Difference between revisions of "2020 CIME II Problems/Problem 8"

Latest revision as of 16:36, 28 December 2023

Problem 8

A committee has an oligarchy, consisting of $A\%$ of the members of the committee. Suppose that $B\%$ of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is $16$ times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of $B-A$ .

Solution

Average work done sets up an equation: $\frac{B}{A} = 16\frac{100-B}{100-A}$ $(100-A)B = 16(100-B)A$ $100B - AB = 1600 A - 16AB$ Let $B-A = C$ and $A+B = D$ : $50C + 50D - \frac{D^2-C^2}{4} = 800D - 800C - 4(D^2-C^2)$ $15D^2 -3000D = 15C^2-3400C$ Complete the squares: $15(D-100)^2 - 1500^2 = 15(C-1700/15)^2 - 1700^2/15$ $(D-100)^2 + \frac{(17^2-15^2) \times 100^2}{15^2} = (C-1700/15)^2$ $C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}$

Note that $\frac{1700+800}{15}>100$ so must use minus. This means that C is maximized if $D=100$ $C = \frac{1700}{15} - \frac{8 \times 100}{15} = 900/15=60$ $B-A$ is at a maximum $60$

Solution 2

As in the first solution, we get $100B - AB = 1600A - 16AB$ . We rearrange and obtain $15AB + 100B - 1600A = 0$ . We divide by $15$ to obtain $AB + \frac{100}{15}B - \frac{1600}{15}A = 0$ . We then subtract $\frac{160000}{225}$ from both sides, and factor to obtain $(A + \frac{100}{15})(B - \frac{1600}{15}) = -\frac{1600000}{225} = -\left(\frac{400}{15}\right)^2$ . If we graph this with $A$ being on the $x$ -axis and $B$ being on the $y$ -axis, this equation is the hyperbola $xy = 1$ , except scaled up by $\frac{400}{15}$ and translated $\frac{100}{15}$ to the left and $\frac{1600}{15}$ up. This graph intersects $(0, 0)$ and $(100, 100)$ , and the maximum difference clearly occurs at the point when the slope of the function is $1$ . This is at $(-\frac{100}{15} + \frac{400}{15}, \frac{1600}{15} - \frac{40}{15}) = (20, 80)$ . Our answer is $\boxed{60}$ .

~mathboy100

Solution 3

Assume WLOG, there are $100$ members and that there are $100$ work that is done. So, there are $A$ members of the oligarchy and they've done $B$ work in total. Thus, on average, each member of the oligarchy does $\dfrac{B}{A}$ work. Then, the average work a nonmember of the oligarchy will be $\dfrac{B}{16A}$ . Thus, $B+(100-A)\cdot \dfrac{B}{16A} = 100$ $\implies 16AB+(100-A)B = 1600A$ $\implies 3AB+20B-320A = 0$ $\implies (3B-20)(3A+20) = -6400$ When $3B-20 = -80$ and $3A+20 = 80$ , we have $A=20, B=80$ , giving $B-A = 80-20 = \boxed{60}$

~sml1809

@@ Line 15: / Line 15: @@
 <cmath> C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}</cmath>
-Note that <math>\frac{1700+800}{15}>1</math> so must use minus. This means that C is maximized if <math>D=100</math>
+Note that <math>\frac{1700+800}{15}>100</math> so must use minus. This means that C is maximized if <math>D=100</math>
 <cmath> C = \frac{1700}{15} - \frac{8 \times 100}{15} = 900/15=60</cmath>
 <math>B-A</math> is at a maximum <math>60</math>
+==Solution 2==
+As in the first solution, we get <math>100B - AB = 1600A - 16AB</math>. We rearrange and obtain <math>15AB + 100B - 1600A = 0</math>. We divide by <math>15</math> to obtain <math>AB + \frac{100}{15}B - \frac{1600}{15}A = 0</math>. We then subtract <math>\frac{160000}{225}</math> from both sides, and factor to obtain <math>(A + \frac{100}{15})(B - \frac{1600}{15}) = -\frac{1600000}{225} = -\left(\frac{400}{15}\right)^2</math>. If we graph this with <math>A</math> being on the <math>x</math>-axis and <math>B</math> being on the <math>y</math>-axis, this equation is the hyperbola <math>xy = 1</math>, except scaled up by <math>\frac{400}{15}</math> and translated <math>\frac{100}{15}</math> to the left and <math>\frac{1600}{15}</math> up. This graph intersects <math>(0, 0)</math> and <math>(100, 100)</math>, and the maximum difference clearly occurs at the point when the slope of the function is <math>1</math>. This is at <math>(-\frac{100}{15} + \frac{400}{15}, \frac{1600}{15} - \frac{40}{15}) = (20, 80)</math>. Our answer is <math>\boxed{60}</math>.
+~mathboy100
+==Solution 3==
+Assume WLOG, there are <math>100</math> members and that there are <math>100</math> work that is done. So, there are <math>A</math> members of the oligarchy and they've done <math>B</math> work in total. Thus, on average, each member of the oligarchy does <math>\dfrac{B}{A}</math> work. Then, the  average work a nonmember of the oligarchy will be <math>\dfrac{B}{16A}</math>. Thus,
+<cmath>B+(100-A)\cdot \dfrac{B}{16A} = 100</cmath>
+<cmath>\implies 16AB+(100-A)B = 1600A</cmath>
+<cmath>\implies 3AB+20B-320A = 0</cmath>
+<cmath>\implies (3B-20)(3A+20) = -6400</cmath>
+When <math>3B-20 = -80</math> and <math>3A+20 = 80</math>, we have <math>A=20, B=80</math>, giving <math>B-A = 80-20 = \boxed{60}</math>
+~sml1809

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