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

Latest revision as of 03:29, 24 July 2024

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 1

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 (Trigonometry Identities)

$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1, \quad \sin^2{(\pi x)} - (1 - \sin^2{(\pi y)} ) > 0, \quad \sin^2{(\pi x)} - \cos^2{(\pi y)} > 0$

$( \sin{(\pi x)} + \cos{(\pi y)} )(\sin{(\pi x)} - \cos{(\pi y)})> 0, \quad \left( \sin{(\pi x)} + \sin{( \frac{\pi}{2} - \pi y)} \right) \left( \sin{(\pi x)} - \sin{(\frac{\pi}{2} - \pi y)} \right)> 0$

$2\sin{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} \cos{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} 2 \sin{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} \cos{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} > 0$

$2\sin{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} \cos{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} 2 \sin{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} \cos{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} > 0$

$\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} \sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} > 0$

If $\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} > 0$ and $\sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} > 0$

$0 < \pi x + \frac{\pi}{2} - \pi y < \pi \quad \text{and} \quad 0 < \pi x - \frac{\pi}{2} + \pi y < \pi$

$0 < x + \frac{1}{2} - y < 1 \quad \text{and} \quad 0 < x - \frac{1}{2} + y < 1$

$x - \frac{1}{2} < y < x + \frac{1}{2} \quad \text{and} \quad -x + \frac{1}{2} < y < -x + \frac{3}{2}$

If $\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} < 0$ and $\sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} < 0$

$\pi < \pi x + \frac{\pi}{2} - \pi y < 2\pi \quad \text{and} \quad \pi < \pi x - \frac{\pi}{2} + \pi y < 2\pi$

$1 < x + \frac{1}{2} - y < 2 \quad \text{and} \quad 1 < x - \frac{1}{2} + y < 2$

$x - \frac{3}{2} < y < x - \frac{1}{2} \quad \text{and} \quad -x + \frac{3}{2} < y < -x + \frac{5}{2}$

Notice that $\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} < 0$ and $\sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} < 0$ cannot be true, because that means $x$ is in the interval $[1, 2]$ . Thus, $y$ is in the the boundaries $x - \frac{1}{2} < y < x + \frac{1}{2}$ , and $-x + \frac{1}{2} < y < -x + \frac{3}{2}$ .

Finishing like Solution 1, 3, or 4 gives $P(a) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}$

~isabelchen

Solution 3 (Calculus finish)

We find the same region as in the first solution or the second 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)

Solution 4 (Calculus Finish)

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

The numerator of $P(a)$ is the area of the triangle with vertices $(0, \frac12), (\frac12, 0), (\frac12, 1)$ plus the area of the rectangle with vertices $(\frac12,0), (a,0), (a,1), (\frac12,1)$ subtract the area of the isosceles right triangle between $x=a, y=0,x - \frac{1}{2}$ and the isosceles right triangle between $x=a, y=1, -x + \frac{3}{2}$ . The denominator of $P(a)$ is the area of rectangle with vertices $(0,0), (a,0), (a,1), (0,1)$ .

$P(a) = \frac{ \frac14 + a - \frac12 -(a - \frac12)^2 }{a} = \frac{a - \frac14 -a^2 + a - \frac14 }{a} = \frac{-a^2 + 2a - \frac12 }{a} = -a + 2 - \frac{1}{2a}$

By the power rule, $P'(a) = -1 - \frac12(-1)\frac{1}{a^2} = -1 + \frac{1}{2a^2}$ . Solving $P'(a) = 0$ , we find that $a = \frac{ \sqrt{2} }{2}$

$\because P'(a) = 0$ when $a = \frac{ \sqrt{2} }{2}$ , $P'(a)$ is positive when $a < \frac{ \sqrt{2} }{2}$ , and $P'(a)$ is negative when $a > \frac{ \sqrt{2} }{2}$

$\therefore P(a)$ has a local maximum when $a = \frac{ \sqrt{2} }{2}$ .

