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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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0 replies
jlacosta
May 1, 2025
0 replies
Inspired by qrxz17
sqing   9
N 29 minutes ago by sqing
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
9 replies
sqing
Yesterday at 8:50 AM
sqing
29 minutes ago
Interesting inequality
sqing   0
42 minutes ago
Source: Own
Let $  a, b,c>0,b+c\geq 3a$. Prove that
$$ \sqrt{\frac{a}{b+c-a}}-\frac{ 2a^2-b^2-c^2}{(a+b)(a+c)}\geq \frac{2}{5}+\frac{1}{\sqrt 2}$$$$ \frac{3}{2}\sqrt{\frac{a}{b+c-a}}-\frac{ 2a^2-b^2-c^2}{(a+b)(a+c)}\geq \frac{2}{5}+\frac{3}{2\sqrt 2}$$
0 replies
sqing
42 minutes ago
0 replies
Inspired by m4thbl3nd3r
sqing   4
N an hour ago by sqing
Source: Own
Let $  a, b,c>0,b+c>a$. Prove that$$\sqrt{\frac{a}{b+c-a}}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)}\geq 1$$$$\frac{a}{b+c-a}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)} \geq  \frac{4\sqrt 2}{3}-1$$
4 replies
sqing
Yesterday at 3:43 AM
sqing
an hour ago
Not so beautiful
m4thbl3nd3r   3
N an hour ago by m4thbl3nd3r
Let $a, b,c>0$ such that $b+c>a$. Prove that $$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$
3 replies
m4thbl3nd3r
Yesterday at 3:23 AM
m4thbl3nd3r
an hour ago
Interesting Geometry
captainmath99   4
N Yesterday at 8:01 PM by captainmath99
Let ABC be a right triangle such that $\angle{C}=90^\circ, CA=6, CB=4$. A circle O with center C has a radius of 2. Let P be a point on the circle O.

a)What is the minimum value of $(AP+\dfrac{1}{2}BP)$?
Answer Check

b) What is the minimum value of $(\dfrac{1}{3}AP+BP)$?
Answer Check
4 replies
captainmath99
May 25, 2025
captainmath99
Yesterday at 8:01 PM
Looking for even one person to study math.
abduqahhor_math   2
N Yesterday at 6:25 PM by EaZ_Shadow
Hi guys,I am looking for a person to study math topics related to olympiad.I have just finished 10th grade
2 replies
abduqahhor_math
Yesterday at 5:22 PM
EaZ_Shadow
Yesterday at 6:25 PM
Great Geometry with Squares on sides of triangles
SomeonecoolLovesMaths   3
N Yesterday at 6:14 PM by sunken rock
Three squares are drawn on the sides of triangle \(ABC\) (i.e., the square on \(AB\) has \(AB\) as one of its sides and lies outside \(ABC\)). Show that the lines drawn from the vertices \(A\), \(B\), and \(C\) to the centers of the opposite squares are concurrent.

IMAGE
3 replies
SomeonecoolLovesMaths
May 22, 2025
sunken rock
Yesterday at 6:14 PM
rare creative geo problem spotted in the wild
abbominable_sn0wman   0
Yesterday at 6:04 PM
The following is the construction of the twindragon fractal.

Let $I_0$ be the solid square region with vertices at
\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]
Recursively, the region $I_{n+1}$ consists of two copies of $I_n$: one copy which is rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and another copy which is also rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and then translated by $\left(\frac{1}{2}, -\frac{1}{2}\right)$.

We have displayed $I_0$ and $I_1$ below.

Let $I_\infty$ be the limiting region of the sequence $I_0, I_1, \dots$.

