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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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[*]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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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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How to judge a number is prime or not?
mingzhehu   0
Today at 2:45 PM
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
Today at 2:45 PM
0 replies
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N Today at 2:40 PM 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
Today at 2:40 PM
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Any nice way to do this?
NamelyOrange   3
N Today at 2:00 PM 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
Today at 2:00 PM
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Inequalities
sqing   3
N Today at 2:00 PM 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
Today at 2:00 PM
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Inequalities
sqing   0
Today at 1:10 PM
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
Today at 1:10 PM
0 replies
J
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that statement is true
pennypc123456789   3
N Today at 12:32 PM 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
Today at 12:32 PM
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
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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
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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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area <ABC=90^o,<ACB=<BDC,<DBC =<ACD+<ADC,AC = BD (2004 Argentina OMA L2 p6)
parmenides51   1
N Jun 14, 2020 by vanstraelen
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 90^o$, $\angle ACB = \angle BDC$, $\angle DBC = \angle ACD + \angle ADC$, $AC = BD$ and $CD = 3$. Calculate the area of the quadrilateral.
1 reply
parmenides51
Jun 11, 2020
vanstraelen
Jun 14, 2020
J
area <ABC=90^o,<ACB=<BDC,<DBC =<ACD+<ADC,AC = BD (2004 Argentina OMA L2 p6)
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Reveal topic tags
geometry
area
angles
equal angles
equal segments
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parmenides51
30629 posts
parmenides51
#1 h  Jun 11, 2020, 7:16 PM
Y by
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 90^o$, $\angle ACB = \angle BDC$, $\angle DBC = \angle ACD + \angle ADC$, $AC = BD$ and $CD = 3$. Calculate the area of the quadrilateral.
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vanstraelen
8949 posts
vanstraelen
#2 h  Jun 14, 2020, 2:29 PM • 1 Y
Y by aie8920
$A(0,\sqrt{3}),B(0,0),C(2,0),D(\frac{1}{2},\frac{3\sqrt{3}}{2})$.

Area $\frac{7\sqrt{3}}{4}$.
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