Difference between revisions of "2012 AMC 10A Problems/Problem 25"

Latest revision as of 16:24, 11 October 2024

Problem

Real numbers $x$ , $y$ , and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$ . The probability that no two of $x$ , $y$ , and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$ . What is the smallest possible value of $n$ ?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

Solution 1

Since $x,y,z$ are all reals located in $[0, n]$ , the number of choices for each one is continuous so we use geometric probability.

WLOG(Without loss of generality), assume that $n\geq x \geq y \geq z \geq 0$ . Then the set of points $(x,y,z)$ is a tetrahedron, or a triangular pyramid. The point $(x,y,z)$ distributes uniformly in this region. If this is not easy to understand, read Solution II.

The altitude of the tetrahedron is $n$ and the base is an isosceles right triangle with a leg length $n$ . The volume is $V_1=\dfrac{n^3}{6}$ , as shown in the first figure in red.

$[asy] import three; unitsize(10cm); size(150); currentprojection=orthographic(1/2,-1,2/3); // three - currentprojection, orthographic draw((1,1,0)--(0,1,0)--(0,0,0),dashed); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1),dashed); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); draw((0,0,0)--(1,0,0)--(1,1,0)--(1,1,1), red); draw((1,1,0)--(0,0,0)--(1,1,1), dashed+red); draw((1,1,1)--(1,0,0), red); [/asy]$

Now we will find the region with points satisfying $|x-y|\geq1$ , $|y-z|\geq1$ , $|z-x|\geq1$ .

Since $n\geq x \geq y \geq z \geq 0$ , we have $x-y\geq1$ , $y-z\geq1$ .

The region of points $(x,y,z)$ satisfying the condition is shown in the second figure in black. It is a tetrahedron, too.

$[asy] import three; unitsize(10cm); size(150); currentprojection=orthographic(1/2, -1, 2/3); // three - currentprojection, orthographic draw((1, 1, 0)--(0, 1, 0)--(0, 0, 0), dashed+green); draw((0, 0, 0)--(0, 0, 1), green); draw((0, 1, 0)--(0, 1, 1), dashed+green); draw((1, 0, 0)--(1, 0, 1), green); draw((0, 0, 1)--(1, 0, 1)--(1, 1, 1)--(0, 1, 1)--cycle, green); draw((1,0,0)--(1,1,0)--(0,0,0)--(1,1,1), dashed+red); draw((0,0,0)--(1,0,0)--(1,1,1), red); draw((1,1,1)--(1,1,0)--(1,0.9,0), red); draw((1, 0.1, 0)--(1, 0.9, 0)--(1, 0.9, 0.8)--cycle); draw((0.2, 0.1, 0)--(1, 0.9, 0.8),dashed); draw((1, 0.1, 0)--(0.2, 0.1, 0)--(1, 0.9, 0),dashed); [/asy]$

The volume of this region is $V_2=\dfrac{(n-2)^3}{6}$ .

So the probability is $p=\dfrac{V_2}{V_1}=\dfrac{(n-2)^3}{n^3}$ .

Substituting $n$ with the values in the choices, we find that when $n=10$ , $p=\frac{512}{1000}>\frac{1}{2}$ , when $n=9$ , $p=\frac{343}{729}<\frac{1}{2}$ . So $n\geq 10$ .

So the answer is $\boxed{\textbf{(D)}\ 10}$ .

Solution 2

Because $x$ , $y$ , and $z$ are chosen independently and at random from the interval $[0,n]$ , which means that $x$ , $y$ , and $z$ distributes uniformly and independently in the interval $[0,n]$ . So the point $(x, y, z)$ distributes uniformly in the cubic $0\leqslant x, y, z \leqslant n$ , as shown in the figure below. The volume of this cubic is $V_0=n^3$ .

As we want to find the probablity of the incident $A=\big\{ |x-y| \geq 1, |y-z| \geq1, |z-x| \geq 1 \big\}$ , we should find the volume of the region of points such that $|x-y|\geq 1$ , $|y-z|\geq 1$ , $|z-x|\geq 1$ and $0\leq x, y, z \leq n$ .

Now we will find the region $\big\{ (x,y,z)\ | \ 0\leq x, y, z \leq n, |x-y|\geq 1, |y-z|\geq 1, |z-x|\geq 1 \big\}$ .

The region can be generated by cutting off 3 slices corresponding to $|x-y|< 1$ , $|y-z|< 1$ , and $|z-x|< 1$ , respectively, from the cubic.

After cutting off a slice corresponding to $|x-y|< 1$ , we get two triangular prisms, as shown in the figure.

