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Difference between revisions of "1991 AIME Problems/Problem 13"

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== Solution ==
 
== Solution ==
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Let <math>r_{}^{}</math>,  and <math>b_{}^{}</math> denote the number of red and blue socks, respectively. Also, let <math>t_{}^{}=r_{}^{}+b_{}^{}</math>. The probability <math>P^{}</math> that when two socks are drawn randomly, without replacement, both are red or both are blue is given by
+
Let <math>r_{}^{}</math>,  and <math>b_{}^{}</math> denote the number of red and blue socks, respectively. Also, let <math>t_{}^{}=r_{}^{}+b_{}^{}</math>. The probability <math>P_{}^{}</math> that when two socks are drawn randomly, without replacement, both are red or both are blue is given by
  
 
<math>
 
<math>
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</math>
 
</math>
  
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Now, since <math>r</math> and <math>t</math> are positive integers, it must be the case that <math>t=n^{2}</math>, with <math>n\in\mathbb{N}</math>. Hence, <math>r=n(n\pm 1)/2</math> would correspond to the general solution. For the present case, <math>t\leq 1991</math> and so one easily finds that <math>n=44</math> is the largest possible integer satisfying the problem conditions.  
+
Now, since <math>r_{}^{}</math> and <math>t_{}^{}</math> are positive integers, it must be the case that <math>t_{}^{}=n^{2}</math>, with <math>n\in\mathbb{N}</math>. Hence, <math>r=n(n\pm 1)/2</math> would correspond to the general solution. For the present case, <math>t\leq 1991</math> and so one easily finds that <math>n=44</math> is the largest possible integer satisfying the problem conditions.  
  
 
In summary, the solution is that the maximum number of red socks is <math>r=990</math>.
 
In summary, the solution is that the maximum number of red socks is <math>r=990</math>.

Revision as of 18:27, 19 April 2007

Problem

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\displaystyle \frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

Solution

Let $r_{}^{}$, and $b_{}^{}$ denote the number of red and blue socks, respectively. Also, let $t_{}^{}=r_{}^{}+b_{}^{}$. The probability $P_{}^{}$ that when two socks are drawn randomly, without replacement, both are red or both are blue is given by

$P=\frac{r(r-1)}{(r+b)(r+b-1)}+\frac{b(b-1)}{(r+b)(r+b-1)}=\frac{r(r-1)+(t-r)(t-r-1)}{t(t-1)}=\frac{1}{2}\,.$

Solving the resulting quadratic equation $r^{2}-rt+t(t-1)/4=0$, for $r_{}^{}$ in terms of $t_{}^{}$, one obtains that

$r=\frac{t\pm\sqrt{t}}{2}\, .$

Now, since $r_{}^{}$ and $t_{}^{}$ are positive integers, it must be the case that $t_{}^{}=n^{2}$, with $n\in\mathbb{N}$. Hence, $r=n(n\pm 1)/2$ would correspond to the general solution. For the present case, $t\leq 1991$ and so one easily finds that $n=44$ is the largest possible integer satisfying the problem conditions.

In summary, the solution is that the maximum number of red socks is $r=990$.

See also

1991 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

</math>

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