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Problem 2
delegat 147
N
32 minutes ago
by math-olympiad-clown
Source: 0
Let
be an integer, and let
be positive real numbers such that
. Prove that
![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](//latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
Proposed by Angelo Di Pasquale, Australia



![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
Proposed by Angelo Di Pasquale, Australia
147 replies

Coloring points of a square, finding a monochromatic hexagon
goodar2006 6
N
41 minutes ago
by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P1
Prove that for each coloring of the points inside or on the boundary of a square with
colors, there exists a monochromatic regular hexagon.

6 replies

Van der Warden Theorem!
goodar2006 7
N
an hour ago
by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P2
Suppose
is the smallest number such that if
, for each coloring of the set
with two colors there exists a monochromatic arithmetic progression of length
. Prove that
.





7 replies
Isosceles triangles among a group of points
goodar2006 2
N
an hour ago
by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part1-P2
Consider a set of
points in plane. Prove that the number of isosceles triangles having their vertices among these
points is
. Find a configuration of
points in plane such that the number of equilateral triangles with vertices among these
points is
.






2 replies
APMO Number Theory
somebodyyouusedtoknow 12
N
an hour ago
by math-olympiad-clown
Source: APMO 2023 Problem 2
Find all integers
satisfying
and
, in which
denotes the sum of all positive divisors of
, and
denotes the largest prime divisor of
.







12 replies
My Unsolved Problem
ZeltaQN2008 0
an hour ago
Source: IDK
Let
be a polynomial with real coefficients.
(a) Suppose that
. Prove that the polynomial
cannot have 2024 real roots.
(b) Suppose that
and
. Prove that
for all real numbers
.

(a) Suppose that


(b) Suppose that




0 replies
Points of a grid
goodar2006 2
N
an hour ago
by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part1-P4
Prove that from an
grid, one can find
points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of
!



2 replies
Classical NT FE
Kimchiks926 6
N
2 hours ago
by math-olympiad-clown
Source: Baltic Way 2022, Problem 16
Let
denote the set of positive integers. Find all functions
satisfying the condition
for all




6 replies
Hagge circle, Thomson cubic, coaxal
kosmonauten3114 0
2 hours ago
Source: My own (maybe well-known)
Let
be a scalene triangle,
its medial triangle, and
a point on the Thomson cubic (=
) of
. (Suppose that
).
Let
be the circumcevian triangle of
wrt
.
Let
be the pedal triangle of
wrt
.
Let
be the reflection in
of
. Define
,
cyclically.
Let
be the reflection in
of
. Define
,
cyclically.
Let
be the reflection in
of
. Define
,
cyclically.
Prove that
,
,
and the orthocentroidal circle of
are coaxal.






Let



Let



Let





Let





Let





Prove that




0 replies
