Geometry
by Lukariman, May 7, 2025, 4:02 PM
This post has been edited 2 times. Last edited by Lukariman, Yesterday at 4:24 PM
Geometry
by Lukariman, May 6, 2025, 12:43 PM
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that 
= 2∠AMP.
= 2∠AMP.Attachments:
This post has been edited 2 times. Last edited by Lukariman, Yesterday at 6:47 AM
Tangent to two circles
by Mamadi, May 2, 2025, 7:01 AM
Two circles 
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .Isosceles Triangle Geo
by oVlad, Apr 12, 2025, 9:38 AM
Consider the isosceles triangle 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
Kingdom of Anisotropy
by v_Enhance, Jul 12, 2022, 1:41 PM
The kingdom of Anisotropy consists of 
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.
Let
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from .
Proposed by Warut Suksompong, Thailand
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.Let
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from .Proposed by Warut Suksompong, Thailand
This post has been edited 1 time. Last edited by v_Enhance, Jul 12, 2022, 1:59 PM
Reason: add authorship
Reason: add authorship
Line passes through fixed point, as point varies
by Jalil_Huseynov, May 17, 2022, 6:48 PM
Let be a right triangle with 
. Point 
lies on the line 
such that is between and 
. Let be the midpoint of 
and let 
be the seconf intersection point of the circumcircle of 
and the circumcircle of 
. Prove that as varies, the line 
passes through a fixed point.
be a right triangle with 
. Point 
lies on the line 
such that is between and 
. Let be the midpoint of 
and let 
be the seconf intersection point of the circumcircle of 
and the circumcircle of 
. Prove that as varies, the line 
passes through a fixed point.Deduction card battle
by anantmudgal09, Mar 7, 2021, 10:32 AM
A Magician and a Detective play a game. The Magician lays down cards numbered from 
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
Prove that the Detective can guarantee a win if and only if she is allowed to ask at least
questions.
Proposed by Anant Mudgal
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.Prove that the Detective can guarantee a win if and only if she is allowed to ask at least
questions.Proposed by Anant Mudgal
IMO 2018 Problem 5
by orthocentre, Jul 10, 2018, 11:19 AM
Let 
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer such that 
for all 
.
Proposed by Bayarmagnai Gombodorj, Mongolia
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer such that 
for all 
.Proposed by Bayarmagnai Gombodorj, Mongolia
This post has been edited 3 times. Last edited by djmathman, Jun 16, 2020, 4:03 AM
Reason: problem author
Reason: problem author
perpendicularity involving ex and incenter
by Erken, Dec 24, 2008, 2:56 PM
Suppose that 
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .q(x) to be the product of all primes less than p(x)
by orl, Aug 10, 2008, 5:19 PM
For an integer 
, let 
be the least prime that does not divide 
, and define 
to be the product of all primes less than . In particular, 
For having 
, define 
. Consider the sequence 
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](//latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
. Find all such that 
.
, let 
be the least prime that does not divide 
, and define 
to be the product of all primes less than . In particular, 
For having 
, define 
. Consider the sequence 
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
. Find all such that 
.This post has been edited 1 time. Last edited by v_Enhance, Apr 5, 2015, 12:49 PM
Reason: tex cleanup
Reason: tex cleanup
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