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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
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Hard Inequality Problem
Omerking   1
N 33 minutes ago by lpieleanu
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$ is given where $a,b,c$ are positive reals. Prove that:
$$\frac{1}{\sqrt{a^3+1}}+\frac{1}{\sqrt{b^3+1}}+\frac{1}{\sqrt{c^3+1}} \le \frac{3}{\sqrt{2}}$$
1 reply
Omerking
Yesterday at 3:51 PM
lpieleanu
33 minutes ago
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USAMO 2000 Problem 5
MithsApprentice   22
N 33 minutes ago by Maximilian113
Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$
22 replies
MithsApprentice
Oct 1, 2005
Maximilian113
33 minutes ago
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f.e with finite number of f(t)=-t
jjkim0336   0
40 minutes ago
Source: own
f:R->R
f(xf(y)+y)=yf(x)+f(f(y)) and there are finite number of t such that f(t)= - t
0 replies
jjkim0336
40 minutes ago
0 replies
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Common external tangents of two circles
a1267ab   55
N 41 minutes ago by awesomeming327.
Source: USA Winter TST for IMO 2020, Problem 2, by Merlijn Staps
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.

Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.

Merlijn Staps
55 replies
a1267ab
Dec 16, 2019
awesomeming327.
41 minutes ago
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tangent plane of any of 4 points // plane passing through other 3, convex
parmenides51   2
N Feb 28, 2025 by kiyoras_2001
Source: IV All-Ukrainian Tournament of Young Mathematicians, Qualifying p9
Given a convex body in space. Prove that four points can be marked on its surface so that the tangent plane to the surface at any of these four points is parallel to the plane passing through the other three.
2 replies
parmenides51
May 19, 2021
kiyoras_2001
Feb 28, 2025
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tangent plane of any of 4 points // plane passing through other 3, convex
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Reveal topic tags
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geometry
3D geometry
Plane
Ukrainian TYM
combinatorial geometry
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Source: IV All-Ukrainian Tournament of Young Mathematicians, Qualifying p9
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parmenides51
30650 posts
parmenides51
#1 h  May 19, 2021, 7:51 PM • 1 Y
Y by centslordm
Given a convex body in space. Prove that four points can be marked on its surface so that the tangent plane to the surface at any of these four points is parallel to the plane passing through the other three.
This post has been edited 1 time. Last edited by parmenides51, May 19, 2021, 7:52 PM
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jasperE3
11269 posts
jasperE3
#2 h  Feb 28, 2025, 4:42 AM
Y by
parmenides51 wrote:
Given a convex body in space. Prove that four points can be marked on its surface so that the tangent plane to the surface at any of these four points is parallel to the plane passing through the other three.

is there some topological solution to this (borsuk-ulam theorem? idk)
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kiyoras_2001
678 posts
kiyoras_2001
#3 h  Feb 28, 2025, 10:43 AM • 1 Y
Y by MS_asdfgzxcvb
Let $\mathcal P\subset \mathbb R^3$ be the given convex body. Let $ABCD$ be a tetrahedron of maximal volume inscribed in $\mathcal P$ (it exists in force of Weirestrass' theorem). Then the vertices of $ABCD$ are the desired points.

Indeed, let $\pi$ be the plane parallel to $ABC$ and tangent to $\mathcal P$ lying in the same side of $ABC$ as $D$. Take $D'\in \pi \cap \mathcal P$ (the/a touchpoint of $\pi$ and $\mathcal P$). Then if $D'\ne D$ we would reveal a tetrahedron $ABCD'$ of larger volume in $\mathcal P$. Hence $D'=D$. Similarly the tangent planes to $\mathcal P$ at $A, B, C$ are parallel to their opposite faces in $ABCD$.
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