[/quote]
,
,
,
be positive real numbers such that 
.![\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.
\]](http://latex.artofproblemsolving.com/b/2/9/b295b0b2a765e41a639657e13b724c60323fa208.png)
.
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
are non-negative integers.
satisfying the following equation:
.
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
,
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
children, no two of the same height, lined up. Let us call a pair of different children 
good if between them there is no child whose height is greater than the height of one of 
are considered the same pair.)
satisfy 
prove in equality.
![\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]](http://latex.artofproblemsolving.com/0/4/7/047adb616a6be9f9ed52f2385fa592fb5464e420.png)
and 
.
be elements of the set 
such that 
,
is a divisor of 
,
is a divisor of 
.
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
is a perfect square for infinitely many positive integers 
?
Y by
In the nation of Peach Blossom Origin, there are 
cities numbered 
to . City 
and city 
has a bidirectional road of lengths 
if and only if 
and 
is a prime number.
As Typhoon Gaemi strikes, all roads are destroyed. The admin Laura wants to repair some roads such that for each pair of city, you can get from one to another just using the repaired roads. However, each unit length of road costs dollar to fix, and Laura wants to minimize the cost. Prove that Laura would have one unique way to do this.
Proposed by CrazyInMath.
cities numbered 
to . City 
and city 
has a bidirectional road of lengths 
if and only if 
and 
is a prime number.As Typhoon Gaemi strikes, all roads are destroyed. The admin Laura wants to repair some roads such that for each pair of city, you can get from one to another just using the repaired roads. However, each unit length of road costs
dollar to fix, and Laura wants to minimize the cost. Prove that Laura would have one unique way to do this.Proposed by CrazyInMath.
