I was going through some Olympiad pass papers and came across this question:
Given four different positive integers $A, B, C, D$ so that $\frac{1}{A}+\frac{1}{B}+ \frac{1}{C}+\frac{1}{D}=1.1$. Find the smallest possible value of $A+B+C+D$.
Does the value $1.1$ have to do with anything? Also, what trick can I use to solve this question? Is there an inverse equation formula of some type I can use?
I tried doing this:
$$\frac{1}{A}+\frac{1}{B}+ \frac{1}{C}+\frac{1}{D}=1.1$$
$$ABC+BCD+ACD+ABD=1.1ABCD$$
But now I don’t know where to continue. Also, this is an Olympiad math question, which means I probably need an answer that can solve the question in 2 minutes or less.
Best Answer
Without loss of generality, suppose $A < B < C < D$.
Notice that if $A \ge 3$, then $B \ge 4$, $C \ge 5$, $D \ge 6$, and then $\tfrac{1}{A}+\tfrac{1}{B}+\tfrac{1}{C}+\tfrac{1}{D} \le \tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{5}+\tfrac{1}{6} = \tfrac{19}{20} < 1.1$. So we need $A = 1$ or $A = 2$.
Case 1: $A = 2$. Then we need $\tfrac{1}{B}+\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{3}{5}$ with $2 < B < C < D$.
Since $\tfrac{3}{B} > \tfrac{1}{B}+\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{3}{5}$, we must have $B < 5$, i.e. $B = 3$ or $B = 4$.
If $B = 4$, we need $\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{7}{20}$ with $4 < C < D$. Since $\tfrac{2}{C} > \tfrac{1}{C}+\tfrac{1}{D} = \tfrac{7}{20}$, we must have $C < \tfrac{40}{7}$, i.e. $C \le 5$. Since $4 < C \le 5$, we must have $C = 5$, but then $D = \tfrac{20}{3}$, which is not an integer. So there are no solutions with $A = 2$ and $B = 4$.
If $B = 3$, we need $\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{4}{15}$ with $3 < C < D$. Since $\tfrac{2}{C} > \tfrac{1}{C}+\tfrac{1}{D} = \tfrac{4}{15}$, we must have $C < \tfrac{15}{2}$, i.e. $C \le 7$. Testing $C = 4, 5, 6, 7$ yields $D = 60, 15, 10, \tfrac{105}{13}$ respectively. In this case, the smallest sum where $C$ and $D$ are integers is $21$ which occurs for $(A,B,C,D) = (2,3,6,10)$.
Case 2: $A = 1$. Then, we need $\tfrac{1}{B}+\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{1}{10}$. But since $\tfrac{3}{D} < \tfrac{1}{B}+\tfrac{1}{C}+\tfrac{1}{D} = \tfrac{1}{10}$, any solution in this case will have $D > 30$, and thus, $A+B+C+D > 30 > 21$. So we will not find a smaller sum in this case.
Therefore, the minimum sum is $21$.
Note that if you just need to get an answer quickly without a rigorous proof, then you can probably just guess and check until you find something reasonably small. In problems with Egyptian fractions (fractions with numerator $1$), the sum $\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{6} = 1$ comes up a lot, namely it is the smallest set of distinct Egyptian fractions that add up to $1$. So it's not too hard to build off of that to get $\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{6}+\tfrac{1}{10} = \tfrac{11}{10}$. I'm not sure if there is an easy way to convince yourself that's the smallest sum though.