Difference between revisions of "2011 AIME I Problems/Problem 15"

(Created page with '== Problem == For some integer <math>m</math>, the polynomial <math>x^3 - 2011x + m</math> has the three integer roots <math>a</math>, <math>b</math>, and <math>c</math>. Find <…')
 
m (Solution)
Line 42: Line 42:
 
<tr><th><math>a</math></th><th><math>a^2 = 2011 + bc</math></th></tr>
 
<tr><th><math>a</math></th><th><math>a^2 = 2011 + bc</math></th></tr>
  
<tr><td><math>45</math></td><td><math>14</math></td><td><math>5</math></td></tr>
+
<tr><td><math>45</math></td><td><math>14</math></td></tr>
 
<tr><td><math>46</math></td><td><math>14  + 91 =105</math></td></tr>
 
<tr><td><math>46</math></td><td><math>14  + 91 =105</math></td></tr>
 
<tr><td><math>47</math></td><td><math>105 + 93 = 198</math></td></tr>
 
<tr><td><math>47</math></td><td><math>105 + 93 = 198</math></td></tr>
Line 54: Line 54:
 
<math>105 = (3)(5)(7)</math>, So <math>46</math> won't work. <math>198/47 > 4</math>,  
 
<math>105 = (3)(5)(7)</math>, So <math>46</math> won't work. <math>198/47 > 4</math>,  
  
<math>198 is not divisible by 5</math>, <math>198/6 = 33</math>, which is too small to get <math>47</math>
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<math>198</math> is not divisible by <math>5</math>, <math>198/6 = 33</math>, which is too small to get <math>47</math>
  
 
<math>293/48 > 6</math>, <math>293</math> is not divisible by <math>7</math> or <math>8</math> or <math>9</math>, we can clearly tell that <math>10</math> is too much
 
<math>293/48 > 6</math>, <math>293</math> is not divisible by <math>7</math> or <math>8</math> or <math>9</math>, we can clearly tell that <math>10</math> is too much
 +
  
 
Hence, <math>a = 49</math>, <math>a^2 -2011 = 390</math>. <math>b = 39</math>, <math>c = 10</math>.
 
Hence, <math>a = 49</math>, <math>a^2 -2011 = 390</math>. <math>b = 39</math>, <math>c = 10</math>.
  
 
Answer: <math>098</math>
 
Answer: <math>098</math>
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 +
== See also ==
 +
 +
{{AIME box|year=2011|num-b=14|after=Last Problem|n=I}}
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 +
[[Category:Intermediate Geometry Problems]]

Revision as of 23:09, 19 March 2011

Problem

For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$.

Solution

With Vieta's formula, we know that $a+b+c = 0$, and $ab+bc+ac = -2011$.



$a,b,c\neq 0$ since any one being zero will make the the other 2 $\pm sqrt{2011}$.

$a = -(b+c)$. WLOG, let $|a| \ge |b| \ge |c|$.

Then if $a > 0$, then $b,c < 0$ and if $a < 0$, $b,c > 0$.


$ab+bc+ac = -2011 = a(b+c)+bc = -a^2+bc$


$a^2 = 2011 + bc$

We know that $b$, $c$ have the same sign. So $|a| \ge 45$. ($44^2<2011$ and $45^2 = 2025$)

Also, $bc$ maximize when $b = c$ if we fixed $a$. Hence, $2011 = a^2 - bc < \frac{3}{4}a^2$.

So $a ^2 < \frac{(4)2011}{3} = 2681+\frac{1}{3}$.

$52^2 = 2704$ so $|a| \le 51$.



Now we have limited a to $45\le |a| \le 51$.

Let's us analyze $a^2 = 2011 + bc$.


Here is a table:

$a$$a^2 = 2011 + bc$
$45$$14$
$46$$14  + 91 =105$
$47$$105 + 93 = 198$
$48$$198 + 95 = 293$
$49$$293 + 97 = 390$


We can tell we don't need to bother with $45$,

$105 = (3)(5)(7)$, So $46$ won't work. $198/47 > 4$,

$198$ is not divisible by $5$, $198/6 = 33$, which is too small to get $47$

$293/48 > 6$, $293$ is not divisible by $7$ or $8$ or $9$, we can clearly tell that $10$ is too much


Hence, $a = 49$, $a^2 -2011 = 390$. $b = 39$, $c = 10$.

Answer: $098$

See also

2011 AIME I (ProblemsAnswer KeyResources)
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Problem 14
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Last Problem
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