The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be thought of as a generalization of the Nesbitt Inequality, which states that for all positive $x,y,z$, $$\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq \frac{3}{2}.$$

However, the Shapiro inequality is false for large $n$. In particular, it is false for $n=14$.

Thus there are two questions:

1. Is the Shapiro inequality true if our $x_i$ are all the positive divisors of some $N$ with $n$ distinct positive divisors? (Not necessarily in some nice order.)

2. Is it true if in question 1 that all the $x_i$ occur in increasing order, or in decreasing order? (Edit: Question 2 is true by Fedor's comment below.)

Note that if $N$ has no small prime factors compared to its number of divisors then the answer to 1 is true for that specific $N$. Proof sketch: If the smallest prime factor of $N$ is $p$, then the second largest prime divisors of $N$ are at most $\frac{N}{p}$ and $\frac{N}{p+2}$. Thus if, $x_j=N$ for some $j$, then $$\frac{x_j}{x_{j+1} + x_{j+2}} \geq \frac{p(p+2)}{2(p+1)} > \frac{p}{2}.$$ So if $n < \frac{p}{2}$ then we get the Shapiro inequality for free just from that term. It seems likely that if one made a similar argument one could improve this bound for how large one needs to assume $p$ is compared to $n$.

The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be thought of as a generalization of the Nesbitt inequality, which states that for all positive $x,y,z$, $$\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq \frac{3}{2}.$$

However, the Shapiro inequality is false for large $n$. In particular, it is false for $n=14$.

Thus there are two questions:

1. Is the Shapiro inequality true if our $x_i$ are all the positive divisors of some $N$ with $n$ distinct positive divisors? (Not necessarily in some nice order.)

2. Is it true if in question 1 that all the $x_i$ occur in increasing order, or in decreasing order? (Edit: Question 2 is true by Fedor's comment below.)

Note that if $N$ has no small prime factors compared to its number of divisors then the answer to 1 is true for that specific $N$. Proof sketch: If the smallest prime factor of $N$ is $p$, then the second largest prime divisors of $N$ are at most $\frac{N}{p}$ and $\frac{N}{p+2}$. Thus if, $x_j=N$ for some $j$, then $$\frac{x_j}{x_{j+1} + x_{j+2}} \geq \frac{p(p+2)}{2(p+1)} > \frac{p}{2}.$$ So if $n < \frac{p}{2}$ then we get the Shapiro inequality for free just from that term. It seems likely that if one made a similar argument one could improve this bound for how large one needs to assume $p$ is compared to $n$.

The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be thought of as a generalization of the Nesbitt Inequality, which states that for all positive $x,y,z$, $$\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq \frac{3}{2}.$$

However, the Shapiro inequality is false for large $n$. In particular, it is false for $n=14$.

There are two closely related questions:

1. Is the Shapiro inequality true if our $x_i$ are all the positive divisors of some $N$ with $n$ distinct positive divisors? (Not necessarily in some nice order.)

2. Is it true if in question 1 that all the $x_i$ occur in increasing order, or in decreasing order?

Note that if $N$ has no small prime factors compared to its number of divisors then the answer to 1 is true for that specific $N$. Proof sketch: If the smallest prime factor of $N$ is $p$, then the second largest prime divisors of $N$ are at most $\frac{N}{p}$ and $\frac{N}{p+2}$. Thus if, $x_j=N$ for some $j$, then $$\frac{x_j}{x_{j+1} + x_{j+2}} \geq \frac{p(p+2)}{2(p+1)} > \frac{p}{2}.$$ So if $n < \frac{p}{2}$ then we get the Shapiro inequality for free just from that term. It seems likely that if one made a similar argument one could improve this bound for how large one needs to assume $p$ is compared to $n$.

The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be thought of as a generalization of the Nesbitt Inequality, which states that for all positive $x,y,z$, $$\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq \frac{3}{2}.$$

However, the Shapiro inequality is false for large $n$. In particular, it is false for $n=14$.

Thus there are two questions:

1. Is the Shapiro inequality true if our $x_i$ are all the positive divisors of some $N$ with $n$ distinct positive divisors? (Not necessarily in some nice order.)

2. Is it true if in question 1 that all the $x_i$ occur in increasing order, or in decreasing order? (Edit: Question 2 is true by Fedor's comment below.)

Note that if $N$ has no small prime factors compared to its number of divisors then the answer to 1 is true for that specific $N$. Proof sketch: If the smallest prime factor of $N$ is $p$, then the second largest prime divisors of $N$ are at most $\frac{N}{p}$ and $\frac{N}{p+2}$. Thus if, $x_j=N$ for some $j$, then $$\frac{x_j}{x_{j+1} + x_{j+2}} \geq \frac{p(p+2)}{2(p+1)} > \frac{p}{2}.$$ So if $n < \frac{p}{2}$ then we get the Shapiro inequality for free just from that term. It seems likely that if one made a similar argument one could improve this bound for how large one needs to assume $p$ is compared to $n$.