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

Latest revision as of 19:24, 5 February 2024

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 $\frac{1}{2}$ 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?

Solutions

Solution 1

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

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

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 $r=\boxed{990}$ .

Solution 2

Let $r$ and $b$ denote the number of red and blue socks such that $r+b\le1991$ . Then by complementary counting, the number of ways to get a red and a blue sock must be equal to $1-\frac12=\frac12=\frac{2rb}{(r+b)(r+b-1)}\implies4rb=(r+b)(r+b-1)$ $=(r+b)^2-(r+b)\implies r^2+2rb+b^2-r-b=4rb\implies r^2-2rb+b^2$ $=(r-b)^2=r+b$ , so $r+b$ must be a perfect square $k^2$ . Clearly, $r=\frac{k^2+k}2$ , so the larger $k$ , the larger $r$ : $k^2=44^2$ is the largest perfect square below $1991$ , and our answer is $\frac{44^2+44}2=\frac12\cdot44(44+1)=22\cdot45=11\cdot90=\boxed{990}$ .

Solution 3

Let $r$ and $b$ denote the number of red and blue socks, respectively. In addition, let $t = r + b$ , the total number of socks in the drawer.

From the problem, it is clear that $\frac{r(r-1)}{t(t-1)} + \frac{b(b-1)}{t(t-1)} = \frac{1}{2}$

Expanding, we get $\frac{r^2 + b^2 - r - b}{t^2 - t} = \frac{1}{2}$

Substituting $t$ for $r + b$ and cross multiplying, we get $2r^2 + 2b^2 - 2r - 2b = r^2 + 2br + b^2 - r - b$

Combining terms, we get $b^2 - 2br + r^2 - b - r = 0$

To make this expression factorable, we add $2r$ to both sides, resulting in $(b - r)^2 - 1(b-r) = (b - r - 1)(b - r) = 2r$

From this equation, we can test values for the expression $(b - r - 1)(b - r)$ , which is the multiplication of two consecutive integers, until we find the highest value of $b$ or $r$ such that $b + r \leq 1991$ .

By testing $(b - r - 1) = 43$ and $(b - r) = 44$ , we get that $r = 43(22) = 946$ and $b = 990$ . Testing values one integer higher, we get that $r = 990$ and $b = 1035$ . Since $990 + 1035 = 2025$ is greater than $1991$ , we conclude that $(946, 990)$ is our answer.

Since it doesn't matter whether the number of blue or red socks is $990$ , we take the lower value for $r$ , thus the maximum number of red socks is $r=\boxed{990}$ .

Solution 4

As above, let $r$ , $b$ , and $t$ denote the number of red socks, the number of blue socks, and the total number of socks, respectively. We see that $\frac{r(r-1)}{t(t-1)}+\frac{b(b-1)}{t(t-1)}=\frac{1}{2}$ , so $r^2+b^2-r-b=\frac{t(t-1)}{2}=r^2+b^2-t=\frac{t^2}{2}-\frac{t}{2}$ .

Seeing that we can rewrite $r^2+b^2$ as $(r+b)^2-2rb$ , and remembering that $r+b=t$ , we have $\frac{t^2}{2}-\frac{t}{2}=t^2-2rb-t$ , so $2rb=\frac{t^2}{2}-\frac{t}{2}$ , which equals $r^2+b^2-t$ .

We now have $r^2+b^2-t=2rb$ , so $r^2-2rb+b^2=t$ and $r-b=\pm\sqrt{t}$ . Adding this to $r+b=t$ , we have $2r=t\pm\sqrt{t}$ . To maximize $r$ , we must use the positive square root and maximize $t$ . The largest possible value of $t$ is the largest perfect square less than 1991, which is $1936=44^2$ . Therefore, $r=\frac{t+\sqrt{t}}{2}=\frac{1936+44}{2}=\boxed{990}$ .

