Difference between revisions of "2020 AMC 10B Problems/Problem 24"

Latest revision as of 09:26, 9 March 2024

The following problem is from both the 2020 AMC 10B #24 and 2020 AMC 12B #21, so both problems redirect to this page.

Problem

How many positive integers $n$ satisfy $\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$ (Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$ .)

$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

Solution 1

We can first consider the equation without a floor function:

$\dfrac{n+1000}{70} = \sqrt{n}$

Multiplying both sides by 70 and then squaring:

$n^2 + 2000n + 1000000 = 4900n$

Moving all terms to the left:

$n^2 - 2900n + 1000000 = 0$

Now we can determine the factors:

$(n-400)(n-2500) = 0$

This means that for $n = 400$ and $n = 2500$ , the equation will hold without the floor function.

Now we can simply check the multiples of 70 around 400 and 2500 in the original equation, which we abbreviate as $L=R$ .

For $n = 330$ , $L=19$ but $18^2 < 330 < 19^2$ so $R=18$

For $n = 400$ , $L=20$ and $R=20$

For $n = 470$ , $L=21$ , $R=21$

For $n = 540$ , $L=22$ but $540 > 23^2$ so $R=23$

Now we move to $n = 2500$

For $n = 2430$ , $L=49$ and $49^2 < 2430 < 50^2$ so $R=49$

For $n = 2360$ , $L=48$ and $48^2 < 2360 < 49^2$ so $R=48$

For $n = 2290$ , $L=47$ and $47^2 < 2360 < 48^2$ so $R=47$

For $n = 2220$ , $L=46$ but $47^2 < 2220$ so $R=47$

For $n = 2500$ , $L=50$ and $R=50$

For $n = 2570$ , $L=51$ but $2570 < 51^2$ so $R=50$

Therefore we have 6 total solutions, $n = 400, 470, 2290, 2360, 2430, 2500 = \boxed{\textbf{(C) 6}}$

Solution 2

This is my first solution here, so please forgive me for any errors.

We are given that $\frac{n+1000}{70}=\lfloor\sqrt{n}\rfloor$

$\lfloor\sqrt{n}\rfloor$ must be an integer, which means that $n+1000$ is divisible by $70$ . As $1000\equiv 20\pmod{70}$ , this means that $n\equiv 50\pmod{70}$ , so we can write $n=70k+50$ for $k\in\mathbb{Z}$ .

Therefore, $\frac{n+1000}{70}=\frac{70k+1050}{70}=k+15=\lfloor\sqrt{70k+50}\rfloor$

Also, we can say that $\sqrt{70k+50}-1 < k+15$ and $k+15\leq\sqrt{70k+50}$

Squaring the second inequality, we get $k^{2}+30k+225\leq70k+50\implies k^{2}-40k+175\leq 0\implies (k-5)(k-35)\leq0\implies 5\leq k\leq 35$ .

Similarly solving the first inequality gives us $k < 19-\sqrt{155}$ or $k > 19+\sqrt{155}$

$\sqrt{155}$ is larger than $12$ and smaller than $13$ , so instead, we can say $k\leq 6$ or $k\geq 32$ .

Combining this with $5\leq k\leq 35$ , we get $k=5,6,32,33,34,35$ are all solutions for $k$ that give a valid solution for $n$ , meaning that our answer is $\boxed{\textbf{(C) 6}}$ . -Solution By Qqqwerw

Solution 3

We start with the given equation $\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor$ From there, we can start with the general inequality that $\lfloor \sqrt{n} \rfloor \leq \sqrt{n} < \lfloor \sqrt{n} \rfloor + 1$ . This means that $\dfrac{n+1000}{70} \leq \sqrt{n} < \dfrac{n+1070}{70}$ Solving each inequality separately gives us two inequalities: $n - 70\sqrt{n} +1000 \leq 0 \rightarrow (\sqrt{n}-50)(\sqrt{n}-20)\leq 0 \rightarrow 20\leq \sqrt{n} \leq 50$ $n-70\sqrt{n}+1070 > 0 \rightarrow \sqrt{n} < 35-\sqrt{155} , \sqrt{n} > 35+\sqrt{155}$ Simplifying and approximating decimals yields 2 solutions for one inequality and 4 for the other. Hence, the answer is $2+4 = \boxed{\textbf{(C) } 6}$ .

