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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
0 replies
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inequality ( 4 var
SunnyEvan   1
N 3 minutes ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
1 reply
SunnyEvan
5 hours ago
SunnyEvan
3 minutes ago
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Inspired by JK1603JK
sqing   6
N 5 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$
6 replies
sqing
Today at 3:31 AM
sqing
5 minutes ago
J
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Geometry problem
kjhgyuio   1
N an hour ago by Mathzeus1024
Source: smo
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
1 reply
kjhgyuio
Apr 1, 2025
Mathzeus1024
an hour ago
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D1018 : Can you do that ?
Dattier   1
N an hour ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
an hour ago
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Inequalities
sqing   0
Today at 3:52 AM
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
$$ a+b+c\leq 3 $$
0 replies
sqing
Today at 3:52 AM
0 replies
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Excalibur Identity
jjsunpu   8
N Today at 2:08 AM by MineCuber
proof is below
8 replies
jjsunpu
Yesterday at 3:27 PM
MineCuber
Today at 2:08 AM
J
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New geometry problem
titaniumfalcon   1
N Today at 1:51 AM by mathprodigy2011
Post any solutions you have, with explanation or proof if possible, good luck!
1 reply
titaniumfalcon
Yesterday at 10:40 PM
mathprodigy2011
Today at 1:51 AM
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Geo Mock #7
Bluesoul   1
N Yesterday at 8:20 PM by vanstraelen
Consider $\triangle{ABC}$ with $\angle{A}=90^{\circ}$ and $AB=10$. Let $D$ be a point on $AB$ such that $BD=6$. Suppose that the angle bisector of $\angle{C}$ is tangent to the circle with diameter $BD$ and say it intersects $AB$ at point $E$. Find the length of $BE$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 8:20 PM
J
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Number of solutions
Ecrin_eren   3
N Yesterday at 8:00 PM by rchokler
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.

3 replies
Ecrin_eren
Yesterday at 11:27 AM
rchokler
Yesterday at 8:00 PM
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Geo Mock #8
Bluesoul   1
N Yesterday at 7:06 PM by vanstraelen
Consider acute triangle $ABC$. Denote $M$ as the midpoint of $AB$, and let $O$ be a point on segment $CM$ such that $\angle{AOB}=120^{\circ}$. Find the length of $CM$ given $AO=5, BO=8, \angle{BAC}=60^{\circ}$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 7:06 PM
J
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An inequality
jokehim   6
N Yesterday at 4:47 PM by imnotgoodatmathsorry
Let $a,b,c \in \mathbb{R}: a+b+c=3$ then prove $$\color{black}{\frac{a^2}{a^{2}-2a+3}+\frac{b^2}{b^{2}-2b+3}+\frac{c^2}{c^{2}-2c+3}\ge \frac{3}{2}.}$$
6 replies
jokehim
Mar 21, 2025
imnotgoodatmathsorry
Yesterday at 4:47 PM
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Geometry problem
Raul_S_Baz   2
N Yesterday at 4:13 PM by Raul_S_Baz
IMAGE
2 replies
Raul_S_Baz
Wednesday at 8:49 PM
Raul_S_Baz
Yesterday at 4:13 PM
J
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Cyclic inequality
JK1603JK   0
Yesterday at 3:40 PM
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\frac{abc-1}{abc-2}\ge \frac{a^2b+b^2c+c^2a}{a^3b+b^3c+c^3a+1}.$$Equality holds at $a=b=1,c=0$ and its cyclic permutations.
0 replies
JK1603JK
Yesterday at 3:40 PM
0 replies
J
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Easy Problem
MathleteMystic   2
N Yesterday at 3:05 PM by Mathematicalprodigy37
Prove that among n integers we can always choose some of them whose sum is a multiple on n.

I do have a solution to this, but could someone write a more descriptive one, please? Something like the logic behind it...
2 replies
MathleteMystic
Yesterday at 1:20 PM
Mathematicalprodigy37
Yesterday at 3:05 PM
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equal angles wanted, intersecting chords
parmenides51   0
Nov 17, 2022
Source: 2022 VWO Flanders MO p1
The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$.

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0 replies
parmenides51
Nov 17, 2022
0 replies
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equal angles wanted, intersecting chords
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Source: 2022 VWO Flanders MO p1
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parmenides51
30629 posts
parmenides51
#1 h  Nov 17, 2022, 12:55 PM
Y by
The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$.

https://cdn.artofproblemsolving.com/attachments/2/2/7ec65246ff5ecfebc25ca13f3709d1791ceb6c.png
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This post has been edited 1 time. Last edited by parmenides51, Dec 23, 2022, 6:51 PM
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