2016 Kmo Final round
by Jackson0423, Apr 22, 2025, 3:58 PM
Let 
with 
.
Find the maximum value of
![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](//latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
with 
.Find the maximum value of
![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
2^x+3^x = yx^2
by truongphatt2668, Apr 22, 2025, 3:38 PM
Prove that the following equation has infinite integer solutions:
Interesting F.E
by Jackson0423, Apr 18, 2025, 4:12 PM
Show that there does not exist a function
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](//latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,
![\[
f(x + y^2) \geq f(x) + y.
\]](//latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
~Korea 2017 P7
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
~Korea 2017 P7
This post has been edited 3 times. Last edited by Jackson0423, Yesterday at 3:23 PM
Reason: Sorry guys..
Reason: Sorry guys..
Geometry Problem
by Itoz, Apr 18, 2025, 11:49 AM
Given 
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
Prove that
is tangent to 
.
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.Prove that
is tangent to 
.Attachments:
This post has been edited 1 time. Last edited by Itoz, Apr 18, 2025, 11:53 AM
combinatorial geo question
by SAAAAAAA_B, Apr 14, 2025, 10:34 PM
Kuba has two finite families 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.
\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of or .
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of
or .
and 
: Prove that :
Inequalities make a comeback
by MS_Kekas, Jan 20, 2025, 3:16 AM
Determine the largest possible constant 
such that for any positive real numbers 
, which are the sides of a triangle, the following inequality holds:
![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](//latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
Proposed by Vadym Solomka
such that for any positive real numbers 
, which are the sides of a triangle, the following inequality holds:![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](http://latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
Proposed by Vadym Solomka
All prime factors under 8
by qwedsazxc, Mar 26, 2023, 6:29 AM
Find all positive integers 
satisfying the following.
satisfying the following.
This post has been edited 1 time. Last edited by qwedsazxc, Mar 26, 2023, 6:56 AM
Factor sums of integers
by Aopamy, Feb 23, 2023, 3:13 AM
Let be a positive integer. A positive integer 
is called a benefactor of if the positive divisors of can be partitioned into two sets and such that is equal to the sum of elements in minus the sum of the elements in . Note that or could be empty, and that the sum of the elements of the empty set is 
.
For example,
is a benefactor of 
because 
.
Show that every positive integer has at least 
benefactors.
be a positive integer. A positive integer 
is called a benefactor of if the positive divisors of can be partitioned into two sets and such that is equal to the sum of elements in minus the sum of the elements in . Note that or could be empty, and that the sum of the elements of the empty set is 
.For example,
is a benefactor of 
because 
.Show that every positive integer
has at least 
benefactors.Nasty Floor Sum with Omega Function
by Kezer, Jul 15, 2017, 9:52 AM
Let 
denote the number of prime factors of , counted with multiplicity. Evaluate ![\[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]](//latex.artofproblemsolving.com/2/0/4/2043dca9f2ff2adfd3c8d49fa4dd278871968db0.png)
denote the number of prime factors of , counted with multiplicity. Evaluate ![\[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]](http://latex.artofproblemsolving.com/2/0/4/2043dca9f2ff2adfd3c8d49fa4dd278871968db0.png)
This post has been edited 3 times. Last edited by Kezer, Jul 15, 2017, 12:55 PM
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