~Rekt4

Solution 4

Let $n$ be uniquely of the form $n=k^2+r$ where $0 \le r \le 2k \; \bigstar$ . Then, $\frac{k^2+r+1000}{70} = k$ Rearranging and completeing the square gives $(k-35)^2 + r = 225$ $\Rightarrow r = (k-20)(50-k)\; \smiley$ This gives us $(k-35)^2 \le (k-35)^2+r=225 \le (k-35)^2 + 2k$ Solving the left inequality shows that $20 \le k \le 50$ . Combing this with the right inequality gives that $(k-35)^2+r=225 \le (k-35)^2 + 2k \le (k-35)^2+100$ which implies either $k \ge 47$ or $k \le 23$ . By directly computing the cases for $k = 20, 21, 22, 23, 47, 48, 49, 50$ using $\smiley$ , it follows that only $k = 22, 23$ yield and invalid $r$ from $\bigstar$ . Since each $k$ corresponds to one $r$ and thus to one $n$ (from $\smiley$ and the original form), there must be 6 such $n$ .

~the_jake314

Solution 5

Since the right-hand-side is an integer, so must be the left-hand-side. Therefore, we must have $n\equiv -20\pmod{70}$ ; let $n=70j-20$ . The given equation becomes $j+14 = \lfloor \sqrt{70j-20} \rfloor$

Since $\lfloor x \rfloor \leq x < \lfloor x \rfloor +1$ for all real $x$ , we can take $x=\sqrt{70j-20}$ with $\lfloor x \rfloor =j+14$ to get $j+14 \leq \sqrt{70j-20} < j+15$ We can square the inequality to get $196+28j+j^{2} \leq 70j-20 < 225 + 30j + j^{2}$ The left inequality simplifies to $(j-36)(j-6) \leq 0$ , which yields $6 \le j \le 36.$ The right inequality simplifies to $(j-20)^2 - 155 > 0$ , which yields $j < 20 - \sqrt{155} < 8 \quad \text{or} \quad j > 20 + \sqrt{155} > 32$

Solving $j < 8$ , and $6 \le j \le 36$ , we get $6 \le j < 8$ , for $2$ values $j\in \{6, 7\}$ .

Solving $j >32$ , and $6 \le j \le 36$ , we get $32 < j \le 36$ , for $4$ values $k\in \{33, \ldots , 36\}$ .

Thus, our answer is $2 + 4 = \boxed{\textbf{(C) }6}$

~KingRavi

Solution 6

Set $x=\sqrt{n}$ in the given equation and solve for $x$ to get $x^2 = 70 \cdot \lfloor x \rfloor - 1000$ . Set $k = \lfloor x \rfloor \ge 0$ ; since $\lfloor x \rfloor^2 \le x^2 < (\lfloor x \rfloor + 1)^2$ , we get $k^2 \le 70k - 1000 < k^2 + 2k + 1.$ The left inequality simplifies to $(k-20)(k-50) \le 0$ , which yields $20 \le k \le 50.$ The right inequality simplifies to $(k-34)^2 > 155$ , which yields $k < 34 - \sqrt{155} < 22 \quad \text{or} \quad k > 34 + \sqrt{155} > 46$ Solving $k < 22$ , and $20 \le k \le 50$ , we get $20 \le k < 22$ , for $2$ values $k\in \{20, 21\}$ .

Solving $k >46$ , and $20 \le k \le 50$ , we get $46 < k \le 50$ , for $4$ values $k\in \{47, \ldots , 50\}$ .

