with 
and the angle bisector 
.
and perpendicular to 
intersects the line passing through 
and parallel to 
.
intersects the line passing through 
and parallel to 
.
be the intersection of the three perpendicular bisectors of triangle 
passes through the midpoint of segment 
.
and 
Prove that
is the infinite sum 
.
since 
,
,
So is 
Y by
Consider a teapot that holds 
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a 
% tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a 
% tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?This post has been edited 1 time. Last edited by smartvong, Mar 26, 2025, 8:44 AM
![\[
\begin{aligned}
&\text{If you notice, we can have at most } (1000 \cdot n\%) \text{ ml of tea.} \\[10pt]
&\text{So we just need to discard } \left(\frac{n}{m} \cdot 1000\right) \text{ ml from our teapot and refill it with water.} \\[10pt]
&\text{This will turn the contents into:} \\
&\text{Tea: } 10m, \quad \text{Water: } 1000 - 10m. \\[10pt]
&\text{After discarding:} \\
&\text{Tea: } 10n, \quad \text{Water: } \frac{1000n}{m} - 10n. \\[10pt]
&\text{Adding back water until the teapot is filled:} \\
&\text{Tea: } 10n, \quad \text{Water: } 1000 - 10n. \\[10pt]
&\text{There is now } n\% \text{ of tea.} \\[10pt]
&\text{So the amount of tea served will be: } 10n.
\end{aligned}
\]](http://latex.artofproblemsolving.com/a/c/b/acb97e68d9f81b0822885d3ee4f8849ab253315e.png)
