Question 79
The 2020 numbers
$$
1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \ldots, 2020 \cdot 2021
$$
are written on a board. A move consists of choosing three numbers $a,b,c$ on the board, erasing them, and writing the number
$$
\frac{a b c}{a b + b c + a c}
$$
a) Verify that
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \left(\frac{a b c}{a b + b c + a c}\right)^{-1}
$$
b) Using
$$
\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
$$
find the value of the sum
$$
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{2020 \cdot 2021}
$$
c) After 1009 moves, two numbers remain. Prove that if one is $4/3$, the other is greater than $4$.
$$
1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \ldots, 2020 \cdot 2021
$$
are written on a board. A move consists of choosing three numbers $a,b,c$ on the board, erasing them, and writing the number
$$
\frac{a b c}{a b + b c + a c}
$$
a) Verify that
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \left(\frac{a b c}{a b + b c + a c}\right)^{-1}
$$
b) Using
$$
\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
$$
find the value of the sum
$$
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{2020 \cdot 2021}
$$
c) After 1009 moves, two numbers remain. Prove that if one is $4/3$, the other is greater than $4$.