Thus, our answer is $2 + 4 = \boxed{\textbf{(C) }6}$

~isabelchen

Solution 7

If $n$ is a perfect square, we can write $n = k^2$ for a positive integer $k$ , so $\lfloor \sqrt{n} \rfloor = \sqrt{n} = k.$ The given equation turns into

\begin{align*} \frac{k^2 + 1000}{70} &= k \\ k^2 - 70k + 1000 &= 0 \\ (k-20)(k-50) &= 0, \end{align*}

so $k = 20$ or $k= 50$ , so $n = 400, 2500.$

If $n$ is not square, then we can say that, for a positive integer $k$ , we have \begin{align*} k^2 < &n < (k+1)^2 \\ k^2 + 1000 < &n + 1000 = 70\lfloor \sqrt{n} \rfloor = 70k< (k+1)^2 + 1000 \\ k^2 + 1000 < &70k < (k+1)^2 + 1000. \end{align*}

To solve this inequality, we take the intersection of the two solution sets to each of the two inequalities $k^2 + 1000 < 70k$ and $70k < (k+1)^2 + 1000$ . To solve the first one, we have

\begin{align*} k^2 - 70k + 1000 &< 0 \\ (k-20)(k-50) &< 0\\ \end{align*} $k\in (20, 50),$ because the portion of the parabola between its two roots will be negative.

The second inequality yields

\begin{align*} 70k &< k^2 + 2k + 1 + 1000 \\ 0 &< k^2 -68k + 1001. \end{align*} This time, the inequality will hold for all portions of the parabola that are not on or between the its two roots, which are $34 + \sqrt{155}>46$ and $34-\sqrt{155}<22$ (they are roughly equal, but this is to ensure that we do not miss any solutions).

Notation wise, we need all integers $k$ such that

$k \in \left(20, 50\right) \cap \left(-\infty,34 - \sqrt{155} \right)$ or $k \in \left(20, 50\right) \cap \left(34 + \sqrt{155}, \infty \right).$

For the first one, since our uppoer bound is a little less than $22$ , the $k$ that works is $21$ . For the second, our lower bound is a little more than $46$ , so the $k$ that work are $47, 48,$ and $49$ .

$\boxed{\textbf{(C) }6}$ total solutions for $n$ , since each value of $k$ corresponds to exactly one value of $n$ .

-Benedict T (countmath1)

