Tangent to two parabolas

I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE

1571 replies

rrusczyk

Mar 24, 2025

SmartGroot

Mar 26, 2025

k a March Highlights and 2025 AoPS Online Class Information

jlacosta 0

Mar 2, 2025

March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies

jlacosta

Mar 2, 2025

0 replies

IMO ShortList 1998, number theory problem 6

orl 28

N 18 minutes ago by Zany9998

Source: IMO ShortList 1998, number theory problem 6

For any positive integer $n$ , let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$ .

28 replies

orl

Oct 22, 2004

Zany9998

18 minutes ago

A projectional vision in IGO

Shayan-TayefehIR 14

N 22 minutes ago by mathuz

Source: IGO 2024 Advanced Level - Problem 3

In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$ , $I_A$ , $I_C$ be the incenter, $A$ -excenter, and $C$ -excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$ , respectively. Prove that $AP=AQ$ .

Proposed Michal Jan'ik - Czech Republic

14 replies

Shayan-TayefehIR

Nov 14, 2024

mathuz

22 minutes ago

(a²-b²)(b²-c²) = abc

straight 3

N 24 minutes ago by straight

Find all triples of positive integers $(a,b,c)$ such that

$(a^2-b^2)(b^2-c^2) = abc.$
If you can't solve this, assume $gcd(a,c) = 1$ . If this is still too hard assume in $a \ge b \ge c$ that $b-c$ is a prime.

3 replies

straight

Mar 24, 2025

straight

24 minutes ago

A checkered square consists of dominos

nAalniaOMliO 1

N 25 minutes ago by BR1F1SZ

Source: Belarusian National Olympiad 2025

A checkered square $8 \times 8$ is divided into rectangles with two cells. Two rectangles are called adjacent if they share a segment of length 1 or 2. In each rectangle the amount of adjacent with it rectangles is written.
Find the maximal possible value of the sum of all numbers in rectangles.

1 reply

nAalniaOMliO

Yesterday at 8:21 PM

BR1F1SZ

25 minutes ago

No more topics!

Tangent to two parabolas

ancamagelqueme 4

N Feb 4, 2023 by ancamagelqueme

Two parabolas have the same tangent at their vertices and are in different half planes with respect to said tangent. Construct the other common tangent.

4 replies

ancamagelqueme

Feb 4, 2023

ancamagelqueme

Feb 4, 2023

Tangent to two parabolas

G H J

Reveal topic tags

geometry

conics

parabola

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ancamagelqueme

104 posts

ancamagelqueme

#1 h Feb 4, 2023, 6:16 PM

Y by

Two parabolas have the same tangent at their vertices and are in different half planes with respect to said tangent. Construct the other common tangent.

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PROF65

2016 posts

PROF65

#2 h Feb 4, 2023, 7:06 PM

Y by

Since they are tangent to each other they have only one common tangent at finite point ( plus projectively the line at infinity )
best regards
RH HAS

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ancamagelqueme

104 posts

ancamagelqueme

#3 h Feb 4, 2023, 7:18 PM

Y by

The vertices of the given parabolas do not have to coincide.

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PROF65

2016 posts

PROF65

#4 h Feb 4, 2023, 8:16 PM • 1 Y

Y by archimedes26

If so then $S$ , the point of intersection of $FF'$ with the common tangent is their center of homothety where $F,F'$ are their foci ; to draw the second common tangent draw the other tangent to one of them through $S$ .

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ancamagelqueme

104 posts

ancamagelqueme

#5 h Feb 4, 2023, 8:29 PM

Y by

You're right. Thank you RH HAS.

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