$P(\frac{ \sqrt{2} }{2}) = - \frac{\sqrt{2} }{2} + 2 - \frac{1}{2 \cdot \frac{\sqrt{2} }{2}} =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}$

~isabelchen

Solution 5 (Calculus)

Comparing the diagram of $\sin^{2}{(\pi x)}$ and $\cos^{2}{(\pi y)}$ , we can see if we let $x=k$ , then $y\in (0+\lvert \dfrac{1}{2}-k\rvert,1-\lvert\dfrac{1}{2}-k\rvert)$ will satisfy the above inequality.

Thus, we can write that $P(a) = \frac{1}{a} (\int_0^{a}1-2\lvert\frac{1}{2}-x\rvert{\rm d}x)$ $P(a) = \frac{1}{a} (\int_0^{\frac{1}{2}}2x{\rm d}x + \int_{\frac{1}{2}}^{a}2-2x{\rm d}x) = -a+2-\frac{1}{2a}$

Then it is easy to find that $P(a)$ is maximized when $a=\dfrac{\sqrt{2}}{2}$ , which give us $P(\dfrac{\sqrt{2}}{2})=\boxed{\textbf{(B)}\ 2 - \sqrt{2}}$

~ERiccc

Video Solution by On The Spot STEM

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

@@ Line 8: / Line 8: @@
 <math>\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}</math>
-==Solution==
+==Solution 1 ==
 Let's start first by manipulating the given inequality.
@@ Line 15: / Line 15: @@
 <cmath>\sin^{2}{(\pi x)}>1-\sin^{2}{(\pi y)}=\cos^{2}{(\pi y)}</cmath>
-Let's consider the boundary cases: <math>\sin{(\pi x)}=\cos{(\pi y)}</math> and <math>\sin{(\pi x)}=-\cos{(\pi y)}</math>
+Let's consider the boundary cases: <math>\sin{(\pi x)}=\pm \cos{(\pi y)}</math>.
-<cmath>\sin{(\pi x)}=\cos{(\pi y)}=\sin{(\frac{\pi}{2}\pm \pi y)}</cmath>
+<cmath>\sin{(\pi x)}=\pm \cos{(\pi y)}=\sin{(\tfrac 12 {\pi}\pm \pi y)}</cmath>
 Solving the first case gives us <cmath>y=\tfrac{1}{2}-x \quad \textrm{and} \quad  y=x-\tfrac{1}{2}.</cmath> Solving the second case gives us <cmath>y=x+\tfrac{1}{2}\quad \textrm{and} \quad y=\tfrac{3}{2}-x.</cmath> If we graph these equations in <math>[0,1]\times[0,1]</math>, we get a rhombus shape.
@@ Line 34: / Line 34: @@
 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.
@@ Line 39: / Line 40: @@
 From the region graph, notice that in order to maximize <math>P(a)</math>, <math>a\geq\tfrac{1}{2}</math>. We can solve the rest with geometric probability.
-When <math>a\geq\tfrac{1}{2}, P(a)</math> consists of a triangle with area <math>\tfrac{1}{4}</math> and a trapezoid with bases <math>1</math> and <math>2-2a</math> and height <math>a-\frac{1}{2}</math>. Finally, to calculate <math>P(a)</math>, we divide this area by <math>a</math>, so <cmath>P(a)=\frac{1}{a}\left(\frac{1}{4}+\frac{(a-\frac{1}{2})(3-2a)}{2}\right)</cmath>
+Instead of maximizing <math>P(a)</math>, we minimize <math>Q(a)=1-P(a)</math>. <math>Q(a)</math> consists of two squares (each broken into two triangles), one of area <math>\tfrac{1}{4}</math> and another of area <math>(a-\tfrac 12)^2</math>. To calculate <math>Q(a)</math>, we divide this area by <math>a</math>, so <cmath>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.</cmath>
-After expanding out, we get <math>P(a)=2-a-\frac{1}{2a}</math>. In order to maximize this expression, we must minimize <math>a+\frac{1}{2a}</math>.
+By AM-GM, <math>a+\frac{1}{2a}\geq 2\sqrt{\frac{a}{2a}}=\sqrt{2}</math>, which we can achieve by setting <math>a=\frac{\sqrt{2}}{2}</math>.