Video Solutions

@@ Line 1: / Line 1: @@
+{{duplicate|[[2020 AMC 10B Problems|2020 AMC 10B #24]] and [[2020 AMC 12B Problems|2020 AMC 12B #21]]}}
 ==Problem==
-How many positive integers <math>n</math> satisfy<cmath>\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?</cmath>(Recall that <math>\lfloor x\rfloor</math> is the greatest integer not exceeding <math>x</math>.)
+How many positive integers <math>n</math> satisfy <cmath>\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?</cmath>(Recall that <math>\lfloor x\rfloor</math> is the greatest integer not exceeding <math>x</math>.)
 <math>\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32</math>
-==Solution==
+==Solution 1==
-(Quick solution if you’re in a hurry)
-(Could someone fix this with LaTeX?)
+We can first consider the equation without a floor function:
-First notice that (n+1000)/70 must be an integer. This means that n is congruent to 50 (mod 70). Testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields n=20 and n=21. After this, the right side becomes greater than the left. Since the left side is linear and the right side is roughly a radical, the graphs must intersect in another place. Using binary search can give a good estimate of the other cases, being n=47, n=48, n=49, and n=50.
+<cmath>\dfrac{n+1000}{70} = \sqrt{n} </cmath>
+Multiplying both sides by 70 and then squaring:
+<cmath>n^2 + 2000n + 1000000 = 4900n</cmath>
+Moving all terms to the left:
+<cmath>n^2 - 2900n + 1000000 = 0</cmath>
+Now we can determine the factors:
+<cmath>(n-400)(n-2500) = 0</cmath>
+This means that for <math>n = 400</math> and <math>n = 2500</math>, the equation will hold without the floor function.
+Now we can simply check the multiples of 70 around 400 and 2500 in the original equation, which we abbreviate as <math>L=R</math>.
+For <math>n = 330</math>, <math>L=19</math> but <math>18^2 < 330 < 19^2</math> so <math>R=18</math>
+For <math>n = 400</math>,  <math>L=20</math> and  <math>R=20</math>
+For <math>n = 470</math>,  <math>L=21</math>,  <math>R=21</math>
+For <math>n = 540</math>, <math>L=22</math> but <math>540 > 23^2</math> so   <math>R=23</math>
+Now we move to <math>n = 2500</math>
+For <math>n = 2430</math>,  <math>L=49</math> and <math>49^2 < 2430 < 50^2</math> so <math>R=49</math>
+For <math>n = 2360</math>,  <math>L=48</math> and <math>48^2 < 2360 < 49^2</math> so <math>R=48</math>
+For <math>n = 2290</math>,  <math>L=47</math> and <math>47^2 < 2360 < 48^2</math> so  <math>R=47</math>
+For <math>n = 2220</math>, <math>L=46</math> but <math>47^2 < 2220</math> so  <math>R=47</math>
+For <math>n = 2500</math>,  <math>L=50</math> and <math>R=50</math>
+For <math>n = 2570</math>, <math>L=51</math> but <math>2570 < 51^2</math> so <math>R=50</math>
+Therefore we have 6 total solutions, <math>n = 400, 470, 2290, 2360, 2430, 2500 = \boxed{\textbf{(C) 6}}</math>
+==Solution 2==
+This is my first solution here, so please forgive me for any errors.
+We are given that <cmath>\frac{n+1000}{70}=\lfloor\sqrt{n}\rfloor</cmath>
+<math>\lfloor\sqrt{n}\rfloor</math> must be an integer, which means that <math>n+1000</math> is divisible by <math>70</math>. As <math>1000\equiv 20\pmod{70}</math>, this means that <math>n\equiv 50\pmod{70}</math>, so we can write <math>n=70k+50</math> for <math>k\in\mathbb{Z}</math>.
+Therefore, <cmath>\frac{n+1000}{70}=\frac{70k+1050}{70}=k+15=\lfloor\sqrt{70k+50}\rfloor</cmath>
+Also, we can say that <math>\sqrt{70k+50}-1 < k+15</math> and <math>k+15\leq\sqrt{70k+50}</math>
+Squaring the second inequality, we get <math>k^{2}+30k+225\leq70k+50\implies k^{2}-40k+175\leq 0\implies (k-5)(k-35)\leq0\implies 5\leq k\leq 35</math>.
+Similarly solving the first inequality gives us <math>k < 19-\sqrt{155}</math> or <math>k > 19+\sqrt{155}</math>
