.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
be positive reals such that 
.
![\[
M = \frac{1 - x^2}{x^2} + 2y^2 + 3x + \frac{24}{y} + 2025.
\]](http://latex.artofproblemsolving.com/6/b/d/6bd846ab0f95b1cf4766c7bf0356a439f3bfc93a.png)
. Prove that
and 
Show that
Let 
Show that
and 
Show that
Let 
Show that
Let 
Show that
Y by cubres
Let 
and 
. Find minimum:
![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](//latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
and 
. Find minimum:![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
This post has been edited 1 time. Last edited by Havu, Apr 24, 2025, 11:19 PM
Reason: Error
Reason: Error
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dirichlet
spiralman 1
N
by clarkculus
Let n be a positive integer. Consider 2n+1 distinct positive integers whose total sum is less than (n+1)(3n+1). Prove that among these 2n+1 numbers, there exist two numbers whose sum is 2n+1.
1 reply
Graphs and Trig
Math1331Math 1
N
by Mathzeus1024
The graph of the function 
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
1 reply
CSMC Question
vicrong 1
N
by Mathzeus1024
Prove that there is exactly one function h with the following properties
- the domain of h is the set of positive integers
- h(n) is a positive integer for every positive integer n, and
- h(h(n)+m) = 1+n+h(m) for all positive integers n and m
- the domain of h is the set of positive integers
- h(n) is a positive integer for every positive integer n, and
- h(h(n)+m) = 1+n+h(m) for all positive integers n and m
1 reply
function
khyeon 1
N
by Mathzeus1024
Find the range of 
so that any two different points in the graph of the quadratic function 
are not symmetrical to the straight line 
so that any two different points in the graph of the quadratic function 
are not symmetrical to the straight line 
1 reply
Find function
trito11 1
N
by Mathzeus1024
Find 
such that
i) f(x)>f(y)
x>y>0
ii) f(2x)
2f(x)x>0
iii)
x,y>0
such thati) f(x)>f(y)
x>y>0ii) f(2x)
2f(x)x>0iii)
x,y>0
1 reply
The centroid of ABC lies on ME [2023 Abel, Problem 1b]
Amir Hossein 2
N
by MITDragon
In the triangle 
, points 
and 
lie on the side 
, with 
. Also, 
is the midpoint of 
. Show that the centroid of lies on 
.
, points 
and 
lie on the side 
, with 
. Also, 
is the midpoint of 
. Show that the centroid of lies on 
.
2 replies
\frac{1}{5-2a}
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and 
such that 
Prove that:
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