[/quote]
,
.![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
?
is defined by![\[ a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.
\]](http://latex.artofproblemsolving.com/e/3/2/e321dcd3384ee3ef58ff8e7412a9d34d3e13eff4.png)
The sequence 
is defined by![\[ b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.
\]](http://latex.artofproblemsolving.com/e/5/4/e546ef04dff00ffc92a5951c4466af1dbd0d474c.png)
Find 
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
are non-negative real numbers and 
Prove that
the equality hold for 
or 
or any
be positive real numbers. Prove 
Prove that
,
.
.
such that for all positive real numbers 
and 
:
and 
.![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](http://latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)
be positive real numbers such that 
.![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
Y by rokkc, PikaPika999
The one-way routes connecting towns 
, 
, 
, 
, 
, and 
are shown in the figure below (not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?
![[asy]
/* AMC8 P14 2024, by NUMANA: BUI VAN HIEU */
import graph;
unitsize(2cm);
real r=0.25;
// Define the nodes and their positions
pair[] nodes = { (0,0), (2,0), (1,1), (3,1), (4,0), (6,0) };
string[] labels = { "A", "M", "X", "Y", "C", "Z" };
// Draw the nodes as circles with labels
for(int i = 0; i < nodes.length; ++i) {
draw(circle(nodes[i], r));
label("$" + labels[i] + "$", nodes[i]);
}
// Define the edges with their node indices and labels
int[][] edges = { {0, 1}, {0, 2}, {2, 1}, {2, 3}, {1, 3}, {1, 4}, {3, 4}, {4, 5}, {3, 5} };
string[] edgeLabels = { "8", "5", "2", "10", "6", "14", "5", "10", "17" };
pair[] edgeLabelsPos = { S, SE, SW, S, SE, S, SW, S, NE};
// Draw the edges with labels
for (int i = 0; i < edges.length; ++i) {
pair start = nodes[edges[i][0]];
pair end = nodes[edges[i][1]];
draw(start + r*dir(end-start) -- end-r*dir(end-start), Arrow);
label("$" + edgeLabels[i] + "$", midpoint(start -- end), edgeLabelsPos[i]);
}
// Draw the curved edge with label
draw(nodes[1]+r * dir(-45)..controls (3, -0.75) and (5, -0.75)..nodes[5]+r * dir(-135), Arrow);
label("$25$", midpoint(nodes[1]..controls (3, -0.75) and (5, -0.75)..nodes[5]), 2S);
[/asy]](//latex.artofproblemsolving.com/9/3/7/937e770e18903819330e195c68a57256dec0b6f0.png)
, 
, 
, 
, 
, and 
are shown in the figure below (not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?![[asy]
/* AMC8 P14 2024, by NUMANA: BUI VAN HIEU */
import graph;
unitsize(2cm);
real r=0.25;
// Define the nodes and their positions
pair[] nodes = { (0,0), (2,0), (1,1), (3,1), (4,0), (6,0) };
string[] labels = { "A", "M", "X", "Y", "C", "Z" };
// Draw the nodes as circles with labels
for(int i = 0; i < nodes.length; ++i) {
draw(circle(nodes[i], r));
label("$" + labels[i] + "$", nodes[i]);
}
// Define the edges with their node indices and labels
int[][] edges = { {0, 1}, {0, 2}, {2, 1}, {2, 3}, {1, 3}, {1, 4}, {3, 4}, {4, 5}, {3, 5} };
string[] edgeLabels = { "8", "5", "2", "10", "6", "14", "5", "10", "17" };
pair[] edgeLabelsPos = { S, SE, SW, S, SE, S, SW, S, NE};
// Draw the edges with labels
for (int i = 0; i < edges.length; ++i) {
pair start = nodes[edges[i][0]];
pair end = nodes[edges[i][1]];
draw(start + r*dir(end-start) -- end-r*dir(end-start), Arrow);
label("$" + edgeLabels[i] + "$", midpoint(start -- end), edgeLabelsPos[i]);
}
// Draw the curved edge with label
draw(nodes[1]+r * dir(-45)..controls (3, -0.75) and (5, -0.75)..nodes[5]+r * dir(-135), Arrow);
label("$25$", midpoint(nodes[1]..controls (3, -0.75) and (5, -0.75)..nodes[5]), 2S);
[/asy]](http://latex.artofproblemsolving.com/9/3/7/937e770e18903819330e195c68a57256dec0b6f0.png)
I guess the routes are not necessarily linear
answer choice 
