2013 Japan MO Finals P5
by parkjungmin, Jun 5, 2025, 4:40 PM
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What the isogonal conjugate on IO
by reni_wee, Jun 5, 2025, 2:11 PM
Given a triangle 
with incircle 
tangent to 
at points 
, respectively. Let 
be a point such that its isogonal conjugate lies on the line 
(where 
is the circumcenter and 
the incenter of ). The line 
intersects segments 
and 
at points 
and 
, respectively, such that the circle with diameter 
meets 
at points 
and 
.
1) Prove that the circle
is tangent to the incircle at some point 
.
2) Similarly define points
corresponding to vertices 
. Prove that the lines 
are concurrent.
with incircle 
tangent to 
at points 
, respectively. Let 
be a point such that its isogonal conjugate lies on the line 
(where 
is the circumcenter and 
the incenter of ). The line 
intersects segments 
and 
at points 
and 
, respectively, such that the circle with diameter 
meets 
at points 
and 
.1) Prove that the circle
is tangent to the incircle at some point 
.2) Similarly define points
corresponding to vertices 
. Prove that the lines 
are concurrent.Might be slightly generalizable
by Rijul saini, Jun 4, 2025, 6:39 PM
Let be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at (
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at (
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.Sharing is a nontrivial task
by bjump, Jan 15, 2024, 5:00 PM
Suppose 
is an arithmetic sequence of positive integers, and 
is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
Ray Li
is an arithmetic sequence of positive integers, and 
is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.Ray Li
This post has been edited 1 time. Last edited by v_Enhance, Jan 22, 2024, 4:13 AM
Most accurate rounding
by popcorn1, Dec 11, 2023, 5:00 PM
Find the smallest constant 
such that the following statement holds: for every integer 
and sequence of non-integer positive real numbers 
satisfying 
it's possible to choose positive integers 
such that
(i) for each
, either 
or 
, and
(ii) we have
(Here 
denotes the floor function, as usual.)
Merlijn Staps
such that the following statement holds: for every integer 
and sequence of non-integer positive real numbers 
satisfying 
it's possible to choose positive integers 
such that(i) for each
, either 
or 
, and(ii) we have
(Here 
denotes the floor function, as usual.)Merlijn Staps
This post has been edited 2 times. Last edited by popcorn1, Dec 11, 2023, 5:21 PM
students in a classroom sit in a round table, possible to split into 3 groups
by parmenides51, Aug 30, 2019, 5:16 PM
Merlijn Has 100 Coins
by tastymath75025, Jun 25, 2019, 5:36 PM
Consider coins with positive real denominations not exceeding 1. Find the smallest 
such that the following holds: if we have any 
such coins with total value 
, then we can always split them into two stacks of coins each such that the absolute difference between the total values of the two stacks is at most 
.
Merlijn Staps
such that the following holds: if we have any 
such coins with total value 
, then we can always split them into two stacks of coins each such that the absolute difference between the total values of the two stacks is at most 
.Merlijn Staps
This post has been edited 2 times. Last edited by djmathman, Jan 4, 2020, 4:02 AM
Prove that circumcircle of the triangle TEF tangent to (O), (K), (L)
by centos6, Nov 30, 2018, 11:40 AM
Let 
be the circumcircle of the triangle 
. 
be the antipode of 
in . Angle bisector of angle 
meets and 
at 
and 
. Let 
be the midpoind of the arc 
. 
. 
and 
be the Thebault circles of the cevian 
. Prove that circumcircle of the triangle 
tangent to , and .
be the circumcircle of the triangle 
. 
be the antipode of 
in . Angle bisector of angle 
meets and 
at 
and 
. Let 
be the midpoind of the arc 
. 
. 
and 
be the Thebault circles of the cevian 
. Prove that circumcircle of the triangle 
tangent to , and .This post has been edited 1 time. Last edited by centos6, Nov 30, 2018, 11:43 AM
IMO ShortList 1998, algebra problem 1
by orl, Oct 22, 2004, 2:46 PM
Let 
be positive real numbers such that 
. Prove that
![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](//latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
be positive real numbers such that 
. Prove that![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
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This post has been edited 1 time. Last edited by orl, Oct 23, 2004, 12:50 PM
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