to the segment $PB, AB = 18, AP = 12$. Calculate $AR$.
2 replies
parmenides51
Oct 17, 2020
jasperE3
Oct 18, 2020
J
computational, perpendiculars, angle bisectors (2010 HSO Высшая проба 11.7)
G H J
Reveal topic tags
geometry
perpendicular
angle bisector
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parmenides51
30629 posts
parmenides51
#1 h  Oct 17, 2020, 9:32 PM
Y by
Point $D$ is taken on the side $AB$ of the triangle $ABC$. In the triangle $ADC$, bisectors $AP$ and $CQ$ are drawn. A point $R$ is taken on the side $AC$ of triangle $ADC$ so that $PR \perp CQ$. It is known that the bisector of angle $D$ of triangle $BCD$ is perpendicular to the segment $PB, AB = 18, AP = 12$. Calculate $AR$.
This post has been edited 1 time. Last edited by parmenides51, Oct 17, 2020, 9:32 PM
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Imajinary
44 posts
Imajinary
#2 h  Oct 18, 2020, 5:23 AM
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This post has been edited 1 time. Last edited by Imajinary, Oct 18, 2020, 5:24 AM
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jasperE3
11165 posts
jasperE3
#3 h  Oct 18, 2020, 6:13 AM
Y by
How do you prove that $\triangle ARP\sim\triangle AQB$?
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