2012 AMC 10A Problems/Problem 25

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

Solution I:

Since $x,y,z$ are all reals lacated in $[0, n]$ , the number of choices for each one is infinite.

Without loss of generality, assume that $n\geqslant x \geqslant y \geqslant z \geqslant 0$ . 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.

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

Now we will find the region with points satisfying $|x-y|\geqslant1$ , $|y-z|\geqslant1$ , $|z-x|\geqslant1$ .

Since $n\geqslant x \geqslant y \geqslant z \geqslant 0$ , we have $x-y\geqslant1$ , $y-z\geqslant1$ , $z-x\geqslant1$ .

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

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

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

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

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

So the answer is D.

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2012 AMC 10A Problems/Problem 25

Problem

Solution