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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Eazy equation clap
giangtruong13   0
2 hours 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
2 hours ago
0 replies
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inequality
revol_ufiaw   3
N 3 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
4 hours ago
MS_asdfgzxcvb
3 hours ago
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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
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N 4 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
4 hours ago
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Any nice way to do this?
NamelyOrange   3
N 4 hours ago by pooh123
Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$?
3 replies
NamelyOrange
Apr 2, 2025
pooh123
4 hours ago
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Inequalities
sqing   3
N 4 hours ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
3 replies
sqing
Yesterday at 3:52 AM
sqing
4 hours ago
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Inequalities
sqing   0
5 hours ago
Let $a,b$ be real numbers such that $ a^2+b^2+a^3 +b^3=4 . $ Prove that
$$a+b \leq 2$$Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 . $ Prove that
$$a+b \leq 2$$
0 replies
sqing
5 hours ago
0 replies
J
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that statement is true
pennypc123456789   3
N 6 hours ago by sqing
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
3 replies
pennypc123456789
Mar 23, 2025
sqing
6 hours ago
J
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Distance vs time swimming problem
smalkaram_3549   1
N Today at 11:54 AM by Lankou
How should I approach a problem where we deal with velocities becoming negative and stuff. I know that they both travel 3 Lengths of the pool before meeting a second time.
1 reply
smalkaram_3549
Today at 2:57 AM
Lankou
Today at 11:54 AM
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.problem.
Cobedangiu   4
N Today at 11:40 AM by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
4 replies
Cobedangiu
Yesterday at 6:20 AM
Lankou
Today at 11:40 AM
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inequalities - 5/4
pennypc123456789   2
N Today at 11:35 AM by sqing
Given real numbers $x, y$ satisfying $|x| \le 3, |y| \le 3$. Prove that:
\[ 0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164. \]
2 replies
pennypc123456789
Today at 8:57 AM
sqing
Today at 11:35 AM
J
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Source of a combinatorics problem
isodynamicappolonius2903   0
Today at 6:47 AM
Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.
0 replies
isodynamicappolonius2903
Today at 6:47 AM
0 replies
J
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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   5
N Today at 3:26 AM by Shiyul
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
5 replies
Shiyul
Yesterday at 3:09 PM
Shiyul
Today at 3:26 AM
J
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Regarding IMO prepartion
omega2007   1
N Today at 2:49 AM by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
1 reply
omega2007
Yesterday at 3:14 PM
omega2007
Today at 2:49 AM
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J
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angles of trapezoid wanted, ratio 1:2, 120'o sum (2007 Zhautykov City VIII P4)
parmenides51   1
N Aug 31, 2019 by Leo142857
The non parallel sides of a trapezoid have ratio of lengths $1: 2$. The sum of the angles of the larger base is $ 120 {} ^\circ $. Find the angles of of this trapezoid.
1 reply
parmenides51
Aug 31, 2019
Leo142857
Aug 31, 2019
J
angles of trapezoid wanted, ratio 1:2, 120'o sum (2007 Zhautykov City VIII P4)
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Reveal topic tags
ratio
trapezoid
geometry
angles
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parmenides51
30629 posts
parmenides51
#1 h  Aug 31, 2019, 12:07 PM • 1 Y
Y by Adventure10
The non parallel sides of a trapezoid have ratio of lengths $1: 2$. The sum of the angles of the larger base is $ 120 {} ^\circ $. Find the angles of of this trapezoid.
This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 11:54 AM
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Leo142857
128 posts
Leo142857
#2 h  Aug 31, 2019, 12:28 PM • 1 Y
Y by Adventure10
First, let’s name the trapezoid as ABCD

Assume that AD is parallel to BC

Then we would also have AB=2BC
Now, we construct DE such that DE is parallel to AB and point E is on BC
Then angleEDC is 60 degrees, leading to triangleDEC being a right triangle ( 30-60-90)
And ABED is also a rectangle
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