The area of the smallest convex polygon which encloses $I_\infty$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a + b$.
0 replies
abbominable_sn0wman
Yesterday at 6:04 PM
0 replies
Inequalities
lgx57   0
Yesterday at 3:55 PM
Let $a,b,c,d,e \ge 0$,$\sum \dfrac{1}{a+4}=1$.Prove that:
$$\sum \dfrac{a}{a^2+4} \le 1$$
Let $x,y,z>0$.Prove that:
$$\sum (y+z)\sqrt{\dfrac{yz}{(z+x)(y+x)}} \ge x+y+z$$
0 replies
lgx57
Yesterday at 3:55 PM
0 replies
Find the number of ordered triples (p,q,r)
Darealzolt   2
N Yesterday at 2:25 PM by elizhang101412
Let \(p,q,r\) be prime numbers such that
\[
\frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839}
\]Hence find the numbers of ordered triples \(\{p,q,r\}\)
2 replies
Darealzolt
Yesterday at 1:52 PM
elizhang101412
Yesterday at 2:25 PM
Find the sum of all the products a_i a_j
Darealzolt   1
N Yesterday at 2:13 PM by alexheinis
Among the 100 constants \( \{ a_1,a_2,a_3,\dots,a_{100} \} \),there are \(39\) equal to \( -1\), and \(61\) equal to \(1\). Find the sum of all the products \(a_i a_j\) , where \(a \leq i < j \leq 100\).
1 reply
Darealzolt
Yesterday at 11:24 AM
alexheinis
Yesterday at 2:13 PM
IOQM P26 2024
SomeonecoolLovesMaths   5
N Yesterday at 1:11 PM by SomeonecoolLovesMaths
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
5 replies
SomeonecoolLovesMaths
Sep 8, 2024
SomeonecoolLovesMaths
Yesterday at 1:11 PM
IOQM 2023-24 P-4
lifeismathematics   8
N Yesterday at 1:02 PM by SomeonecoolLovesMaths
Let $x, y$ be positive integers such that
$$
x^4=(x-1)\left(y^3-23\right)-1 .
$$Find the maximum possible value of $x+y$.
8 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
Yesterday at 1:02 PM
IOQM 2023-24 P-2
lifeismathematics   7
N Yesterday at 12:10 PM by SomeonecoolLovesMaths
Find the number of elements in the set
$$
\left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\}
$$
7 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
Yesterday at 12:10 PM
JBMO Shortlist 2023 A4
Orestis_Lignos   6
N May 9, 2025 by MR.1
Source: JBMO Shortlist 2023, A4
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that

$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
6 replies
Orestis_Lignos
Jun 28, 2024
MR.1
May 9, 2025
JBMO Shortlist 2023 A4
G H J
G H BBookmark kLocked kLocked NReply
Source: JBMO Shortlist 2023, A4
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Orestis_Lignos
558 posts
#1 • 1 Y
Y by xytunghoanh
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that

$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
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ehuseyinyigit
840 posts
#3 • 2 Y
Y by aqusha_mlp12, xytunghoanh
By Cauchy-Schwarz or Jensen, we have
$$\sum_{cyc}{\sqrt{\dfrac{a}{b+c+d^2+a^3}}}\leq \sqrt{4\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}}\leq 2$$It sufficies to show
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq 1$$But by C-S $$\left(b+c+d^2+a^3\right)\left(b+c+1+\dfrac{1}{a}\right)\geq \left(a+b+c+d\right)^2$$then
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq \dfrac{\sum\limits_{cyc}{a\left(b+c+1+\dfrac{1}{a}\right)}}{\left(a+b+c+d\right)^2}$$$$=\dfrac{\sum\limits_{sym}{(ab)}+ac+bd+a+b+c+d+4}{\left(a+b+c+d\right)^2}\overbrace{\leq}^{?} 1$$which is easy since
$$a^2+b^2+c^2+d^2+\left(a+c\right)\left(b+d\right)\geq a+b+c+d+4$$$$\Longleftrightarrow a^2+b^2+c^2+d^2\geq a+b+c+d$$where the last inequality is true by $abcd=1$.
.
This post has been edited 1 time. Last edited by ehuseyinyigit, Aug 1, 2024, 10:58 PM
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ehuseyinyigit
840 posts
#4 • 1 Y
Y by Assassino9931
I think that main idea in the problem is the same with JBMO 2019 #A.5. These problems are very related in their basis.
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Duk168
3 posts
#6
Y by
ehuseyinyigit wrote:
By Cauchy-Schwarz or Jensen, we have
$$\sum_{cyc}{\sqrt{\dfrac{a}{b+c+d^2+a^3}}}\leq \sqrt{4\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}}\leq 2$$It sufficies to show
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq 1$$But by C-S $$\left(b+c+d^2+a^3\right)\left(b+c+1+\dfrac{1}{a}\right)\geq \left(a+b+c+d\right)^2$$then
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq \dfrac{\sum\limits_{cyc}{a\left(b+c+1+\dfrac{1}{a}\right)}}{\left(a+b+c+d\right)^2}$$$$=\dfrac{\sum\limits_{sym}{(ab)}+ac+bd+a+b+c+d+4}{\left(a+b+c+d\right)^2}\overbrace{\leq}^{?} 1$$which is easy since
$$a^2+b^2+c^2+d^2+\left(a+c\right)\left(b+d\right)\geq a+b+c+d+4$$$$\Longleftrightarrow a^2+b^2+c^2+d^2\geq a+b+c+d$$where the last inequality is true by $abcd=1$.
.