In order to observe the object clearly, we rotate the object by the $z$ axis, as shown.

We can draw the slice corresponding to $|y-z|< 1$ on the object.

After cutting off the slice corresponding to $|y-z|< 1$ , we have 4 pieces left.

After cutting off the slice corresponding to $|z-x|< 1$ , we have 6 congruent triangular prisms.

Here we draw all the pictures in colors in order to explain the solution clearly. That does not mean that the students should do it in the examination. They can draw a figure with lines only, as shown below.

Every triangular pyramid has an altitude $n-2$ and a base of isoceless right triangle with leg length $n-2$ , so the volume is $(n-2)^3/6$ . Then the volume of the region $\big\{ (x,y,z)\ | \ 0\leqslant x, y, z \leqslant n, |x-y|\geqslant 1, |y-z|\geqslant 1, |z-x|\geqslant 1 \big\}$ is $V_A=6\times(n-2)^3/6$ = $(n-2)^3$ .

So the probability of the incident $A$ is $P(A)=\dfrac{V_A}{V_0}$ = $\dfrac{(n-2)^3}{n^3}$ .

Then we can get the answer the same way as Solution I.

The answer is $\boxed{\textbf{(D)}\ 10}$ .

If there is no choice for selection, we can also find the minimum value of the integer $n$ if we do not substitute $n$ by the possible values one by one.

Let $P(A)>1/2$ , i.e., $\dfrac{(n-2)^3}{n^3}>\dfrac{1}{2}$ , so $\dfrac{n-2}{n}>\dfrac{1}{\sqrt[^3\!]{2}}$ , or $1-\dfrac{2}{n}>\dfrac{1}{\sqrt[^3\!]{2}}$ , hence $n>\dfrac{2\sqrt[^3\!]{2}}{\sqrt[^3\!]{2}-1}$ .

Now we will estimate the value of $\dfrac{2\sqrt[^3\!]{2}}{\sqrt[^3\!]{2}-1}$ without a calculator.

Since $a^3-1$ = $(a-1)(a^2+a+1)$ , so $\dfrac{2\sqrt[^3\!]{2}}{\sqrt[^3\!]{2}-1}$ = $\dfrac{2\sqrt[^3\!]{2}\times\left( \sqrt[^3\!]{2}^2+\sqrt[^3\!]{2}+1\right)}{\left( \sqrt[^3\!]{2}-1\right)\left( \sqrt[^3\!]{2}^2+\sqrt[^3\!]{2}+1\right)}$ = $\dfrac{2\times\left( 2+\sqrt[^3\!]{2}^2+\sqrt[^3\!]{2}\right)}{ \sqrt[^3\!]{2}^3-1}$ = $2\times\left( 2+\sqrt[^3\!]{4}+\sqrt[^3\!]{2}\right)$ .

Now we would get the approximation of $\sqrt[^3\!]{4}$ and $\sqrt[^3\!]{2}$ .

In order to avoid compicated computation, we get the approximation with one decimal digit only.

Estimation of $\sqrt[^3\!]{2}$ .

Since $1.5^3=2.25\times1.5>2$ , so $1<\sqrt[^3\!]{2}<1.5$ .

The mean of 1 and 1.5 with one decimal digit is about 1.3 .

As $1.3^3=1.69\times 1.3=2.197>2$ , so $1<\sqrt[^3\!]{2}<1.3$ .

The mean of 1 and 1.3 with one decimal digit is about 1.2.

As $1.2^3=1.44\times 1.2=1.728<2$ , so $1.2<\sqrt[^3\!]{2}<1.3$ .

Estimation of $\sqrt[^3\!]{4}$ .

As $\sqrt[^3\!]{4}=\sqrt[^3\!]{2}^2$ , so $1.2^2<\sqrt[^3\!]{4}<1.3^2$ , then $1.24<\sqrt[^3\!]{4}<1.69$ .

As $1.5^3=2.25\times 1.5=3.375<4$ , so $1.5<\sqrt[^3\!]{4}<1.69$ .

The mean of 1.5 and 1.69 with one decimal digit is about 1.6.

As $1.6^3=(16/10)^3=(2^4/10)^3=2^{12}/10^3=4\times 2^10/10^3=4\times 1.024>4$ , so $1.5<\sqrt[^3\!]{4}<1.6$ .

Then $2\times(2+1.5+1.2)<2\times\left(2+\sqrt[^3\!]{4}+\sqrt[^3\!]{2}\right)<2\times(2+1.6+1.3)$ , i.e., $9.4<2\times\left(2+\sqrt[^3\!]{4}+\sqrt[^3\!]{2}\right)<9.8$ ,

As $n>2\times\left(2+\sqrt[^3\!]{4}+\sqrt[^3\!]{2}\right)$ , So the minimal value of integer $n$ is 10.

