G
Topic
First Poster
Last Poster
Geometry
MathsII-enjoy 3
N
by MathsII-enjoy
Given triangle 
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of 
onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of 
onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
3 replies
+1 w
IMO 2018 Problem 2
juckter 98
N
by ezpotd
Find all integers 
for which there exist real numbers 
satisfying 
, 
and
for 
.
Proposed by Patrik Bak, Slovakia
for which there exist real numbers 
satisfying 
, 
and
for 
.Proposed by Patrik Bak, Slovakia
98 replies
Element sum of k others
akasht 19
N
by ezpotd
Source: ISL 2022 A2
Let 
be an integer. Find the smallest integer 
with the property that there exists a set of 
distinct real numbers such that each of its elements can be written as a sum of 
other distinct elements of the set.
be an integer. Find the smallest integer 
with the property that there exists a set of 
distinct real numbers such that each of its elements can be written as a sum of 
other distinct elements of the set.
19 replies
Least swaps to get any labeling of a regular 99-gon
Photaesthesia 9
N
by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let 
be a regular 
-gon. Assign integers between 
and to the vertices of such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of . Find the smallest integer such that one can achieve every other assignment from a given one with no more than operations.
Proposed by Zhenhua Qu
be a regular 
-gon. Assign integers between 
and to the vertices of such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of . Find the smallest integer such that one can achieve every other assignment from a given one with no more than operations.Proposed by Zhenhua Qu
9 replies
Angles in a triangle with integer cotangents
Stear14 0
In a triangle , the point is the midpoint of and is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.
(a) Show that the cotangent of the angle
is also an integer and equals 
.
(b) Show that there are infinitely many possible triples
, some of which consisting of Fibonacci numbers.
, the point is the midpoint of and is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.(a) Show that the cotangent of the angle
is also an integer and equals 
.(b) Show that there are infinitely many possible triples
, some of which consisting of Fibonacci numbers.
0 replies
R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3 1
N
by maromex
Source: p24734470
Find all functions 
such that for all positive real numbers 
and 
:
such that for all positive real numbers 
and 
:
1 reply
An important lemma of isogonal conjugate points
buratinogigle 6
N
by buratinogigle
Source: Own
Let and 
be two isogonal conjugate with respect to triangle . Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of , respectively. Prove that 
and 
bisect two arcs containing 
and not containing , respectively, of 
.
and 
be two isogonal conjugate with respect to triangle . Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of , respectively. Prove that 
and 
bisect two arcs containing 
and not containing , respectively, of 
.
6 replies
A difficult problem [tangent circles in right triangles]
ThAzN1 48
N
by Autistic_Turk
Source: IMO ShortList 1998, geometry problem 8; Yugoslav TST 1999
Let be a triangle such that 
and 
. The tangent at to the circumcircle 
of triangle meets the line at . Let be the reflection of in the line , let 
be the foot of the perpendicular from to 
, and let 
be the midpoint of the segment 
. Let the line 
intersect the circle again at 
.
Prove that the line
is tangent to the circumcircle of triangle 
.
be a triangle such that 
and 
. The tangent at to the circumcircle 
of triangle meets the line at . Let be the reflection of in the line , let 
be the foot of the perpendicular from to 
, and let 
be the midpoint of the segment 
. Let the line 
intersect the circle again at 
.Prove that the line
is tangent to the circumcircle of triangle 
.48 replies
1 viewing
IMO 2008, Question 2
delegat 63
N
by ezpotd
Source: IMO Shortlist 2008, A2
(a) Prove that
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](//latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers , , 
, each different from , and satisfying 
.
(b) Prove that equality holds above for infinitely many triples of rational numbers, , , each different from , and satisfying .
Author: Walther Janous, Austria
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers , , 
, each different from , and satisfying 
.(b) Prove that equality holds above for infinitely many triples of rational numbers
, , , each different from , and satisfying .Author: Walther Janous, Austria
63 replies
USAMO 2003 Problem 1
MithsApprentice 69
N
by de-Kirschbaum
Prove that for every positive integer there exists an -digit number divisible by 
all of whose digits are odd.
there exists an -digit number divisible by 
all of whose digits are odd.
69 replies
