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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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
Apr 2, 2025
0 replies
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Symmedian line
April   91
N 3 minutes ago by BS2012
Source: All Russian Olympiad - Problem 9.2, 10.2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
91 replies
April
May 10, 2009
BS2012
3 minutes ago
J
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Integral-Summation Duality
Mathandski   2
N 15 minutes ago by Mathandski
Source: Friend at school gave it to me
Given a continuous function $f$ such that $f(2x) = 3 f(x)$ and $\int_0^1 f(x) \, dx = 1$, evaluate $\int_1^2 f(x) \, dx$.
2 replies
Mathandski
5 hours ago
Mathandski
15 minutes ago
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The old one is gone.
EeEeRUT   10
N 17 minutes ago by Pompombojam
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
10 replies
EeEeRUT
Apr 16, 2025
Pompombojam
17 minutes ago
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poly roots equation
NicoN9   2
N 21 minutes ago by alextheadventurer
Source: My acquaintance
Let $a, b, c, d$ be positive integers. Let $\alpha$, $\beta$, $\gamma$ be three roots of the polynomial $P(x)=ax^3+bx^2+cx+d$. Given that $\alpha^2+\beta^2+\gamma^2=-1$, $\alpha^3+\beta^3+\gamma^3=-3$, and $2a^2+c^2=27$, find the values of $a$, $b$, $c$, $d$.
2 replies
NicoN9
Sunday at 11:15 AM
alextheadventurer
21 minutes ago
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No more topics!
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Turkey NMO 1999, P-5, nice geometric inequality on altitudes
efoski1687   1
N Dec 23, 2010 by Luis González
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that
$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
1 reply
efoski1687
Dec 23, 2010
Luis González
Dec 23, 2010
J
Turkey NMO 1999, P-5, nice geometric inequality on altitudes
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Reveal topic tags
inequalities
geometry
circumcircle
trigonometry
geometry proposed
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efoski1687
97 posts
efoski1687
#1 h  Dec 23, 2010, 8:45 PM • 2 Y
Y by Adventure10, Mango247
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that
$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
Z K Y
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Luis González
4148 posts
Luis González
#2 h  Dec 23, 2010, 11:53 PM • 2 Y
Y by Adventure10, Mango247
Powers $P_A,P_B,P_C$ of $A,B,C$ WRT 9-point circle $\odot(DEF)$ are

$P_A= \frac{_1}{^2} AB \cdot AC \cdot \cos A \ , \ P_B= \frac{_1}{^2} BC \cdot BA \cdot \cos B \ , \ P_C= \frac{_1}{^2} CA \cdot CB \cdot \cos C $

Lengths of altitudes $AD,CE,CF$ are given by

$AD=\frac{AB \cdot AC}{2R} \ \ , \ \ \ BE=\frac{BC \cdot BA}{2R} \ \ , \ \ CF=\frac{CA \cdot CB}{2R}$

Substituting the latter expressions into the proposed inequality yields

$\cos A+ \cos B + \cos C \le \frac{3}{2} \ ,$ which follows from Jensen's inequality for $\cos \theta \ , (0, \frac{_{\pi}}{^{2}})$
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