Appendix

This solution is motivated by the suggestive formula $\frac{(n-2)^{3}}{n^{3}}$ .

The problem generalizes easily to $k$ -dimensional real space $\mathbb{R}^{k}$ . In the general $k$ -dimensional case, we are asked to find the probability that a randomly chosen $k$ -tuple $(x_{1},\dotsc,x_{k}) \in [0,n]^{k}$ satisfies $|x_{i} - x_{j}| > 1$ for all $i \ne j$ . To avoid repetition, let us say that $(x_{1},\dotsc,x_{k})$ is spaced-out if $|x_{i} - x_{j}| > 1$ for all $i \ne j$ .

Let $C$ be the $k$ -dimensional hyper-cube of side length $n$ : $C = [0,n]^{k} = \big\{(x_{1},\dotsc,x_{k}) \in \mathbb{R}^{k} \;:\; 0 \leqslant x_{i} \leqslant n \text{ for all }i \big\} \;.$ Then $C$ has volume $n^{k}$ . Let $S$ be the set of spaced-out $k$ -tuples $(x_{1},\dotsc,x_{k})$ . The desired probability is Vol $(S)/n^{k}$ .

The set of $k$ -tuples $(x_{1},\dotsc,x_{k})$ such that there exist distinct indices $i, j$ such that $x_{i} = x_{j}$ has volume $0$ , so we may restrict our attention to $k$ -tuples such that $x_{i} \ne x_{j}$ for all $i \ne j$ .

Further, the condition that $(x_{1},\dotsc,x_{k})$ is spaced-out is "invariant upon permuting the indices"; in other words, if $\sigma$ is a permutation of the set of indices $\{1,\dotsc,k\}$ , then $(x_{1},\dotsc,x_{k})$ is spaced-out if and only if $(x_{\sigma(1)},\dotsc,x_{\sigma(k)})$ is spaced-out. Therefore, we may consider the set of spaced-out $k$ -tuples $(x_{1},\dotsc,x_{k})$ which additionally satisfy $x_{1} < \dotsb < x_{k}$ . Let us denote this set by $T$ . This condition is equivalent to $0 \leqslant x_{1} < x_{2} - 1 < \dotsb < x_{i}-(i-1) < \dotsb < x_{k}-(k-1) \leqslant n-(k-1) \;.$ Let us choose new variables $y_{i} = x_{i} - (i-1)$ for $i = 1,\dotsc,k$ . This change of variables is just a translation of each $(x_{1},\dotsc,x_{k})$ by the vector $(0,1,\dots,k-1)$ ; in the above solutions, it corresponds to taking the 6 tetrahedrons and gluing them together to form a cube.

We now compute the volume of the set of $(y_{1},\dotsc,y_{k}) \in [0,n-(k-1)]^{k}$ which satisfy $y_{1} < \dotsb < y_{k}$ . As above, we can disregard any $(y_{1},\dotsc,y_{k})$ such that $y_{i} = y_{j}$ for some $i \ne j$ . Given any $(y_{1},\dotsc,y_{k})$ such that $y_{i} \ne y_{j}$ for all $i \ne j$ , there exists exactly one permutation $\sigma$ of the indices $\{1,\dotsc,k\}$ such that $y_{\sigma(1)} < \dotsb < y_{\sigma(k)}$ . Since there are $k!$ permutations of $\{1,\dotsc,k\}$ , the desired volume is equal to $\frac{1}{k!}$ times the volume of the $k$ -dimensional hyper-cube of side length $n-(k-1)$ , which is $\frac{1}{k!}(n-(k-1))^{k}$ . Hence $T$ has volume $\frac{1}{k!}(n-(k-1))^{k}$ as well and $S$ has volume $(n-(k-1))^{k}$ . Hence the desired probability is $\frac{(n-(k-1))^{k}}{n^{k}}$ .

Solution 3

If $x$ , $y$ , and $z$ are separated by at least one, then by subtracting the minimum space between the three variables (which is $2$ ), $x$ , $y$ , and $z$ can be chosen randomly in the interval $[0,n-2]$ . The probability is hence $\dfrac{(n-2)^3}{n^3} > \dfrac{1}{2}.$ $\boxed{\textbf{(D)} 10}$ is the minimum value in the answer choices that satisfies this inequality.