+<math>\sqrt{155}</math> is larger than <math>12</math> and smaller than <math>13</math>, so instead, we can say <math>k\leq 6</math> or <math>k\geq 32</math>.
+Combining this with <math>5\leq k\leq 35</math>, we get <math>k=5,6,32,33,34,35</math> are all solutions for <math>k</math> that give a valid solution for <math>n</math>, meaning that our answer is <math>\boxed{\textbf{(C) 6}}</math>.
+-Solution By Qqqwerw
+==Solution 3==
+We start with the given equation<cmath>\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor</cmath>From there, we can start with the general inequality that <math>\lfloor \sqrt{n} \rfloor \leq \sqrt{n} < \lfloor \sqrt{n} \rfloor + 1</math>. This means that<cmath>\dfrac{n+1000}{70} \leq \sqrt{n} < \dfrac{n+1070}{70}</cmath>Solving each inequality separately gives us two inequalities:<cmath>n - 70\sqrt{n} +1000 \leq 0 \rightarrow (\sqrt{n}-50)(\sqrt{n}-20)\leq 0 \rightarrow 20\leq \sqrt{n} \leq 50</cmath><cmath>n-70\sqrt{n}+1070 > 0 \rightarrow \sqrt{n} < 35-\sqrt{155} , \sqrt{n} > 35+\sqrt{155}</cmath>Simplifying and approximating decimals yields 2 solutions for one inequality and 4 for the other. Hence, the answer is <math>2+4 = \boxed{\textbf{(C) } 6}</math>.
+~Rekt4
+==Solution 4==
+Let <math>n</math> be uniquely of the form <math>n=k^2+r</math> where <math>0 \le r \le 2k \; \bigstar</math>. Then, <cmath> \frac{k^2+r+1000}{70} = k</cmath> Rearranging and completeing the square gives <cmath>(k-35)^2 + r = 225</cmath> <cmath>\Rightarrow r = (k-20)(50-k)\; \smiley</cmath> This gives us <cmath>(k-35)^2 \le (k-35)^2+r=225 \le (k-35)^2 + 2k</cmath> Solving the left inequality shows that <math>20 \le k \le 50</math>. Combing this with the right inequality gives that <cmath>(k-35)^2+r=225 \le (k-35)^2 + 2k \le (k-35)^2+100 </cmath> which implies either <math>k \ge 47</math> or <math>k \le 23</math>. By directly computing the cases for <math>k = 20, 21, 22, 23, 47, 48, 49, 50</math> using <math>\smiley</math>, it follows that only <math> k = 22, 23</math> yield and invalid <math>r</math> from <math>\bigstar</math>. Since each <math>k</math> corresponds to one <math>r</math> and thus to one <math>n</math> (from <math>\smiley</math> and the original form), there must be 6 such <math>n</math>.
+~the_jake314
+==Solution 5==
+Since the right-hand-side is an integer, so must be the left-hand-side. Therefore, we must have <math>n\equiv -20\pmod{70}</math>; let <math>n=70j-20</math>. The given equation becomes<cmath>j+14 = \lfloor \sqrt{70j-20} \rfloor</cmath>
+Since <math>\lfloor x \rfloor \leq x < \lfloor x \rfloor +1</math> for all real <math>x</math>, we can take <math>x=\sqrt{70j-20}</math> with <math>\lfloor x \rfloor =j+14</math> to get
+<cmath>j+14 \leq \sqrt{70j-20} < j+15</cmath>
+We can square the inequality to get<cmath>196+28j+j^{2} \leq 70j-20 < 225 + 30j + j^{2}</cmath>
+The left inequality simplifies to <math>(j-36)(j-6) \leq 0</math>, which yields <cmath>6 \le j \le 36.</cmath>
+The right inequality simplifies to <math>(j-20)^2 - 155 > 0</math>, which yields <cmath>j < 20 - \sqrt{155} < 8 \quad \text{or} \quad j > 20 + \sqrt{155} > 32</cmath>
+Solving <math>j < 8</math>, and <math>6 \le j \le 36</math>, we get <math>6 \le j < 8</math>, for <math>2</math> values <math>j\in \{6, 7\}</math>.
+Solving <math>j >32</math>, and <math>6 \le j \le 36</math>, we get <math>32 < j \le 36</math>, for <math>4</math> values <math>k\in \{33, \ldots , 36\}</math>.
+Thus, our answer is <math>2 + 4 = \boxed{\textbf{(C) }6}</math>
+~KingRavi
+==Solution 6 ==
+Set <math>x=\sqrt{n}</math> in the given equation and solve for <math>x</math> to get <math>x^2 = 70 \cdot \lfloor x \rfloor - 1000</math>. Set <math>k = \lfloor x \rfloor \ge 0</math>; since <math>\lfloor x \rfloor^2 \le x^2 < (\lfloor x \rfloor + 1)^2</math>, we get  <cmath>k^2 \le  70k - 1000 < k^2 + 2k + 1.</cmath>