Solution by Zeroman

Solution 5

Let $r$ be the number of socks that are red, and $t$ be the total number of socks. We get:

$2(r(r-1)+(t-r)(t-r-1))=t(t-1)$ Expanding the left hand side and the right hand side, we get: $4r^2-4rt+2t^2-2t = t^2-t$

And, moving terms, we will get that: $4r^2-4rt+t^2 = t$

We notice that the left side is a perfect square. $(2r-t)^2 = t$

Thus $t$ is a perfect square. And, the higher $t$ is, the higher $r$ will be. So, we should set $t = 44^2 = 1936$

And, we see, $2r-1936 = \pm44$ We will use the positive root, to get that $2r-1936 = 44$ , $2r = 1980$ , and $r = \boxed{990}$ .

- AlexLikeMath

Solution 6

Let $r$ and $b$ denote the red socks and blue socks, respectively. Thus the equation in question is:

$\frac{r(r-1)+b(b-1)}{(r+b)(r+b-1)}=\frac{1}{2}$

$\Rightarrow 2r^2-2r+2b^2-2b=r^2+2rb+b^2-r-b$

$\Rightarrow r^2+b^2-r-b-2rb=0$

$\Rightarrow (r-b)^2=r+b\le 1991$

Because we wish to maximize $r$ , we have $r\ge b$ and thus $r-b\le 44$ as both $r$ and $b$ must be integers and ${44}^2=1936$ is the largest square less than or equal to $1991$ . We now know that the maximum difference between the number of socks is $44$ . Now we return to an earlier equation:

$r^2+b^2-r-b-2rb=0$

$\Rightarrow r^2-(2b+1)r+(b^2-b)=0$

Solving by the Quadratic formula, we have:

$r=\frac{2b+1\pm\sqrt{4b^2+4b+1-4b^2+4b}}{2}$

$\Rightarrow r=\frac{2b+1\pm\sqrt{8b+1}}{2}$

$\Rightarrow r-b=\frac{1\pm\sqrt{8b+1}}{2}$

$\Rightarrow 44\ge r-b=\frac{1\pm\sqrt{8b+1}}{2}$

$\Rightarrow 44\ge\frac{1\pm\sqrt{8b+1}}{2}$

$\Rightarrow \pm\sqrt{8b+1}\le 87$

$\Rightarrow b\le 946$ (Here we use the positive sign to maximize $r$ .)

Thus the optimal case would be $b=946$ as $r$ increases only when $b$ increases. In addition, $\sqrt{8b+1}=87$ as we are using the equality case. We then plug back in:

$r=\frac{2b+1\pm\sqrt{8b+1}}{2}$

$\Rightarrow r=\frac{1893+87}{2}$ (using the established $\sqrt{8b+1}=87$ )

