Difference between revisions of "2020 AMC 12B Problems/Problem 25"

Revision as of 08:45, 27 October 2023

Problem

For each real number $a$ with $0 \leq a \leq 1$ , let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$ , respectively, and let $P(a)$ be the probability that

$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$ What is the maximum value of $P(a)?$

$\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$

Solution

Let's start first by manipulating the given inequality.

$\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1$ $\sin^{2}{(\pi x)}>1-\sin^{2}{(\pi y)}=\cos^{2}{(\pi y)}$

Let's consider the boundary cases: $\sin{(\pi x)}=\pm \cos{(\pi y)}$ .

$\sin{(\pi x)}=\pm \cos{(\pi y)}=\sin{(\tfrac 12 {\pi}\pm \pi y)}$

Solving the first case gives us $y=\tfrac{1}{2}-x \quad \textrm{and} \quad y=x-\tfrac{1}{2}.$ Solving the second case gives us $y=x+\tfrac{1}{2}\quad \textrm{and} \quad y=\tfrac{3}{2}-x.$ If we graph these equations in $[0,1]\times[0,1]$ , we get a rhombus shape. $[asy] defaultpen(fontsize(9)+0.8); size(200); pen p=fontsize(10); pair A,B,C,D,A1,A2,B1,B2,C1,C2,D1,D2,I,L; A=MP("(0,0)",origin,down+left,p); B=MP("(1,0)",right,down+right,p); C=MP("(1,1)",right+up,up+right,p); D=MP("(0,1)",up,up+left,p); A1=MP("",extension(A,B,(0.5,0),(0,0.5)),2*down,p); dot(A1); A2=MP("",extension(A,D,(0.5,0),(0,0.5)),2*left,p); dot(A2); B1=MP("",extension(B,C,(0.5,0),(0,-0.5)),2*right,p); dot(B1); B2=MP("",extension(C,D,(0.5,1),(0,0.5)),2*up,p); dot(B2); real a=0.7; draw(A1--B1--B2--A2--cycle, gray+0.6); draw(a*right--a*right+up, royalblue); draw(A1--B2, royalblue+dashed); draw(A--B--C--D--A, black+1.2); dot("$(a,0)$",(a,0),down); dot("$(a,1)$",(a,1),up); [/asy]$ Testing points in each section tells us that the inside of the rhombus satisfies the inequality in the problem statement.

From the region graph, notice that in order to maximize $P(a)$ , $a\geq\tfrac{1}{2}$ . We can solve the rest with geometric probability.

Instead of maximizing $P(a)$ , we minimize $Q(a)=1-P(a)$ . $Q(a)$ consists of two squares (each broken into two triangles), one of area $\tfrac{1}{4}$ and another of area $(a-\tfrac 12)^2$ . To calculate $Q(a)$ , we divide this area by $a$ , so $Q(a) = \frac{1}{a}\left(\frac{1}{4}+(a-\tfrac 12)^2\right) = \frac{1}{a}\left(\frac{1}{2}+a^2-a\right)= a+\frac 1{2a}-1.$

By AM-GM, $a+\frac{1}{2a}\geq 2\sqrt{\frac{a}{2a}}=\sqrt{2}$ , which we can achieve by setting $a=\frac{\sqrt{2}}{2}$ .

Therefore, the maximum value of $P(a)$ is $1-\min(Q(a))$ , which is $1-(\sqrt{2}-1) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}$

Solution 2 (Calculus finish)

We find the same region as in the first solution, and again notice we must have $a \geq \frac{1}{2}$ .

We now express $P$ as a function of $b=(1-a)$ . The triangle on the right of the line $x=b$ is an isosceles right triangle with altitude $b$ , so it has area $b^2$ . The total area of the region to the left of $x=b$ has area $1-b$ . So $P(b) = \frac{1/2 - b^2}{1-b}$ Differentiating using the quotient rule, we find $P$ has local extrema at $P'(b) = \frac{(1-h)(-2h)-(-1/2 - h^2)(-1)}{(1-h)^2} = 0$ Setting the numerator equal to $0$ and solving the quadratic, we find $P$ has extrema at $b = 1 \pm \frac{\sqrt{2}}{2}$ . Only $b=1-\frac{\sqrt 2}{2}$ is within the desired region. Plugging in, we get $P(1-\frac{\sqrt 2}{2}) = \boxed{\textbf{(B)}\ 2 - \sqrt{2}}$ as our solution. We also need to check $P(b=0) = 1/2$ . But $1/2 < 2 - \sqrt{2}$ , and if this isn't immediately obvious, $1/2$ isn't an answer choice anyways.

~jd9 (AMGM scary)

Video Solution by On The Spot STEM

https://www.youtube.com/watch?v=5goLUdObBrY

@@ Line 54: / Line 54: @@
 Differentiating using the quotient rule, we find <math>P</math> has local extrema at
 <cmath>P'(b) = \frac{(1-h)(-2h)-(-1/2 - h^2)(-1)}{(1-h)^2} = 0</cmath>
-Setting the numerator equal to <math>0</math> and solving the quadratic, we find <math>P</math> has extrema at <math>b = 1 \pm \frac{\sqrt{2}}{2}</math>. Only <math>b=1-\frac{\sqrt 2}{2}</math> is within the desired region. Plugging in <math>P(1-\frac{\sqrt 2}{2}) = \boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math> as our solution.
+Setting the numerator equal to <math>0</math> and solving the quadratic, we find <math>P</math> has extrema at <math>b = 1 \pm \frac{\sqrt{2}}{2}</math>. Only <math>b=1-\frac{\sqrt 2}{2}</math> is within the desired region. Plugging in, we get <math>P(1-\frac{\sqrt 2}{2}) = \boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math> as our solution. We also need to check <math>P(b=0) = 1/2</math>. But <math>1/2 < 2 - \sqrt{2}</math>, and if this isn't immediately obvious, <math>1/2</math> isn't an answer choice anyways.
-We also need to check <math>P(b=0) = 1/2</math>. But <math>1/2 < 2 - \sqrt{2}</math>, and if this isn't immediately obvious, <math>1/2</math> isn't an answer choice anyways.
 ~jd9 (AMGM scary)

2020 AMC 12B (Problems • Answer Key • Resources)
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