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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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A functional equation from MEMO
square_root_of_3   25
N a minute ago by Bardia7003
Source: Middle European Mathematical Olympiad 2022, problem I-1
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$holds for all real numbers $x$ and $y$.
25 replies
square_root_of_3
Sep 1, 2022
Bardia7003
a minute ago
J
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   23
N a minute ago by ErTeeEs06
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
23 replies
falantrng
6 hours ago
ErTeeEs06
a minute ago
J
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Arbitrary point on BC and its relation with orthocenter
falantrng   15
N a minute ago by Steve12345
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
15 replies
falantrng
6 hours ago
Steve12345
a minute ago
J
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Interesting inequality
sqing   7
N 2 minutes ago by Kunihiko_Chikaya
Let $ a,b> 0 $ and $ a+b+ab=1. $ Prove that
$$\frac{1}{1+a^2} + \frac{1}{1+b^2} +a+b\leq \frac{5}{\sqrt{2}}-1 $$
7 replies
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sqing
Today at 3:35 AM
Kunihiko_Chikaya
2 minutes ago
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No more topics!
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constant circle
sam-n   1
N Mar 29, 2004 by chaogold
Source: iran (2003)
XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).
1 reply
sam-n
Mar 27, 2004
chaogold
Mar 29, 2004
J
constant circle
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Reveal topic tags
vector
geometry
angle bisector
geometry solved
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Source: iran (2003)
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sam-n
793 posts
sam-n
#1 h  Mar 27, 2004, 11:55 AM • 2 Y
Y by Adventure10, Mango247
XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).
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chaogold
38 posts
chaogold
#2 h  Mar 29, 2004, 1:41 AM • 2 Y
Y by Adventure10, Mango247
consider rays OA OB OD
where OD bisect the angle BOA.
Draw a circle with center O and radius 1.
Then inverse A B D with respect to the circle.
We got A'B'D',and here A'B'D'O are cyclic.
Also <A'OD'=<D'OB',by Ptolemy's Theorem.
We have (A'O)+(B'O)=k(D'O)
this leads to 1/OA+1/OB=k/OD
So,if 1/OA+1/OB is a constant 1,line AB pass through
a fixed point D on the angle bisector of <BOA.

Take the midpoint C of OD as center,1 the radius.
Because OC is the angle bisector,
AC/BC=AO/BO=a/b (Let OA=a,OB=b)
Let T U be the midpoint of OA,OB
the vector <OC>=b/(a+b)<OT>+a/(a+b)<OU>
and the radius 1=[b/(a+b)](a/2)+[a/(a+b)](b/2)
(here we used 1/a+1/b=1)
By now we can easily check that with the circle we draw,
and the circles with diameter OA,OB.
They have two common tangents all the time.

Best regards,
Chao
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