<s>[INST] Olympiad inequality 
Let $a,b,c,d \in [0,1]$.
  Prove 
  $$ \frac {a}{1+b}+\frac{b}{1+c}+\frac{c}{1+d}+\frac{d}{1+a}+abcd\le3$$

Hi. Im a school student so ive been unable to look for ways to apply this problem. I attempted to use AM-GN on 1+a etc but the bound $2\sqrt{a}$ was too large and was easily larger than 3. Thus id like some help solving this problem.
 [/INST] For $a,b,c,d \in [0,1]$, let
$$
f(a,b,c,d)
=
\frac{a}{1+b}
+
\frac{b}{1+c}
+
\frac{c}{1+d}
+
\frac{d}{1+a}
+
abcd
$$
The goal is to prove $f(a,b,c,d) \le 3$, for all $a,b,c,d \in [0,1]$.

For fixed $b,c,d \in [0,1]$, let $g(a) = f(a,b,c,d)$.

Then 
$$g''(a) = \frac{2d}{(a+1)^3}$$
hence $g''(a) \ge 0$, for all $a \in [0,1]$.

It follows that $g$ is convex on $[0,1]$, hence the maximum value of $g$ on $[0,1]$ is achieved at one of the endpoints.

Thus, for fixed $b,c,d$, the function $f(a,b,c,d)$ is maximized at $a=0$ or $a=1$.

The same reasoning applies to each of the variables $b,c,d$, with the other $3$ variables held fixed.

Thus, to prove $f(a,b,c,d) \le 3$, for all $a,b,c,d \in [0,1]$, it suffices to verify that $f(a,b,c,d) \le 3$, for all $a,b,c,d \in \{0,1\}$.

If any of $a,b,c,d$ is zero, the expression 
$$
\frac{a}{1+b}
+
\frac{b}{1+c}
+
\frac{c}{1+d}
+
\frac{d}{1+a}
+
abcd
$$
reduces to a sum of $3$ fractions, each of which is at most $1$, hence the sum is at most $3$.

If $a,b,c,d$ are all equal to $1$, the expression 
$$
\frac{a}{1+b}
+
\frac{b}{1+c}
+
\frac{c}{1+d}
+
\frac{d}{1+a}
+
abcd
$$
evaluates to exactly $3$.

Thus, the claimed inequality is proved.
</s>