can you expl why
a^2 + b^2 + c^2 + d^2 >= a + b + c + d for abcd = 1
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arqady
30261 posts
#7 • 1 Y
Y by ehuseyinyigit
Duk168 wrote:

can you expl why
a^2 + b^2 + c^2 + d^2 >= a + b + c + d for abcd = 1
It's Muirhead
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zaidova
89 posts
#8
Y by
Duk168 wrote:
ehuseyinyigit wrote:
By Cauchy-Schwarz or Jensen, we have
$$\sum_{cyc}{\sqrt{\dfrac{a}{b+c+d^2+a^3}}}\leq \sqrt{4\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}}\leq 2$$It sufficies to show
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq 1$$But by C-S $$\left(b+c+d^2+a^3\right)\left(b+c+1+\dfrac{1}{a}\right)\geq \left(a+b+c+d\right)^2$$then
$$\sum_{cyc}{\dfrac{a}{b+c+d^2+a^3}}\leq \dfrac{\sum\limits_{cyc}{a\left(b+c+1+\dfrac{1}{a}\right)}}{\left(a+b+c+d\right)^2}$$$$=\dfrac{\sum\limits_{sym}{(ab)}+ac+bd+a+b+c+d+4}{\left(a+b+c+d\right)^2}\overbrace{\leq}^{?} 1$$which is easy since
$$a^2+b^2+c^2+d^2+\left(a+c\right)\left(b+d\right)\geq a+b+c+d+4$$$$\Longleftrightarrow a^2+b^2+c^2+d^2\geq a+b+c+d$$where the last inequality is true by $abcd=1$.
.

can you expl why
a^2 + b^2 + c^2 + d^2 >= a + b + c + d for abcd = 1


u can also use c.s like;
$(1+1+1+1)(a^2+b^2+c^2+d^2) \ge (a+b+c+d)^2$
$a^2+b^2+c^2+d^2 \ge  \frac{(a+b+c+d)^2}{4}$
$(a+b+c+d)^2 \ge 4*(a+b+c+d)$ ==> $a+b+c+d \ge 4$ which is true by $AM-GM$
This post has been edited 2 times. Last edited by zaidova, Mar 4, 2025, 7:55 PM
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MR.1
138 posts
#10 • 1 Y
Y by MIC38
first notice that $a^2+b^2+c^2+d^2\geq a+b+c+d$ from $abcd=1$
by Cauchy-Schwarz we have:$(b+c+d^2+a^3)(b+c+1+\frac{1}{a})\geq (a+b+c+d)^2$
so $\sqrt{\frac{a}{b+c+d^2+a^3}}\leq\sqrt{\frac{b+c+1+\frac{1}{a}}{(a+b+c+d)^2}}=\frac{\sqrt{b+c+a+\frac{1}{a}}}{a+b+c+d}$
so $LHS\leq \sum_{cyc}{\frac{\sqrt{b+c+1+\frac{1}{a}}}{a+b+c+d}}\leq \sqrt{{\sum_{cyc}{\frac{4(b+c+1+\frac{1}{a})}{(a+b+c+d)^2}}}}$
so we have to prove that $\sum_{sym}{b+c+1+\frac{1}{a}}\leq (a+b+c+d)^2\implies$
$4+ab+bc+cd+ad+2ac+2bd+(a+b+c+d)\leq (a+b+c+d)^2$
$4+ab+bc+cd+ad+2ac+2bd+(a+b+c+d)\leq a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd)$
$a^2+b^2+c^2+d^2+\left(a+c\right)\left(b+d\right)\geq a+b+c+d+4$
$a^2+b^2+c^2+d^2\geq a+b+c+d$ and we are done(i hate my life) :noo: :stretcher:
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