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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)!

Be sure to mark your calendars for the following events:
[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
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Eazy equation clap
giangtruong13   0
an hour ago
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
an hour ago
0 replies
J
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inequality
revol_ufiaw   3
N 2 hours 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
3 hours ago
MS_asdfgzxcvb
2 hours ago
J
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How to judge a number is prime or not?
mingzhehu   0
3 hours 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
mingzhehu
3 hours ago
0 replies
J
H
What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N 3 hours 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
3 hours ago
J
No more topics!
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max area, 2 congruent equilaterals, square (2012 UQ/QAMT PS Competition 9-10 p4)
parmenides51   1
N Jan 25, 2021 by vanstraelen
Two congruent non-overlapping equilateral triangles are placed wholly within a square of side length $1$. What is the maximum area they could cover?
1 reply
parmenides51
Jan 19, 2021
vanstraelen
Jan 25, 2021
J
max area, 2 congruent equilaterals, square (2012 UQ/QAMT PS Competition 9-10 p4)
G H J
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parmenides51
30629 posts
parmenides51
#1 h  Jan 19, 2021, 7:25 AM
Y by
Two congruent non-overlapping equilateral triangles are placed wholly within a square of side length $1$. What is the maximum area they could cover?
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vanstraelen
8949 posts
vanstraelen
#2 h  Jan 25, 2021, 8:43 PM • 1 Y
Y by teomihai
Square $ABCD\ :\ A(0,0),B(1,0)C(1,1),D(0,1)$.

Equilateral triangle $AEF\ :\ E(a,0),F(\frac{a}{2},\frac{\sqrt{3}a}{2})$.
Equilateral triangle $GCH\ :\ G(1-a,1)$.

$F \in EG \Rightarrow a=\frac{1+\frac{1}{\sqrt{3}}}{2}$.
Area equilateral triangle $AEF\ =\ \frac{\frac{2\sqrt{3}}{3}+1}{8}$.
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