Minimize $LCM(a,b,c,d)$ given $a+b+c+d=1000$ Minimize $$LCM(a,b,c,d)$$ given $\; a, b, c, d\; $ are distinct positive integers, and $$a+b+c+d=1000$$
Any hint?

Tried the AM-GM inequality $a+b+c+d>4\sqrt[4]{abcd}=4\sqrt[4]{GL}$ but couldn't use this.
 A: Suppose without loss of generality that $a<b<c<d$.  Write $M$ for the minimum possible value of $\text{lcm}(a,b,c,d)$.  Since $$120+160+240+480=1000\text{ and }\text{lcm}(120,160,240,480)=480\,,$$ we get $$M\leq 480\,.$$  Suppose that $M$ is attained when $(a,b,c,d)=(A,B,C,D)$.    
Set $k:=\text{lcm}(A,B,C)$ and $t:=\gcd(k,D)$.  Clearly, $t\mid k$.  We have
$$M=\text{lcm}(A,B,C,D)=\text{lcm}\big(\text{lcm}(A,B,C),D\big)=\frac{kD}{t}\,.$$
If $k\geq 4t$, then $M\geq 4D$.  From $$1000=A+B+C+D\leq (D-3)+(D-2)+(D-1)+D=4D-6\,,$$ we have $D\geq\frac{1006}{4}$ or $D\geq 252$; i.e.,$$M\geq 4D\geq 1008\,,$$ which is a contradiction.  Hence, $k\in \{t,2t,3t\}$.  
If $k=3t$, then $A$, $B$, and $C$ are divisors of $3D$ that are less than $D$.  Hence, $3D\geq 4C$, $3D\geq 5B$, and $3D\geq 6A$.  Hence, $A+B+C+D=1000$ implies that
$$1000=A+B+C+D\leq \frac{D}{2}+\frac{3D}{5}+\frac{3D}{4}+D=\frac{57}{20}D\,,$$
whence $D\geq \frac{20000}{57}$, or $D\geq 351$.  However, this means
$$M=3D\geq 1053\,,$$
which is absurd.
If $k=2t$, then $A$, $B$, and $C$ are divisors of $2D$ that are less than $D$.  Hence, $2D\geq 3C$, $2D\geq 4B$, and $2D\geq 5A$.  Ergo,
$$1000=A+B+C+D\leq \frac{2D}{5}+\frac{D}{2}+\frac{2D}{3}+D=\frac{77}{30}D\,,$$
and so $D\geq \frac{30000}{77}$, or $D\geq 390$.  Nonetheless,
$$M=2D\geq 780$$
leads to another contradiction.
Now, we have concluded that $k=t$.  Hence, $A$, $B$, and $C$ are proper divisors of $D$.  This shows that $D\geq 2C$, $D\geq 3B$, and $D\geq 4A$.  Therefore,
$$1000=A+B+C+D\leq \frac{D}{4}+\frac{D}{3}+\frac{D}{2}+D=\frac{25}{12}D\,,$$
or $D\geq 480$.  That means
$$M=D\geq 480\,.$$
Consequently, $M=480$. The only possible $(a,b,c,d)\in\mathbb{Z}_{>0}^4$ satisfying the conditions $a<b<c<d$, $a+b+c+d=1000$, and $\text{lcm}(a,b,c,d)=M$ is $$(a,b,c,d)=(120,160,240,480)\,.$$

P.S.  It can also be shown that the maximum value of $\text{lcm}(a,b,c,d)$ is $3905625009$.  This happens iff $(a,b,c,d)=(247,249,251,253)$.