+Therefore, the maximum value of <math>P(a)</math> is <math>1-\min(Q(a))</math>, which is <math>1-(\sqrt{2}-1) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math>
+==Solution 2 (Trigonometry Identities)==
+<cmath>\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1, \quad \sin^2{(\pi x)} - (1 - \sin^2{(\pi y)} ) > 0, \quad \sin^2{(\pi x)} - \cos^2{(\pi y)} > 0</cmath>
+<cmath>( \sin{(\pi x)} + \cos{(\pi y)} )(\sin{(\pi x)} - \cos{(\pi y)})> 0, \quad \left( \sin{(\pi x)} + \sin{( \frac{\pi}{2} - \pi y)} \right) \left( \sin{(\pi x)} - \sin{(\frac{\pi}{2} - \pi y)} \right)> 0</cmath>
+<cmath>2\sin{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} \cos{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} 2 \sin{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} \cos{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} > 0 </cmath>
+<cmath>2\sin{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} \cos{\left( \frac{ \pi x + \frac{\pi}{2} - \pi y }{2} \right)} 2 \sin{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} \cos{\left( \frac{ \pi x - \frac{\pi}{2} + \pi y }{2} \right)} > 0 </cmath>
+<cmath>\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} \sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} > 0 </cmath>
+If <math>\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} > 0</math> and <math> \sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} > 0 </math>
+<cmath>0 < \pi x + \frac{\pi}{2} - \pi y < \pi \quad \text{and} \quad 0 < \pi x - \frac{\pi}{2} + \pi y < \pi</cmath>
+<cmath>0 < x + \frac{1}{2} - y < 1 \quad \text{and} \quad 0 < x - \frac{1}{2} + y < 1</cmath>
+<cmath>x - \frac{1}{2} < y < x + \frac{1}{2} \quad \text{and} \quad -x + \frac{1}{2} < y < -x + \frac{3}{2}</cmath>
+If <math>\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} < 0</math> and <math> \sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} < 0 </math>
+<cmath>\pi < \pi x + \frac{\pi}{2} - \pi y < 2\pi \quad \text{and} \quad \pi < \pi x - \frac{\pi}{2} + \pi y < 2\pi</cmath>
+<cmath>1 < x + \frac{1}{2} - y < 2 \quad \text{and} \quad 1 < x - \frac{1}{2} + y < 2</cmath>
+<cmath> x - \frac{3}{2} < y < x - \frac{1}{2} \quad \text{and} \quad -x + \frac{3}{2} < y < -x + \frac{5}{2}</cmath>
+Notice that <math>\sin{\left(\pi x + \frac{\pi}{2} - \pi y\right)} < 0</math> and <math> \sin{\left( \pi x - \frac{\pi}{2} + \pi y \right)} < 0 </math> cannot be true, because that means <math>x</math> is in the interval <math>[1, 2]</math>. Thus, <math>y</math> is in the the boundaries <math>x - \frac{1}{2} < y < x + \frac{1}{2}</math>, and <math>-x + \frac{1}{2} < y < -x + \frac{3}{2}</math>.
+Finishing like Solution 1, 3, or 4 gives <math>P(a) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math>
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+==Solution 3 (Calculus finish)==
+We find the same region as in the first solution or the second solution, and again notice we must have <math>a \geq \frac{1}{2}</math>.
