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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
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Cyclic ine
m4thbl3nd3r   2
N a minute ago by m4thbl3nd3r
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
2 replies
m4thbl3nd3r
Today at 3:34 PM
m4thbl3nd3r
a minute ago
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Sequence inequality
BR1F1SZ   0
12 minutes ago
Source: 2025 Francophone MO Seniors P1
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$,
\[ a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}. \]Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.
0 replies
BR1F1SZ
12 minutes ago
0 replies
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GCD and LCM operations
BR1F1SZ   0
20 minutes ago
Source: 2025 Francophone MO Juniors P4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.

Note: The number $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.
0 replies
BR1F1SZ
20 minutes ago
0 replies
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Concurrency of two lines and a circumcircle
BR1F1SZ   0
24 minutes ago
Source: 2025 Francophone MO Juniors P3
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
0 replies
BR1F1SZ
24 minutes ago
0 replies
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Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   1
N 6 hours ago by Lankou
For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?
1 reply
ThisIsJoe
May 8, 2025
Lankou
6 hours ago
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2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N Today at 3:29 PM by Pengu14
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
11 replies
parmenides51
Jun 17, 2019
Pengu14
Today at 3:29 PM
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Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N Today at 2:38 PM by sqing
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$Proposed by Ngo Van Trang, Vietnam
3 replies
Adventure1000
May 7, 2025
sqing
Today at 2:38 PM
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Malaysia MO IDM UiTM 2025
smartvong   1
N Today at 2:20 PM by jasperE3
MO IDM UiTM 2025 (Category C)

Contest Description

Preliminary Round
Section A
1. Given that $2^a + 2^b = 2016$ such that $a, b \in \mathbb{N}$. Find the value of $a$ and $b$.

2. Find the value of $a, b$ and $c$ such that $$\frac{ab}{a + b} = 1, \frac{bc}{b + c} = 2, \frac{ca}{c + a} = 3.$$
3. If the value of $x + \dfrac{1}{x}$ is $\sqrt{3}$, then find the value of
$$x^{1000} + \frac{1}{x^{1000}}$$.

Section B
1. Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integer $a, b$:
$$f(2a) + 2f(b) = f(f(a + b))$$
2. The side lengths $a, b, c$ of a triangle $\triangle ABC$ are positive integers. Let
$$T_n = (a + b + c)^{2n} - (a - b + c)^{2n} - (a + b - c)^{2n} - (a - b - c)^{2n}$$for any positive integer $n$.
If $\dfrac{T_2}{2T_1} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $\triangle ABC$.

Final Round
Section A
1. Given that the equation $x^2 + (b - 3)x - 2b^2 + 6b - 4 = 0$ has two roots, where one is twice of the other, find all possible values of $b$.

2. Let $$f(y) = \dfrac{y^2}{y^2 + 1}.$$Find the value of $$f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + \cdots + f\left(\frac{2000}{2001}\right) + f\left(\frac{2001}{2001}\right) + f\left(\frac{2001}{2000}\right) + \cdots + f\left(\frac{2001}{2}\right) + f\left(\frac{2001}{1}\right).$$
3. Find the smallest four-digit positive integer $L$ such that $\sqrt{3\sqrt{L}}$ is an integer.

Section B
1. Given that $\tan A : \tan B : \tan C$ is $1 : 2 : 3$ in triangle $\triangle ABC$, find the ratio of the side length $AC$ to the side length $AB$.

2. Prove that $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$
1 reply
smartvong
Today at 1:01 PM
jasperE3
Today at 2:20 PM
J
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Nice problem
gasgous   2
N Today at 1:47 PM by vincentwant
Given that the product of three integers is $60$.What is the least possible positive sum of the three integers?
2 replies
gasgous
Today at 1:30 PM
vincentwant
Today at 1:47 PM
J
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Angle Formed by Points on the Sides of a Triangle
xeroxia   1
N Today at 1:28 PM by vanstraelen

In triangle $ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$BD = 20$, $DC = 15$, $CE = 13$, $EA = 8$, $AF = 6$, $FB = 22$.

What is the measure of $\angle EDF$?


1 reply
xeroxia
Today at 10:28 AM
vanstraelen
Today at 1:28 PM
J
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Inequalities
sqing   1
N Today at 1:08 PM by sqing
Let $ a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{38}{23}$$Let $ a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2(2\sqrt{3}+1)}{5}$$
1 reply
sqing
Today at 12:50 PM
sqing
Today at 1:08 PM
J
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Is this true?
Entrepreneur   1
N Today at 12:56 PM by revol_ufiaw
Define the $\text{\textcolor{red}{Pell Sequence}}$ as $$P_0=0,P_1=1,\;P_{n+2}=2P_{n+1}+P_n.$$Prove that $4P_{2k}^2+1$ is prime for all $k\in\mathbb N.$
1 reply
Entrepreneur
Today at 9:32 AM
revol_ufiaw
Today at 12:56 PM
J
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Geometry
AlexCenteno2007   4
N Today at 12:05 PM by Raul_S_Baz
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.
4 replies
AlexCenteno2007
Apr 28, 2025
Raul_S_Baz
Today at 12:05 PM
J
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Inequalities
sqing   1
N Today at 11:45 AM by sqing
Let $ 0\leq x,y,z\leq 2. $ Prove that
$$-48\leq (x-yz)( 3y-zx)(z-xy)\leq 9$$$$-144\leq (3x-yz)(y-zx)(3z-xy)\leq\frac{81}{64}$$$$-144\leq (3x-yz)(2y-zx)(3z-xy)\leq\frac{81}{16}$$
1 reply
sqing
Yesterday at 8:50 AM
sqing
Today at 11:45 AM
J
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J
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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
J
hard problem
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Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
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Cobedangiu
70 posts
Cobedangiu
#1 h  Apr 21, 2025, 1:51 PM • 1 Y
Y by RainbowJessa
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
Z K Y
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m4thbl3nd3r
287 posts
m4thbl3nd3r
#2 h  Apr 21, 2025, 2:21 PM • 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
145 posts
giangtruong13
#3 h  Apr 21, 2025, 2:38 PM • 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
95 posts
Jackson0423
#4 h  Apr 21, 2025, 3:17 PM • 1 Y
Y by RainbowJessa
use the constants
Z K Y
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Cobedangiu
70 posts
Cobedangiu
#5 h  Apr 21, 2025, 4:00 PM • 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 .-.
Z K Y
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arqady
30244 posts
arqady
#6 h  Apr 22, 2025, 8:20 AM • 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 $(0,+\infty).$
Z K Y
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IceyCold
208 posts
IceyCold
#7 h  Apr 23, 2025, 3:45 PM • 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
30244 posts
arqady
#8 h  Apr 23, 2025, 5:25 PM • 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
682 posts
ReticulatedPython
#9 h  Apr 24, 2025, 3:03 PM • 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
IceyCold
#10 h  Apr 28, 2025, 12:37 PM • 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-
Z K Y
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Cobedangiu
70 posts
Cobedangiu
#11 h  Apr 28, 2025, 4:16 PM • 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?
Z K Y
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ReticulatedPython
682 posts
ReticulatedPython
#12 h  Apr 28, 2025, 7:10 PM • 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
Edward_Tur
#13 h  Apr 28, 2025, 7:43 PM
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
IceyCold
#14 h  Apr 30, 2025, 1:21 AM
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
IceyCold
#15 h  Apr 30, 2025, 1:28 AM
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
30244 posts
arqady
#17 h  May 1, 2025, 3:37 AM
Y by
ReticulatedPython wrote:

Yeah can you share the method with us?
See here
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