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3 var inequality
SunnyEvan   13
N 2 hours ago by Nguyenhuyen_AG
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
13 replies
SunnyEvan
May 17, 2025
Nguyenhuyen_AG
2 hours ago
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trigonometric inequality
MATH1945   13
N 3 hours ago by sqing
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
13 replies
MATH1945
May 26, 2016
sqing
3 hours ago
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Iran TST Starter
M11100111001Y1R   2
N 3 hours ago by sami1618
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[ a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i} \]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[ a_n < \frac{1}{1404} \]
2 replies
M11100111001Y1R
Tuesday at 7:36 AM
sami1618
3 hours ago
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Twin Prime Diophantine
awesomeming327.   23
N 4 hours ago by HDavisWashu
Source: CMO 2025
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
23 replies
awesomeming327.
Mar 7, 2025
HDavisWashu
4 hours ago
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Troublesome median in a difficult inequality
JG666   2
N 4 hours ago by navid
Source: 2022 Spring NSMO Day 2 Problem 3
Determine the minimum value of $\lambda\in\mathbb{R}$, such that for any positive integer $n$ and non-negative reals $x_1, x_2, \cdots, x_n$, the following inequality always holds:
$$\sum_{i=1}^n(m_i-a_i)^2\leqslant \lambda\cdot\sum_{i=1}^nx_i^2,$$Here $m_i$ and $a_i$ denote the median and arithmetic mean of $x_1, x_2, \cdots, x_i$, respectively.

Duanyang ZHANG, High School Affiliated to Renmin University of China
2 replies
JG666
May 22, 2022
navid
4 hours ago
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Concurrent lines, angle bisectors
legogubbe   0
4 hours ago
Source: ???
Hi AoPS!

Let $ABC$ be an isosceles triangle with $AB=AC$, and $M$ an arbitrary point on side $BC$. The internal angle bisector of $\angle MAB$ meets the circumcircle of $\triangle ABC$ again at $P \neq A$, and the internal angle bisector of $\angle CAM$ meets it again at $Q \neq A$. Show that lines $AM$, $BQ$ and $CP$ are concurrent.
0 replies
legogubbe
4 hours ago
0 replies
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Fractional Inequality
sqing   33
N 4 hours ago by Learning11
Source: Chinese Girls Mathematical Olympiad 2012, Problem 1
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that
$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$
33 replies
sqing
Aug 10, 2012
Learning11
4 hours ago
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Geometry angle chasing olympiads
Foxellar   1
N 5 hours ago by Ianis
Let \( \triangle ABC \) be a triangle such that \( \angle ABC = 120^\circ \). Points \( X, Y, Z \) lie on segments \( BC, CA, AB \), respectively, such that lines \( AX, BY, \) and \( CZ \) are the angle bisectors of triangle \( ABC \). Find the measure of angle \( \angle XYZ \).
1 reply
Foxellar
5 hours ago
Ianis
5 hours ago
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Iran Inequality
mathmatecS   17
N 5 hours ago by Learning11
Source: Iran 1998
When $x(\ge1),$ $y(\ge1),$ $z(\ge1)$ satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2,$ prove in equality.
$$\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$$
17 replies
mathmatecS
Jun 11, 2015
Learning11
5 hours ago
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Problem 4
codyj   87
N 5 hours ago by ezpotd
Source: IMO 2015 #4
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Proposed by Greece
87 replies
codyj
Jul 11, 2015
ezpotd
5 hours ago
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Fixed line
Hieutran2000   5
N Nov 13, 2017 by Jon-Snow
Let the triangle $ABC$, $P$ is in $BC$. The circle with the diameter $AP$ ( denote $(AP)$) intersects $AC, AB$ at $E, F$ respectively. Let $T$ be the tangents of $(AP)$ at $E, F$. Prove that $T$ lies in a fixed line when $P$ moves in $BC$.
5 replies
Hieutran2000
Nov 8, 2017
Jon-Snow
Nov 13, 2017
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Fixed line
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Hieutran2000
34 posts
Hieutran2000
#1 h  Nov 8, 2017, 10:17 AM • 1 Y
Y by Adventure10
Let the triangle $ABC$, $P$ is in $BC$. The circle with the diameter $AP$ ( denote $(AP)$) intersects $AC, AB$ at $E, F$ respectively. Let $T$ be the tangents of $(AP)$ at $E, F$. Prove that $T$ lies in a fixed line when $P$ moves in $BC$.
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Hieutran2000
34 posts
Hieutran2000
#3 h  Nov 8, 2017, 4:21 PM • 1 Y
Y by Adventure10
Anyone has solution?
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jestrada
520 posts
jestrada
#5 h  Nov 10, 2017, 1:17 PM • 2 Y
Y by Adventure10, Mango247
Did you mean "$T$ is the intersection of the tangents of circle $(AP)$ at $E, F$"?
This post has been edited 1 time. Last edited by jestrada, Nov 10, 2017, 1:17 PM
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PROF65
2016 posts
PROF65
#6 h  Nov 10, 2017, 3:46 PM • 2 Y
Y by Adventure10, Mango247
not only for $P$ in $BC$ but for any point in the plane it ' s easy to prove that $T$ is on the $A$-bisector
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ManuelKahayon
148 posts
ManuelKahayon
#7 h  Nov 11, 2017, 2:30 PM • 1 Y
Y by Adventure10
Assume that he means \(T\) is the intersection of the tangents from \((AP)\) at \(E\) and \(F\). Let \(D\) be the foot of the perpendicular from \(A\) to \(BC\). Notice that \((AP) = (APD)\). Notice that \(D\) is the center of the spiral similarity of all possible line segments \(EF\). Also, let \(O\) be the center of \((APD)\). Notice that \(\angle EOF = 2\angle EAF = 2\angle BAC\), which implies \(\angle EOF\) is constant, implying that \(D\) is also the center of the spiral similarity of all possible triangles \(EOF\). But since \((E,O,F)\) uniquely determine \(T\), it follows that \(D\) is the center of the spiral similarity of all possible quadrilaterals \(EOFT\), implying that \(T\) varies on a line (as spiral similarities map points along lines).
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Jon-Snow
16 posts
Jon-Snow
#8 h  Nov 13, 2017, 7:26 AM • 2 Y
Y by Adventure10, Mango247
(as spiral similarities map points along lines).
Can u explain this for me?
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