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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
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Weird Inequality Problem
Omerking   2
N 8 minutes ago by sqing
Following inequality is given:
$$3\geq ab+bc+ca\geq \dfrac{1}{3}$$Find the range of values that can be taken by :
$1)a+b+c$
$2)abc$

Where $a,b,c$ are positive reals.
2 replies
1 viewing
Omerking
4 hours ago
sqing
8 minutes ago
J
H
Fair division of loot among pirates
kiyoras_2001   0
9 minutes ago
Source: Kvant №4 1973, In a quite harbor
Three pirates share the loot, consisting of 10 piastres, 10 doubloons and a barrel of wine.

Though they have a container for pouring wine, each pirate has his own opinion about the comparative value of piastres, doubloons and wine. However, everyone agrees that a barrel of wine costs more than four piastres and more than four doubloons.

Prove that the pirates will be able to divide the loot so that each pirate gets a part worth (in his opinion) no less than the part of each of the others.
0 replies
kiyoras_2001
9 minutes ago
0 replies
J
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Inspired by my own results
sqing   1
N 17 minutes ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a+b+a^2+b^2=1. $ Prove that
$$ \frac{1}{a^2+1}+\frac{1}{b^2+1} -ab\geq\frac{3(2-7\sqrt 3)}{26}$$$$ \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 }+ab\leq\frac{58+5\sqrt 3 }{26}$$$$ \frac{29+9\sqrt 3 }{13}\geq \frac{1}{a^2+1}+\frac{1}{b^2+1} -a-b\geq\frac{29-9\sqrt 3 }{13}$$$$ \frac{3+17\sqrt 3 }{13}\geq \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 }+a+b\geq\frac{3-17\sqrt 3 }{13}$$
1 reply
sqing
27 minutes ago
sqing
17 minutes ago
J
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Functional equation
Amin12   46
N 17 minutes ago by FredAlexander
Source: Iranian 3rd round 2016 first Algebra exam
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,
$(f(a)+b) f(a+f(b))=(a+f(b))^2$
46 replies
Amin12
Aug 13, 2016
FredAlexander
17 minutes ago
J
No more topics!
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∆ABC and ∆AUV are similar
FelixD   1
N Apr 26, 2010 by MJ GEO
Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.

(41th Austrian Mathematical Olympiad, regional competition, problem 3)
1 reply
FelixD
Apr 26, 2010
MJ GEO
Apr 26, 2010
J
∆ABC and ∆AUV are similar
G H J
Reveal topic tags
geometry
circumcircle
geometry proposed
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G H BBookmark kLocked kLocked NReply
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FelixD
588 posts
FelixD
#1 h  Apr 26, 2010, 12:29 PM • 2 Y
Y by Adventure10, Mango247
Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.

(41th Austrian Mathematical Olympiad, regional competition, problem 3)
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MJ GEO
304 posts
MJ GEO
#2 h  Apr 26, 2010, 3:10 PM • 2 Y
Y by Adventure10, Mango247
i think its trivial.$VUA=B$ and $UVA=C$ (becuase $UV$ is perpendicular to $AD$). :)
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