+We now express <math>P</math> as a function of <math>b=(1-a)</math>. The triangle on the right of the line <math>x=b</math> is an isosceles right triangle with altitude <math>b</math>, so it has area <math>b^2</math>. The total area of the region to the left of <math>x=b</math> has area <math>1-b</math>. So
+<cmath>P(b) = \frac{1/2 - b^2}{1-b}</cmath>
+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, 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.
+~jd9 (AMGM scary)
+==Solution 4 (Calculus Finish)==
+We find the same region as in the first solution or the second solution, and again notice <math>a \geq \frac{1}{2}</math>.
+The numerator of <math>P(a)</math> is the area of the triangle with vertices <math>(0, \frac12), (\frac12, 0), (\frac12, 1)</math> plus the area of the rectangle with vertices <math>(\frac12,0), (a,0), (a,1), (\frac12,1)</math> subtract the area of the isosceles right triangle between <math>x=a, y=0,x - \frac{1}{2}</math> and the isosceles right triangle between <math>x=a, y=1, -x + \frac{3}{2}</math>. The denominator of <math>P(a)</math> is the area of rectangle with vertices <math>(0,0), (a,0), (a,1), (0,1)</math>.
+<cmath>P(a) = \frac{ \frac14 + a - \frac12 -(a - \frac12)^2 }{a} = \frac{a - \frac14 -a^2 + a - \frac14 }{a} = \frac{-a^2 + 2a - \frac12 }{a} = -a + 2 - \frac{1}{2a}</cmath>
+By the power rule, <math>P'(a) = -1 - \frac12(-1)\frac{1}{a^2} = -1 + \frac{1}{2a^2}</math>. Solving <math>P'(a) = 0</math>, we find that <math>a = \frac{ \sqrt{2} }{2}</math>
-By AM-GM, <math>a+\frac{1}{2a}\geq 2\sqrt{\frac{a}{2a}}=\sqrt{2}</math>, which we can achieve by setting <math>a=\frac{\sqrt{2}}{2}</math>.
+<math>\because P'(a) = 0</math> when <math>a = \frac{ \sqrt{2} }{2}</math>, <math>P'(a)</math> is positive when <math>a < \frac{ \sqrt{2} }{2}</math>, and <math>P'(a)</math> is negative when <math>a > \frac{ \sqrt{2} }{2}</math>
+<math>\therefore P(a)</math> has a local maximum when <math>a = \frac{ \sqrt{2} }{2}</math>.
+<cmath>P(\frac{ \sqrt{2} }{2}) = - \frac{\sqrt{2} }{2} + 2 - \frac{1}{2 \cdot  \frac{\sqrt{2} }{2}} =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</cmath>
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+==Solution 5 (Calculus)==
+Comparing the diagram of <math>\sin^{2}{(\pi x)}</math> and <math>\cos^{2}{(\pi y)}</math>, we can see if we let <math>x=k</math>, then <math>y\in (0+\lvert \dfrac{1}{2}-k\rvert,1-\lvert\dfrac{1}{2}-k\rvert)</math> will satisfy the above inequality.
+Thus, we can write that
+<cmath>P(a) = \frac{1}{a} (\int_0^{a}1-2\lvert\frac{1}{2}-x\rvert{\rm d}x)</cmath>
+<cmath>P(a) = \frac{1}{a} (\int_0^{\frac{1}{2}}2x{\rm d}x + \int_{\frac{1}{2}}^{a}2-2x{\rm d}x) = -a+2-\frac{1}{2a}</cmath>
+Then it is easy to find that <math>P(a)</math> is maximized when <math>a=\dfrac{\sqrt{2}}{2}</math>, which give us <math>P(\dfrac{\sqrt{2}}{2})=\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math>
-Therefore, the maximum value of <math>P(a)</math> is <math>P\left(\frac{\sqrt{2}}{2}\right)=\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math>
+~ERiccc
-==Video Solution==
+==Video Solution by On The Spot STEM==
-On The Spot STEM:
 https://www.youtube.com/watch?v=5goLUdObBrY

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