+The left inequality simplifies to <math>(k-20)(k-50) \le 0</math>, which yields <cmath>20 \le k \le 50.</cmath>
+The right inequality simplifies to <math>(k-34)^2 > 155</math>, which yields <cmath>k < 34 - \sqrt{155} < 22 \quad \text{or} \quad k > 34 + \sqrt{155} > 46</cmath>
+Solving <math>k < 22</math>, and <math>20 \le k \le 50</math>, we get <math>20 \le k < 22</math>, for <math>2</math> values <math>k\in \{20, 21\}</math>.
+Solving <math>k >46</math>, and <math>20 \le k \le 50</math>, we get <math>46 < k \le 50</math>, for <math>4</math> values <math>k\in \{47, \ldots , 50\}</math>.
+Thus, our answer is <math>2 + 4 = \boxed{\textbf{(C) }6}</math>
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+==Solution 7 ==
+If <math>n</math> is a perfect square, we can write <math>n = k^2</math> for a positive integer <math>k</math>, so <math>\lfloor \sqrt{n} \rfloor = \sqrt{n} = k.</math> The given equation turns into
+\begin{align*}
+\frac{k^2 + 1000}{70} &= k \\
+k^2 - 70k + 1000 &= 0 \\
+(k-20)(k-50) &= 0,
+\end{align*}
+so <math>k = 20</math> or <math>k= 50</math>, so <math>n = 400, 2500.</math>
+If <math>n</math> is not square, then we can say that, for a positive integer <math>k</math>, we have
+\begin{align*}
+k^2 < &n < (k+1)^2 \\
+k^2 + 1000 < &n + 1000  = 70\lfloor \sqrt{n} \rfloor = 70k< (k+1)^2 + 1000 \\
+k^2 + 1000 < &70k < (k+1)^2 + 1000.
+\end{align*}
+To solve this inequality, we take the intersection of the two solution sets to each of the two inequalities
+<math>k^2 + 1000 < 70k</math> and <math>70k < (k+1)^2 + 1000</math>. To solve the first one, we have
+\begin{align*}
+k^2 - 70k + 1000 &< 0 \\
+(k-20)(k-50) &< 0\\
+\end{align*}
+<math>k\in (20, 50),</math> because the portion of the parabola between its two roots will be negative.
+The second inequality yields
+\begin{align*}
+k &< k^2 + 2k + 1 + 1000 \\
+&< k^2 -68k + 1001.
+\end{align*}
+This time, the inequality will hold for all portions of the parabola that are not on or between the its two roots, which are <math>34 + \sqrt{155}>46</math> and <math>34-\sqrt{155}<22</math> (they are roughly equal, but this is to ensure that we do not miss any solutions).
+Notation wise, we need all integers <math>k</math> such that
+<cmath>k \in \left(20, 50\right) \cap \left(-\infty,34 - \sqrt{155} \right)</cmath>
+or
+<cmath>k \in \left(20, 50\right) \cap \left(34 + \sqrt{155}, \infty \right).</cmath>
+For the first one, since our uppoer bound is a little less than <math>22</math>, the <math>k</math> that works is <math>21</math>. For the second, our lower bound is a little more than <math>46</math>, so the <math>k</math> that work are <math>47, 48,</math> and  <math>49</math>.
+<math>\boxed{\textbf{(C) }6}</math> total solutions for <math>n</math>, since each value of <math>k</math> corresponds to exactly one value of <math>n</math>.
+-Benedict T (countmath1)
+==Video Solutions==
+===Video Solution 1===
+On The Spot STEM:
+https://youtu.be/BEJybl9TLMA
+===Video Solution 2===
+https://www.youtube.com/watch?v=VWeioXzQxVA&list=PLLCzevlMcsWNcTZEaxHe8VaccrhubDOlQ&index=9 ~ MathEx
+===Video Solution 3 by the Beauty of Math===
+https://youtu.be/4RVYoeiyC4w?t=62
 ==See Also==
 {{AMC10 box|year=2020|ab=B|num-b=23|num-a=25}}
+{{AMC12 box|year=2020|ab=B|num-b=20|num-a=22}}
+[[Category:Intermediate Number Theory Problems]]
 {{MAA Notice}}

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Difference between revisions of "2020 AMC 10B Problems/Problem 24"

Latest revision as of 09:26, 9 March 2024

Contents

Problem

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Video Solutions

Video Solution 1

Video Solution 2

Video Solution 3 by the Beauty of Math

See Also