G
Topic
First Poster
Last Poster
Bounds on degree of polynomials
Phorphyrion 4
N
by Kingsbane2139
Source: 2020 Israel Olympic Revenge P3
For each positive integer 
, define 
to be the least positive integer for which the following holds:
For any partition of
into 
disjoint subsets 
, all of the same size, let 
. Then there exist 
for which
![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](//latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
a) Prove that there is a constant
so that 
for all .
b) Prove that for infinitely many, one has 
.
, define 
to be the least positive integer for which the following holds:For any partition of
into 
disjoint subsets 
, all of the same size, let 
. Then there exist 
for which![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](http://latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
a) Prove that there is a constant
so that 
for all .b) Prove that for infinitely many
, one has 
.
4 replies
+1 w
A point on BC
jayme 7
N
by jayme
Source: Own ?
Dear Mathlinkers,
1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.
Prove : B, C and T are collinear.
Sincerely
Jean-Louis
1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.
Prove : B, C and T are collinear.
Sincerely
Jean-Louis
7 replies
Zack likes Moving Points
pinetree1 73
N
by NumberzAndStuff
Source: USA TSTST 2019 Problem 5
Let 
be an acute triangle with orthocenter 
and circumcircle 
. A line through intersects segments 
and 
at 
and 
, respectively. Let 
be the circumcenter of 
, and suppose line 
intersects again at a point 
. Prove that line 
and the line through perpendicular to 
meet on .
Gunmay Handa
be an acute triangle with orthocenter 
and circumcircle 
. A line through intersects segments 
and 
at 
and 
, respectively. Let 
be the circumcenter of 
, and suppose line 
intersects again at a point 
. Prove that line 
and the line through perpendicular to 
meet on .Gunmay Handa
73 replies
Domain and Inequality
Kunihiko_Chikaya 1
N
by Mathzeus1024
Source: 2018 The University of Tokyo entrance exam / Humanities, Problem 1
Define on a coordinate plane, the parabola 
and the domain 
Suppose that two lines
passing through the origin touch 
.
(1) Let
be a mobile point on the parabola . Let denote 
the distances between the point and the lines respectively. Find the coordinate of the point giving the minimum value of 
(2) Draw the domain of the set of the points
on a coordinate plane such that for all points 
over the domain , the inequality 
holds.
and the domain 
Suppose that two lines
passing through the origin touch 
.(1) Let
be a mobile point on the parabola . Let denote 
the distances between the point and the lines respectively. Find the coordinate of the point giving the minimum value of 
(2) Draw the domain of the set of the points
on a coordinate plane such that for all points 
over the domain , the inequality 
holds.
1 reply
JBMO TST Bosnia and Herzegovina 2020 P1
Steve12345 3
N
by AylyGayypow009
Determine all four-digit numbers 
which are perfect squares and for which the equality holds:
.
which are perfect squares and for which the equality holds:
.
3 replies
Problem3
samithayohan 116
N
by fearsum_fyz
Source: IMO 2015 problem 3
Let be an acute triangle with 
. Let 
be its circumcircle, its orthocenter, and the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let be the point on such that 
. Assume that the points , 
, , and are all different and lie on in this order.
Prove that the circumcircles of triangles
and 
are tangent to each other.
Proposed by Ukraine
be an acute triangle with 
. Let 
be its circumcircle, its orthocenter, and the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let be the point on such that 
. Assume that the points , 
, , and are all different and lie on in this order.Prove that the circumcircles of triangles
and 
are tangent to each other.Proposed by Ukraine
116 replies
geometry problem
invt 0
In a triangle with 
, denote its incenter and midpoint of by 
, , respectively. Let 
be the reflected point of wrt 
. Let the lines 
and 
meet at 
. Suppose that 
. Prove that 
.
with 
, denote its incenter and midpoint of by 
, , respectively. Let 
be the reflected point of wrt 
. Let the lines 
and 
meet at 
. Suppose that 
. Prove that 
.
0 replies
the locus of $P$
littletush 10
N
by SuperBarsh
Source: Italy TST 2009 p2
is a triangle in the plane. Find the locus of point 
for which 
form a triangle whose area is equal to one third of the area of triangle .
10 replies
Abelkonkurransen 2025 3b
Lil_flip38 3
N
by Adywastaken
Source: abelkonkurransen
An acute angled triangle 
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
3 replies
I got stuck in this combinatorics
artjustinhere237 3
N
by GreekIdiot
Let 
, where 
is a positive integer.
Prove that there exists a subset of
with exactly 
elements such that the sum of its elements is a prime number.
, where 
is a positive integer.Prove that there exists a subset of
with exactly 
elements such that the sum of its elements is a prime number.
3 replies