$\Rightarrow r=\frac{1980}{2}$

$\Rightarrow r=\boxed{990}$

~eevee9406

@@ Line 1: / Line 1: @@
 == 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 <math>\displaystyle \frac{1}{2}</math> 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?
+A drawer contains a mixture of red socks and blue socks, at most <math>1991</math> in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\frac{1}{2}</math> 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 ==
+== Solutions ==
-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
+=== Solution 1 ===
+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>
+<cmath>
-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}\,.
+\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}.
-</math>
+</cmath>
-Solving the resulting quadratic equation <math>r^{2}-rt+t(t-1)/4=0</math>, for <math>r_{}^{} </math> in terms of <math>t_{}^{}</math>, one obtains that
+Solving the resulting quadratic equation <math>r^{2}-rt+t(t-1)/4=0</math>, for <math>r</math> in terms of <math>t</math>, one obtains that
-<math>
+<cmath>
 r=\frac{t\pm\sqrt{t}}{2}\, .
-</math>
+</cmath>
-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=\boxed{990}</math>.
+=== Solution 2 ===
+Let <math>r</math> and <math>b</math> denote the number of red and blue socks such that <math>r+b\le1991</math>. Then by complementary counting, the number of ways to get a red and a blue sock must be equal to <math>1-\frac12=\frac12=\frac{2rb}{(r+b)(r+b-1)}\implies4rb=(r+b)(r+b-1)</math>
+<math>=(r+b)^2-(r+b)\implies r^2+2rb+b^2-r-b=4rb\implies r^2-2rb+b^2</math>
+<math>=(r-b)^2=r+b</math>, so <math>r+b</math> must be a perfect square <math>k^2</math>. Clearly, <math>r=\frac{k^2+k}2</math>, so the larger <math>k</math>, the larger <math>r</math>: <math>k^2=44^2</math> is the largest perfect square below <math>1991</math>, and our answer is <math>\frac{44^2+44}2=\frac12\cdot44(44+1)=22\cdot45=11\cdot90=\boxed{990}</math>.
+===Solution 3===
+Let <math>r</math> and <math>b</math> denote the number of red and blue socks, respectively. In addition, let <math>t = r + b</math>, the total number of socks in the drawer.
+From the problem, it is clear that <math>\frac{r(r-1)}{t(t-1)} + \frac{b(b-1)}{t(t-1)} = \frac{1}{2}</math>
+Expanding, we get <math>\frac{r^2 + b^2 - r - b}{t^2 - t} = \frac{1}{2}</math>
+Substituting <math>t</math> for <math>r + b</math> and cross multiplying, we get <math>2r^2 + 2b^2 - 2r - 2b = r^2 + 2br + b^2 - r - b</math>
+Combining terms, we get <math>b^2 - 2br + r^2 - b - r = 0</math>
+To make this expression factorable, we add <math>2r</math> to both sides, resulting in <math>(b - r)^2 - 1(b-r) = (b - r - 1)(b - r) = 2r</math>
+From this equation, we can test values for the expression <math>(b - r - 1)(b - r)</math>, which is the multiplication of two consecutive integers, until we find the highest value of <math>b</math> or <math>r</math> such that <math>b + r \leq 1991</math>.
+By testing <math>(b - r - 1) = 43</math> and <math>(b - r) = 44</math>, we get that <math>r = 43(22) = 946</math> and <math>b = 990</math>. Testing values one integer higher, we get that <math>r = 990</math> and <math>b = 1035</math>. Since <math>990 + 1035 = 2025</math> is greater than <math>1991</math>, we conclude that <math>(946, 990)</math> is our answer.
+Since it doesn't matter whether the number of blue or red socks is <math>990</math>, we take the lower value for <math>r</math>, thus the maximum number of red socks is <math>r=\boxed{990}</math>.
+=== Solution 4 ===
+As above, let <math>r</math>, <math>b</math>, and <math>t</math> denote the number of red socks, the number of blue socks, and the total number of socks, respectively. We see that <math>\frac{r(r-1)}{t(t-1)}+\frac{b(b-1)}{t(t-1)}=\frac{1}{2}</math>, so <math>r^2+b^2-r-b=\frac{t(t-1)}{2}=r^2+b^2-t=\frac{t^2}{2}-\frac{t}{2}</math>.
