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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
sum (a^2 + b^2)/2ab + 2(ab + bc + ca)/3 >=5
parmenides51   9
N a few seconds ago by mudok
Source: 2023 Greece JBMO TST p3/ easy version of Shortlist 2022 A6 https://artofproblemsolving.com/community/c6h3099025p28018726
Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$When equality holds?
9 replies
1 viewing
parmenides51
May 17, 2024
mudok
a few seconds ago
Cool minimum
giangtruong13   0
9 minutes ago
Source: my friend
Let $x,y>0$ such that: $x>y>1$$(xy+1)^2+(x+y)^2\le2(x+y)(x^2-xy+y^2+1)$. Find min:
$P=\dfrac{\sqrt{x-y}}{y-1}$
0 replies
giangtruong13
9 minutes ago
0 replies
angle chasing, tangents at circumcircle of a right triangle
parmenides51   1
N 13 minutes ago by CovertQED
Source: China Northern MO 2015 10.2 CNMO
It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.
1 reply
parmenides51
Oct 28, 2022
CovertQED
13 minutes ago
Inequality
lgx57   3
N 13 minutes ago by lgx57
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
3 replies
lgx57
Yesterday at 3:14 PM
lgx57
13 minutes ago
When to look at solutions - pre calc
omerrob13   1
N 4 hours ago by abartoha
Hey all.
I am doing the precalc book, and unfortunately, im getting into the habit of looking in the solutions quite fast on a problem I did not able to make any progress on.
My goal is mainly to develop problem solving and reasonning skills.

I divide the problems in AOPS to 2:

- Challenge problems at the end of the of each chapter.
- The problems that teach you the material itself, and the problems at the end of each section (1.1,1.2, etc...)

For non challenging problems, It takes around 20 mins of me not be able to solve a problem, and look at the solutions for it

Is it too little?
My goal is mainly to develop problem solving and reasoning skills.
I'm not sure if it's too little time to bring to a regular problem, or its ok to give 20 mins to a problem and continue if making no progress.
1 reply
omerrob13
5 hours ago
abartoha
4 hours ago
f_n(x)=\sum sin(nx)/n
Urumqi   4
N 6 hours ago by Urumqi
$F_n(x)=\sum_{k=1}^{n}\frac{\sin (kx)}{k}$, prove that for all $x \in (0,\pi), F_n(x)>0$.

Thanks.
4 replies
Urumqi
Today at 2:13 AM
Urumqi
6 hours ago
How many pairs
Ecrin_eren   3
N Today at 8:21 AM by Ecrin_eren


Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:

p + 2^p + 3 = n^2 ?



3 replies
Ecrin_eren
Friday at 3:08 PM
Ecrin_eren
Today at 8:21 AM
Hard Inequality
William_Mai   8
N Today at 8:17 AM by DAVROS
Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$.
Find the minimum value of $P = ab + 2bc + 3ca$.

Source: Pham Le Van
8 replies
William_Mai
Yesterday at 2:13 PM
DAVROS
Today at 8:17 AM
How many triangles
Ecrin_eren   2
N Today at 8:15 AM by Ecrin_eren


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


2 replies
Ecrin_eren
May 2, 2025
Ecrin_eren
Today at 8:15 AM
Sum of digits is 18
Ecrin_eren   12
N Today at 8:12 AM by Ecrin_eren
How many 5 digit numbers are there such that sum of its digits is 18
12 replies
Ecrin_eren
Yesterday at 1:10 PM
Ecrin_eren
Today at 8:12 AM
Looking for users and developers
derekli   4
N Today at 4:28 AM by derekli
Guys I've been working on a web app that lets you grind high school lvl math. There's AMCs, AIME, BMT, HMMT, SMT etc. Also, it's infinite practice so you can keep grinding without worrying about finding new problems. Please consider helping me out by testing and also consider joining our developer team! :P :blush:

