,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
and 
. Prove that
Where 
and 
Prove that
Where 
Y by xyzz, Adventure10, Mango247
Let 
be nonnegative real numbers such that 
Prove that
be nonnegative real numbers such that 
Prove that
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Topic
First Poster
Last Poster
Inequality while on a trip
giangtruong13 6
N
by GeoMorocco
Source: Trip
I find this inequality while i was on a trip, it was pretty fun and i have some new experience:
Let
such that: 
. Find the maximum: 
Let
such that: 
. Find the maximum: 
6 replies
A Characterization of Rectangles
buratinogigle 1
N
by lbh_qys
Source: VN Math Olympiad For High School Students P8 - 2025
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.
1 reply
A Segment Bisection Problem
buratinogigle 1
N
by Giabach298
Source: VN Math Olympiad For High School Students P9 - 2025
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 
.
1 reply
2017 PAMO Shortlsit: Power of a prime is a sum of cubes
DylanN 3
N
by AshAuktober
Source: 2017 Pan-African Shortlist - N2
For which prime numbers 
can we find three positive integers 
, 
and 
such that 
?
can we find three positive integers 
, 
and 
such that 
?
3 replies
Hard number theory
Hip1zzzil 14
N
by bonmath
Source: FKMO 2025 P6
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 .
14 replies
Constant Angle Sum
i3435 6
N
by bin_sherlo
Source: AMASCIWLOFRIAA1PD (mock oly geo contest) P3
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
6 replies
NEPAL TST 2025 DAY 2
Tony_stark0094 8
N
by cursed_tangent1434
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.
8 replies
NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094 6
N
by cursed_tangent1434
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?
6 replies
NEPAL TST DAY-2 PROBLEM 1
Tony_stark0094 9
N
by cursed_tangent1434
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 
.
9 replies
3 + abcd >= a + b + c + d
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Source: dedicated to ductrung...
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Y by
A splendid and very instructive comment. Especially since you, along I and another colleague from China are the authors of the generalization of this problem. So the point of this "bump" is known by you only.
Good.
See the problem 494 from https://ssmr.ro/gazeta/gma/2019/gma1-2-2019-continut.pdf
Good.
See the problem 494 from https://ssmr.ro/gazeta/gma/2019/gma1-2-2019-continut.pdf
This post has been edited 1 time. Last edited by mihaig, Jan 18, 2021, 9:03 AM
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#5
Jan 26, 2021, 6:17 AM
Y by
mihaig wrote:
A splendid and very instructive comment. Especially since you, along I and another colleague from China are the authors of the generalization of this problem. So the point of this "bump" is known by you only.
Good.
See the problem 494 from https://ssmr.ro/gazeta/gma/2019/gma1-2-2019-continut.pdf
Good.
See the problem 494 from https://ssmr.ro/gazeta/gma/2019/gma1-2-2019-continut.pdf
If I am not violating any policy of AOPS, I would like to ask...Is he a professor? Is his surname sqing and has he written any books or articles?
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Y by
Quantum_fluctuations wrote:
If I am not violating any policy of AOPS, I would like to ask...Is he a professor? Is his surname sqing and has he written any books or articles?
I'm afraid I don't even know how to start a proof to your proposed problems.
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