Inspired by Omerking
by sqing, Apr 16, 2025, 5:11 AM
Let 
and 
Prove that
Where 
and 
Prove that
Where 
This post has been edited 1 time. Last edited by sqing, 3 hours ago
Interesting inequalities
by sqing, Apr 16, 2025, 3:36 AM
Let 
and 
. Prove that
Where 
and 
. Prove that
Where 
This post has been edited 1 time. Last edited by sqing, 2 hours ago
A Segment Bisection Problem
by buratinogigle, Apr 16, 2025, 1:36 AM
In triangle 
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.Attachments:
A Characterization of Rectangles
by buratinogigle, Apr 16, 2025, 1:35 AM
Prove that if a convex quadrilateral 
satisfies the equation
![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](//latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.NEPAL TST 2025 DAY 2
by Tony_stark0094, Apr 12, 2025, 8:40 AM
Consider an acute triangle 
. Let and 
be the feet of the altitudes from 
to and from 
to 
respectively.
Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .
If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
. Let and 
be the feet of the altitudes from 
to and from 
to 
respectively.Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
This post has been edited 1 time. Last edited by Tony_stark0094, Apr 13, 2025, 12:37 AM
Reason: typo
Reason: typo
NEPAL TST DAY 2 PROBLEM 2
by Tony_stark0094, Apr 12, 2025, 8:37 AM
Kritesh manages traffic on a 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.
Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.
What is the minimum value of that guarantees that Kritesh's job is possible?
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.What is the minimum value of
that guarantees that Kritesh's job is possible?
This post has been edited 2 times. Last edited by Tony_stark0094, Apr 13, 2025, 3:12 AM
Reason: typo
Reason: typo
NEPAL TST DAY-2 PROBLEM 1
by Tony_stark0094, Apr 12, 2025, 8:34 AM
Let the sequence 
be defined by
![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](//latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that
![\[
a_n^{2025} >n^{2024}
\]](//latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
This post has been edited 1 time. Last edited by Tony_stark0094, Apr 13, 2025, 12:36 AM
Reason: typo
Reason: typo
Hard number theory
by Hip1zzzil, Mar 30, 2025, 5:08 AM
Positive integers 
satisfy both of the following conditions.
For a positive integer
, if 
, then 
.
There exist integers
that satisfies the equation 
and 
.
Prove that there exist integers that satisfies the equation 
, for each integer 
.
satisfy both of the following conditions.For a positive integer
, if 
, then 
.There exist integers
that satisfies the equation 
and 
.Prove that there exist integers
that satisfies the equation 
, for each integer 
.This post has been edited 4 times. Last edited by Hip1zzzil, Mar 30, 2025, 1:07 PM
Reason: Better
Reason: Better
Constant Angle Sum
by i3435, May 11, 2021, 1:06 PM
Let be a triangle with circumcircle 
, -angle bisector 
, and -median 
. Suppose that meets 
at and meets again at 
. A line 
parallel to meets , at 
, 
respectively, so that is between and . The circle with diameter 
meets again at 
.
As varies, show that 
is constant.
MP8148
be a triangle with circumcircle 
, -angle bisector 
, and -median 
. Suppose that meets 
at and meets again at 
. A line 
parallel to meets , at 
, 
respectively, so that is between and . The circle with diameter 
meets again at 
.As
varies, show that 
is constant.MP8148
2017 PAMO Shortlsit: Power of a prime is a sum of cubes
by DylanN, May 5, 2019, 8:46 PM
For which prime numbers 
can we find three positive integers , 
and 
such that 
?
can we find three positive integers , 
and 
such that 
?Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
September 2017
February 2017
September 2016
July 2016
April 2016
March 2016
December 2015
November 2015
August 2015
April 2015
March 2015
February 2015
January 2015
October 2014
September 2014
August 2014
July 2014