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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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N.S. condition of passing a fixed point for a function
Kunihiko_Chikaya   1
N 28 minutes ago by Mathzeus1024
Let $ f(t)$ be a function defined in any real numbers $ t$ with $ f(0)\neq 0.$ Prove that on the $ x-y$ plane, the line $ l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $ y$ axis in regardless of the value of $ t$ if only if $ f(t)$ is a linear function in $ t$.
1 reply
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Kunihiko_Chikaya
Sep 6, 2009
Mathzeus1024
28 minutes ago
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Dot product
SomeonecoolLovesMaths   4
N 31 minutes ago by quasar_lord
How to prove that dot product is distributive?
4 replies
SomeonecoolLovesMaths
Yesterday at 6:06 PM
quasar_lord
31 minutes ago
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Geometry Angle Chasing
Sid-darth-vater   5
N 2 hours ago by sunken rock
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
5 replies
Sid-darth-vater
Apr 21, 2025
sunken rock
2 hours ago
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About my new website
Samujjal101   21
N 2 hours ago by Samujjal101
Hi everybody!
I'm registering some of the finest minds in math into my website.. it's not completely developed.. but still if you want we would be very grateful to have you!
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Maths-matchmaker is a website for connecting math minds together with a mission to unite together. It uses a matching algorithm to match 1:1 with like minded peers based on their interests or topics in math
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Samujjal101
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IK/IA +IL/IB +IM/IC >= 3/2, angle bisectors (Formula of Unity 2019 1st gr 10,11)
parmenides51   1
N May 6, 2020 by jessica2003
Angle bisectors $AK, BL$, and $CM$ of a triangle $ABC$ intersect in the point $I$ (the points $K, L, M$ lie at the sides of the triangle). Prove that $\frac{IK}{IA} +\frac{IL}{IB} +\frac{IM}{IC }\ge \frac32$ .

1 reply
parmenides51
May 5, 2020
jessica2003
May 6, 2020
J
IK/IA +IL/IB +IM/IC >= 3/2, angle bisectors (Formula of Unity 2019 1st gr 10,11)
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geometry
geometric inequality
angle bisector
Formula of Unity
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parmenides51
30650 posts
parmenides51
#1 h  May 5, 2020, 6:53 PM
Y by
Angle bisectors $AK, BL$, and $CM$ of a triangle $ABC$ intersect in the point $I$ (the points $K, L, M$ lie at the sides of the triangle). Prove that $\frac{IK}{IA} +\frac{IL}{IB} +\frac{IM}{IC }\ge \frac32$ .
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jessica2003
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jessica2003
#2 h  May 6, 2020, 3:38 AM
Y by
Hint: using Mass Point.
Let's set AB=c, AC=b and BC=a, and set the mass of A=a, then we can calculate the masses of B and C, which are b and c, respectively.
Now, the LHS is equal to
$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$
which is >=3/2.
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