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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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jlacosta
Apr 2, 2025
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TST Junior Romania 2025
ant_   1
N a minute ago by wassupevery1
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
1 reply
ant_
Yesterday at 5:01 PM
wassupevery1
a minute ago
J
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Very tight inequalities
KhuongTrang   2
N 30 minutes ago by SunnyEvan
Source: own
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{35a+12b+2}+\frac{1}{35b+12c+2}+\frac{1}{35c+12a+2}\ge \frac{4}{39}.}$$$$\color{black}{\frac{1}{4a+9b+6}+\frac{1}{4b+9c+6}+\frac{1}{4c+9a+6}\le \frac{2}{9}.}$$When does equality hold?
2 replies
KhuongTrang
May 17, 2024
SunnyEvan
30 minutes ago
J
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Sum of First, Second, and Third Powers
Brut3Forc3   47
N 43 minutes ago by cubres
Source: 1973 USAMO Problem 4
Determine all roots, real or complex, of the system of simultaneous equations
\begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}
47 replies
Brut3Forc3
Mar 7, 2010
cubres
43 minutes ago
J
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Modified Sum of floors
prMoLeGend42   2
N an hour ago by cubres
Find the closed form of : $\sum _{k=0}^{n-1} \left\lfloor \frac{ak+b}{n}\right \rfloor$ where $\gcd(a,n)=1$
2 replies
+1 w
prMoLeGend42
Yesterday at 9:09 AM
cubres
an hour ago
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circle with diameter AH
rogue   1
N May 24, 2008 by mr.danh
Source: Ukrainian journal contest, problem 324, by Yuriy Biletskyy
Let $ H$ be the orthocenter of acute-angled triangle $ ABC.$ Circle $ \omega$ with diameter $ AH$ and circumcircle of triangle $ BHC$ intersect at point $ P\ne H.$ Prove that the straight line $ AP$ pass through the midpoint of $ BC.$
1 reply
rogue
May 24, 2008
mr.danh
May 24, 2008
J
circle with diameter AH
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Source: Ukrainian journal contest, problem 324, by Yuriy Biletskyy
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rogue
553 posts
rogue
#1 h  May 24, 2008, 9:43 AM • 2 Y
Y by Adventure10, Mango247
Let $ H$ be the orthocenter of acute-angled triangle $ ABC.$ Circle $ \omega$ with diameter $ AH$ and circumcircle of triangle $ BHC$ intersect at point $ P\ne H.$ Prove that the straight line $ AP$ pass through the midpoint of $ BC.$
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mr.danh
635 posts
mr.danh
#2 h  May 24, 2008, 10:24 AM • 2 Y
Y by Adventure10, Mango247
rogue wrote:
Let $ H$ be the orthocenter of acute-angled triangle $ ABC$. M is the midpoint of side BC. P is on the median AM such that $ HP\perp AM$. Prove that P lies on the circumcircle $ (\omega)$ of triangle BHC.
$ D=BH\cap AC, E=CH\cap AB, F=AH\cap BC$.
$ AP.AM=AH.AF=AE.AB=AD.AC$, hence BMPE and CMPD are two cyclic quadrilateral.
Then, $ \widehat{BPC}=\widehat{BPM}+\widehat{CPM}=\widehat{BEM}+\widehat{CDM}=\widehat{ABC}+\widehat{ACB}=180^o-\widehat{BAC}=\widehat{BHC}$
So B,H,P,C lie on a circle.
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