Solution 4

Imagine Points $x$ , $y$ , $z$ as the "starting points" of three "blocks" of real numbers that have length $1$ . We are just trying to find the probability that those three "blocks" do not overlap. To do this we can set each unit of $1$ into $\mu$ equal little increments, and take the limit of the probability as $\mu$ approaches $\infty$ . (This is because there are indefinitely many real numbers in a given interval.) We see that the total number of arrangements for the three blocks of $1$ , without considering the rule that no two blocks shall overlap is $(n\mu + 1)^3$ . We see that the number of ways to arrange the three blocks such that no three of them are overlapping is simply $\dbinom{(n-2)\mu + 3}{3}$ . Taking the limit as $\mu$ approaches infinity, we obtain our closed form to be $\lim_{\mu \to \infty} \frac{\dbinom{(n-2)\mu + 3}{3}}{ (n\mu + 1)^3 }.$ Dividing leading coefficients, we obtain $\lim_{\mu \to \infty} \frac{\dbinom{(n-2)\mu + 3}{3}}{ (n\mu + 1)^3 } = \frac{\mu(n-2)^3}{\mu n^3} = \frac{(n-2)^3}{n^3}$ . Solving the inequality $\frac{(n-2)^3}{n^3} > \frac{1}{2}$ , we get the least value of $n$ as $n=\boxed{10}$ .

-fidgetboss_4000

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=HOtyh9qycrI

@@ Line 7: / Line 7: @@
 ==Solution 1==
-Since <math>x,y,z</math> are all reals located in <math>[0, n]</math>, the number of choices for each one is continuous.
+Since <math>x,y,z</math> are all reals located in <math>[0, n]</math>, the number of choices for each one is continuous so we use geometric probability.
 WLOG([[Without loss of generality]]), assume that <math>n\geq x \geq y \geq z \geq 0</math>. Then the set of points <math>(x,y,z)</math> is a tetrahedron, or a triangular pyramid. The point <math>(x,y,z)</math> distributes uniformly in this region. If this is not easy to understand, read Solution II.
@@ Line 19: / Line 19: @@
 currentprojection=orthographic(1/2,-1,2/3);
 // three - currentprojection, orthographic
-draw((1,1,0)--(0,1,0)--(0,0,0),dashed+green);
+draw((1,1,0)--(0,1,0)--(0,0,0),dashed);
-draw((0,0,0)--(0,0,1),green);
+draw((0,0,0)--(0,0,1));
-draw((0,1,0)--(0,1,1),dashed+green);
+draw((0,1,0)--(0,1,1),dashed);
-draw((1,1,0)--(1,1,1),green);
+draw((1,1,0)--(1,1,1));
-draw((1,0,0)--(1,0,1),green);
+draw((1,0,0)--(1,0,1));
-draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,green);
+draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle);
 draw((0,0,0)--(1,0,0)--(1,1,0)--(1,1,1), red);
@@ Line 90: / Line 90: @@
 In order to observe the object clearly, we rotate the object by the <math>z</math> axis, as shown.
-[[File:3.png]]
 We can draw the slice corresponding to <math>|y-z|< 1</math> on the object.
@@ Line 185: / Line 183: @@
 ==Solution 3==
-If <math>x</math>, <math>y</math>, and <math>z</math> are separated by at least one, then by subtracting the minimum space between the three variables (which is <math>2</math>), <math>x</math>, <math>y</math>, and <math>z</math> can be chosen randomly in the interval <math>[0,n-2]</math>. The probability is hence <math>\dfrac{(n-2)^3}{n^3} > \dfrac{1}{2}</math>, which simplifies to <math>n^3-12n^2+24n-16>0</math>. <math>\boxed{\textbf{(D)} 10}</math> is the minimum which satisfies this inequality.
+If <math>x</math>, <math>y</math>, and <math>z</math> are separated by at least one, then by subtracting the minimum space between the three variables (which is <math>2</math>), <math>x</math>, <math>y</math>, and <math>z</math> can be chosen randomly in the interval <math>[0,n-2]</math>. The probability is hence <math>\dfrac{(n-2)^3}{n^3} > \dfrac{1}{2}.</math> <math>\boxed{\textbf{(D)} 10}</math> is the minimum value in the answer choices that satisfies this inequality.
 ==Solution 4==
@@ Line 195: / Line 193: @@
 ==Video Solution by Richard Rusczyk==
-https://www.youtube.com/watch?v=f1nxu8MWWKc
+https://www.youtube.com/watch?v=HOtyh9qycrI
-~dolphin7
 == See Also ==
-{{AMC10 box|year=2012|ab=A|num-b=24|after=Last Problem.}}
+{{AMC10 box|year=2012|ab=A|num-b=24|after=Last Problem}}
 {{MAA Notice}}

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Problem
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