Link: https://stellarlearning.app/competitive
4 replies
derekli
Today at 12:57 AM
derekli
Today at 4:28 AM
Summation
Saucepan_man02   2
N Today at 3:54 AM by P162008
$\sum_{r=1}^{\infty}\frac{12r^2+1}{64r^6-48r^4+12r^2-1}$
2 replies
Saucepan_man02
Mar 6, 2025
P162008
Today at 3:54 AM
Geometry Basic
AlexCenteno2007   4
N Today at 3:18 AM by giratina3
Let $ABC$ be an isosceles triangle such that $AC=BC$. Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$. Prove that CD$ \parallel$AB
4 replies
AlexCenteno2007
Apr 28, 2025
giratina3
Today at 3:18 AM
Geometry
AlexCenteno2007   2
N Today at 3:00 AM by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
2 replies
AlexCenteno2007
Apr 28, 2025
AlexCenteno2007
Today at 3:00 AM
hard problem
Cobedangiu   15
N May 1, 2025 by arqady
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
15 replies
Cobedangiu
Apr 21, 2025
arqady
May 1, 2025
hard problem
G H J
G H BBookmark kLocked kLocked NReply
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Cobedangiu
66 posts
#1 • 1 Y
Y by RainbowJessa
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
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m4thbl3nd3r
285 posts
#2 • 1 Y
Y by RainbowJessa
Cobedangiu wrote:
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

Tangent line :whistling:
Z K Y
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giangtruong13
141 posts
#3 • 1 Y
Y by RainbowJessa
giangtruong13 wrote:
Bài này giống với bài BĐT trong đề thi HSG Thái Bình năm 2024-2025
Z K Y
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Jackson0423
66 posts
#4 • 1 Y
Y by RainbowJessa
use the constants
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Cobedangiu
66 posts
#5 • 1 Y
Y by RainbowJessa
giangtruong13 wrote:
giangtruong13 wrote:
Bài này giống với bài BĐT trong đề thi HSG Thái Bình năm 2024-2025
nó có thể giải đc chỉ với Schwarz .-.
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arqady
30231 posts
#6 • 1 Y
Y by RainbowJessa
Cobedangiu wrote:
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
It's just Popoviciu for $f(x)=\frac{1}{x}$ on $(0,+\infty).$
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IceyCold
208 posts
#7 • 1 Y
Y by RainbowJessa
m4thbl3nd3r wrote:
Cobedangiu wrote:
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

Tangent line :whistling:

mhmm,Tangent Line
I like
Z K Y
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arqady
30231 posts
#8 • 1 Y
Y by RainbowJessa
IceyCold wrote:
mhmm,Tangent Line
I like
Did you try? I think, it does not help here.
Z K Y
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ReticulatedPython
617 posts
#9 • 1 Y
Y by RainbowJessa
Interesting problem. I suspect that AM-GM might be applicable here, since equality is achieved at $a=b=c=1$ (which is the AM-GM equality condition).
This post has been edited 1 time. Last edited by ReticulatedPython, Apr 24, 2025, 3:04 PM
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IceyCold
208 posts
#10 • 1 Y
Y by RainbowJessa
arqady wrote:
IceyCold wrote:
mhmm,Tangent Line
I like
Did you try? I think, it does not help here.

It was one of our test.The graders marked my method correct,so I hope darn well I am right lol-
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Cobedangiu
66 posts
#11 • 1 Y
Y by RainbowJessa
IceyCold wrote:
arqady wrote:
IceyCold wrote:
mhmm,Tangent Line
I like
Did you try? I think, it does not help here.

It was one of our test.The graders marked my method correct,so I hope darn well I am right lol-

Can you write your method?
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ReticulatedPython
617 posts
#12 • 1 Y
Y by RainbowJessa
IceyCold wrote:
arqady wrote:
IceyCold wrote:
mhmm,Tangent Line
I like
Did you try? I think, it does not help here.

It was one of our test.The graders marked my method correct,so I hope darn well I am right lol-

Yeah can you share the method with us?
Z K Y
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Edward_Tur
127 posts
#13
Y by
Cobedangiu wrote:
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

$a=\frac{3x}{x+y+z},...$
$\sum_{sym} x^4y^2-x^4yz+x^3y^3-x^2y^2z^2\ge0.$
This post has been edited 1 time. Last edited by Edward_Tur, Apr 28, 2025, 7:44 PM
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IceyCold
208 posts
#14
Y by
Cobedangiu wrote:
Can you write your method?
Fakesolve
This post has been edited 3 times. Last edited by IceyCold, Apr 30, 2025, 1:36 AM
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IceyCold
208 posts
#15
Y by
IceyCold wrote:
Cobedangiu wrote:
Can you write your method?
Show that $\frac{4}{3-c} \le \frac{1}{c} -2c + 3 $.

This is equivalent to $\frac{(c-1)^2(2c+3)}{c(c-3)} \le 0$,trivially true.

Never mind,I see the issue now.
Sorry for a waste of time-
This post has been edited 1 time. Last edited by IceyCold, Apr 30, 2025, 1:34 AM
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arqady
30231 posts
#17
Y by
ReticulatedPython wrote:

Yeah can you share the method with us?
See here
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