what the yap
by KevinYang2.71, Mar 20, 2025, 12:00 PM
Alice the architect and Bob the builder play a game. First, Alice chooses two points 
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
Note:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
For every city 
distinct from 
and 
, there exists 
such
distinct from 
and 
, there exists 
such
that 
is directly similar to either 
or 
.
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
is directly similar to either 
or 
.
Note:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.[TEST RELEASED] Mock Geometry Test for College Competitions
by Bluesoul, Feb 24, 2025, 9:42 AM
This post has been edited 12 times. Last edited by Bluesoul, 4 hours ago
2024 AMC 10B Discussion Thread
by LauraZed, Nov 13, 2024, 5:09 PM
This post has been edited 7 times. Last edited by LauraZed, Nov 13, 2024, 6:20 PM
[TEST RELEASED] OMMC Year 4
by DottedCaculator, Apr 23, 2024, 2:31 PM
FINAL LEADERBOARD: https://docs.google.com/spreadsheets/u/0/d/12RamVH-gQIPN4wibYZVqkx1F2JQuy5Li_8IJ8TqVEyg/htmlview#gid=409219165
Hello to all creative problem solvers,
Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists?
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?
Check out the fourth annual iteration of the
Online Monmouth Math Competition!
Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.
Our website: https://www.ommcofficial.org/
Our Discord (5000+ members): https://tinyurl.com/joinommc
Test portal: https://ommc-test-portal.vercel.app/
This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.
We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!
You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc
Timeline:
Main Round: May 19th - May 26th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.
Final Round: May 28th - May 30th
The top placing teams will qualify for this invitational round (7 questions). The final round consists of 7 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.
Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.
We hope for your participation, and good luck!
OMMC staff
OMMC’S 2024 EVENTS ARE SPONSORED BY:
Hello to all creative problem solvers,
Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists?
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?
Check out the fourth annual iteration of the
Online Monmouth Math Competition!
Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.
Our website: https://www.ommcofficial.org/
Our Discord (5000+ members): https://tinyurl.com/joinommc
Test portal: https://ommc-test-portal.vercel.app/
This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.
We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!
You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc
Timeline:
Main Round: May 19th - May 26th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.
Final Round: May 28th - May 30th
The top placing teams will qualify for this invitational round (7 questions). The final round consists of 7 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.
Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.
We hope for your participation, and good luck!
OMMC staff
OMMC’S 2024 EVENTS ARE SPONSORED BY:
- Nontrivial Fellowship
- Citadel
- SPARC
- Jane Street
- And counting!
This post has been edited 5 times. Last edited by DottedCaculator, Jul 31, 2024, 1:21 AM
A beautiful theorem about tangency of circles
by math_pi_rate, Aug 19, 2018, 2:22 PM
So a week before I went for the Sharygin Finals, I striked upon this beautiful theorem, called Casey's Theorem (also known as Generalised Ptolemy's Theorem). So here is the theorem, in all its might:
THEOREM (Casey) Given four circles
, let 
denote the length of a common tangent (either internal or external) between 
and 
. Then the four circles are tangent to a fifth circle 
(or line) if and only if for appropriate choice of signs, 
.
For a proof, see this handout by Luis Gonzales or have a look at Problem 239 and 240 of Problems in Plane Geometry by I. F. Sharygin.
Anyway, I'll just post a few problems which are reduced to mere computations by this theorem (mainly the if part).
PROBLEM 1 (Sharygin Finals 2017 Problem 9.4) Points
and 
are chosen on lateral sides 
of an isosceles triangle 
and point 
is chosen on 
such that 
is a parallelogram. Let the lines 
and meet at point 
, and let 
be the intersection points of with the perpendicular line from to . Prove that the circle with center and radius 
and the circumcircle of triangle 
are tangent.
PROBLEM 2 (Tuymaada 2018 Senior/Junior League Problem 8) Quadrilateral
with perpendicular diagonals is inscribed in a circle with centre 
. The tangents to this circle at and together with line 
form the triangle 
. Prove that the circumcircles of 
and are tangent.
PROBLEM 3 (Source=buratinogigle) Let be a triangle inscribed in circle 
The circle 
passes through 
with is on perpendicular bisector of 
Similarly, we have the circles 
and 
Prove that the circle is tangent to 
and 
internally then is tangent to 
.
PROBLEM 4 (Source=buratinogigle) Let be a triangle inscribed in circle with altitude 
. Incircle 
touches at . is midpoint of . is projection of on 
. Prove that circle 
is tangent to .
PROBLEM 5 (Source=buratinogigle) Let be a triangle with circumcenter and altitude 
meets at and meets the circle 
again at 
is the midpoint of 
is the projection of on line Prove that the circle 
is tangent to the circle .
THEOREM (Casey) Given four circles
, let 
denote the length of a common tangent (either internal or external) between 
and 
. Then the four circles are tangent to a fifth circle 
(or line) if and only if for appropriate choice of signs, 
.For a proof, see this handout by Luis Gonzales or have a look at Problem 239 and 240 of Problems in Plane Geometry by I. F. Sharygin.
Anyway, I'll just post a few problems which are reduced to mere computations by this theorem (mainly the if part).
PROBLEM 1 (Sharygin Finals 2017 Problem 9.4) Points
and 
are chosen on lateral sides 
of an isosceles triangle 
and point 
is chosen on 
such that 
is a parallelogram. Let the lines 
and meet at point 
, and let 
be the intersection points of with the perpendicular line from to . Prove that the circle with center and radius 
and the circumcircle of triangle 
are tangent.
and 
.
,
and 
.
and circle 
,
.PROBLEM 2 (Tuymaada 2018 Senior/Junior League Problem 8) Quadrilateral
with perpendicular diagonals is inscribed in a circle with centre 
. The tangents to this circle at and together with line 
form the triangle 
. Prove that the circumcircles of 
and are tangent.
.
is tangent to 
.
.
and circle 
is tangent to PROBLEM 3 (Source=buratinogigle) Let
be a triangle inscribed in circle 
The circle 
passes through 
with is on perpendicular bisector of 
Similarly, we have the circles 
and 
Prove that the circle is tangent to 
and 
internally then is tangent to 
.
denote the length of the common external tangent of 
be the radius of 
denote the angles of 
.
and 
.
,PROBLEM 4 (Source=buratinogigle) Let
be a triangle inscribed in circle with altitude 
. Incircle 
touches at . is midpoint of . is projection of on 
. Prove that circle 
is tangent to .
.
denote the sidelengths of 
and circle 
.PROBLEM 5 (Source=buratinogigle) Let
be a triangle with circumcenter and altitude 
meets at and meets the circle 
again at 
is the midpoint of 
is the projection of on line Prove that the circle 
is tangent to the circle .
-
.
be the antipode of 
in 
.
is tangent to 
be the 
at 
.
be the circle with diameter 
.
denote its semiperimeter.
This post has been edited 3 times. Last edited by math_pi_rate, Sep 27, 2018, 12:57 PM
January 2020
September 2018