Max value
by Hip1zzzil, May 17, 2025, 3:55 AM
Three distinct nonzero real numbers 
satisfy:
(i)
(ii)
Find the maximum value of
.
satisfy:(i)
(ii)
Find the maximum value of
.Pythagoras...
by Hip1zzzil, May 17, 2025, 3:41 AM
Find the sum of all 
s such that:
There exists two odd positive integers
such that 
s such that:There exists two odd positive integers
such that 
This post has been edited 1 time. Last edited by Hip1zzzil, 2 hours ago
Reason: E
Reason: E
Guangxi High School Mathematics Competition 2025 Q12
by sqing, May 17, 2025, 2:58 AM
Let 
. Prove that
. Prove that
This post has been edited 1 time. Last edited by sqing, 2 hours ago
Prove that the triangle is isosceles.
by TUAN2k8, May 16, 2025, 9:55 AM
Given acute triangle 
with two altitudes 
and 
.Let 
be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
with two altitudes 
and 
.Let 
be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.Inequality on APMO P5
by Jalil_Huseynov, May 17, 2022, 6:50 PM
Let 
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
such that:
2018 Hong Kong TST2 problem 4
by YanYau, Oct 21, 2017, 11:54 AM
In triangle with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and 
to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.
with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and 
to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.APMO 2016: one-way flights between cities
by shinichiman, May 16, 2016, 8:52 PM
The country Dreamland consists of 
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.
Warut Suksompong, Thailand
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.Warut Suksompong, Thailand
This post has been edited 1 time. Last edited by MellowMelon, May 18, 2017, 3:31 AM
Reason: add proposer
Reason: add proposer
Circles intersecting each other
by rkm0959, Mar 22, 2015, 5:25 AM
There are 
distinct circles in a plane, with radius 
.
Prove that you can select
circles, which form a set , which satisfy the following.
For two arbitrary circles in, they intersect with each other or
For two arbitrary circles in, they don't intersect with each other.
distinct circles in a plane, with radius 
.Prove that you can select
circles, which form a set , which satisfy the following.For two arbitrary circles in
, they intersect with each other orFor two arbitrary circles in
, they don't intersect with each other.This post has been edited 1 time. Last edited by rkm0959, Jun 6, 2015, 2:14 PM
IMO ShortList 2002, algebra problem 3
by orl, Sep 28, 2004, 1:19 PM
Let 
be a cubic polynomial given by 
, where are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be a cubic polynomial given by 
, where are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.Attachments:
This post has been edited 1 time. Last edited by orl, Oct 25, 2004, 12:23 AM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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