+Seeing that we can rewrite <math>r^2+b^2</math> as <math>(r+b)^2-2rb</math>, and remembering that <math>r+b=t</math>, we have <math>\frac{t^2}{2}-\frac{t}{2}=t^2-2rb-t</math>, so <math>2rb=\frac{t^2}{2}-\frac{t}{2}</math>, which equals <math>r^2+b^2-t</math>.
+We now have <math>r^2+b^2-t=2rb</math>, so <math>r^2-2rb+b^2=t</math> and <math>r-b=\pm\sqrt{t}</math>. Adding this to <math>r+b=t</math>, we have <math>2r=t\pm\sqrt{t}</math>. To maximize <math>r</math>, we must use the positive square root and maximize <math>t</math>. The largest possible value of <math>t</math> is the largest perfect square less than 1991, which is <math>1936=44^2</math>. Therefore, <math>r=\frac{t+\sqrt{t}}{2}=\frac{1936+44}{2}=\boxed{990}</math>.
+Solution by Zeroman
+===Solution 5===
+Let <math>r</math> be the number of socks that are red, and <math>t</math> be the total number of socks. We get:
+<math>2(r(r-1)+(t-r)(t-r-1))=t(t-1)</math>
+Expanding the left hand side and the right hand side, we get:
+<math>4r^2-4rt+2t^2-2t = t^2-t</math>
+And, moving terms, we will get that:
+<math>4r^2-4rt+t^2 = t</math>
+We notice that the left side is a perfect square.
+<math>(2r-t)^2 = t</math>
+Thus <math>t</math> is a perfect square. And, the higher <math>t</math> is, the higher <math>r</math> will be. So, we should set <math>t = 44^2 = 1936</math>
+And, we see, <math>2r-1936 = \pm44</math> We will use the positive root, to get that <math>2r-1936 = 44</math>, <math>2r = 1980</math>, and <math>r = \boxed{990}</math>.
+- AlexLikeMath
+==Solution 6==
+Let <math>r</math> and <math>b</math> denote the red socks and blue socks, respectively. Thus the equation in question is:
+<math>\frac{r(r-1)+b(b-1)}{(r+b)(r+b-1)}=\frac{1}{2}</math>
+<math>\Rightarrow 2r^2-2r+2b^2-2b=r^2+2rb+b^2-r-b</math>
+<math>\Rightarrow r^2+b^2-r-b-2rb=0</math>
+<math>\Rightarrow (r-b)^2=r+b\le 1991</math>
+Because we wish to maximize <math>r</math>, we have <math>r\ge b</math> and thus <math>r-b\le 44</math> as both <math>r</math> and <math>b</math> must be integers and <math>{44}^2=1936</math> is the largest square less than or equal to <math>1991</math>. We now know that the maximum difference between the number of socks is <math>44</math>. Now we return to an earlier equation:
+<math>r^2+b^2-r-b-2rb=0</math>
+<math>\Rightarrow r^2-(2b+1)r+(b^2-b)=0</math>
+Solving by the [[Quadratic formula]], we have:
+<math>r=\frac{2b+1\pm\sqrt{4b^2+4b+1-4b^2+4b}}{2}</math>
+<math>\Rightarrow r=\frac{2b+1\pm\sqrt{8b+1}}{2}</math>
+<math>\Rightarrow r-b=\frac{1\pm\sqrt{8b+1}}{2}</math>
+<math>\Rightarrow 44\ge r-b=\frac{1\pm\sqrt{8b+1}}{2}</math>
+<math>\Rightarrow 44\ge\frac{1\pm\sqrt{8b+1}}{2}</math>
+<math>\Rightarrow \pm\sqrt{8b+1}\le 87</math>
+<math>\Rightarrow b\le 946</math> (Here we use the positive sign to maximize <math>r</math>.)
+Thus the optimal case would be <math>b=946</math> as <math>r</math> increases only when <math>b</math> increases. In addition, <math>\sqrt{8b+1}=87</math> as we are using the equality case. We then plug back in:
+<math>r=\frac{2b+1\pm\sqrt{8b+1}}{2}</math>
+<math>\Rightarrow r=\frac{1893+87}{2}</math> (using the established <math>\sqrt{8b+1}=87</math>)
+<math>\Rightarrow r=\frac{1980}{2}</math>
+<math>\Rightarrow r=\boxed{990}</math>
+~eevee9406
 == See also ==
-{{AIME box|year=1991|num-b=12|num-a=14}}</math>
+{{AIME box|year=1991|num-b=12|num-a=14}}
+[[Category:Intermediate Combinatorics Problems]]
+{{MAA Notice}}

1991 AIME (Problems • Answer Key • Resources)
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