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

Latest revision as of 21:23, 20 November 2024

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 1

From Vieta's formulas, we know that $a+b+c = 0$ , and $ab+bc+ac = -2011$ . Thus $a = -(b+c)$ . All three of $a$ , $b$ , and $c$ are non-zero: say, if $a=0$ , then $b=-c=\pm\sqrt{2011}$ (which is not an integer). $\textsc{wlog}$ , let $|a| \ge |b| \ge |c|$ . If $a > 0$ , then $b,c < 0$ and if $a < 0$ , then $b,c > 0,$ from the fact that $a+b+c=0$ . We have $-2011=ab+bc+ac = a(b+c)+bc = -a^2+bc$ Thus $a^2 = 2011 + bc$ . We know that $b$ , $c$ have the same sign, so product $bc$ is always positive. So $|a| \ge 45 = \lceil \sqrt{2011} \rceil$ .

Also, if we fix $a$ , $b+c$ is fixed, so $bc$ is maximized when $b = c$ . Hence, $2011 = a^2 - bc > \tfrac{3}{4}a^2 \qquad \Longrightarrow \qquad a ^2 < \tfrac{4}{3}\cdot 2011 = 2681+\tfrac{1}{3}$ So $|a| \le 51$ . Thus we have bounded $a$ as $45\le |a| \le 51$ , i.e. $45\le |b+c| \le 51$ since $a=-(b+c)$ . Let's analyze $bc=(b+c)^2-2011$ . Here is a table:

$\|a\|$	$bc=a^2-2011$
$45$	$14$
$46$	$105$
$47$	$198$
$48$	$293$
$49$	$390$

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: $\boxed{098}$

Solution 2

Starting off like the previous solution, we know that $a + b + c = 0$ , and $ab + bc + ac = -2011$ .

Therefore, $c = -b-a$ .

Substituting, $ab + b(-b-a) + a(-b-a) = ab-b^2-ab-ab-a^2 = -2011$ .

Factoring the perfect square, we get: $ab-(b+a)^2=-2011$ or $(b+a)^2-ab=2011$ .

Therefore, a sum ( $a+b$ ) squared minus a product ( $ab$ ) gives $2011$ ..

We can guess and check different $a+b$ ’s starting with $45$ since $44^2 < 2011$ .

$45^2 = 2025$ therefore $ab = 2025-2011 = 14$ .

Since no factors of $14$ can sum to $45$ ( $1+14$ being the largest sum), a + b cannot equal $45$ .

$46^2 = 2116$ making $ab = 105 = 3 * 5 * 7$ .

$5 * 7 + 3 < 46$ and $3 * 5 * 7 > 46$ so $46$ cannot work either.

We can continue to do this until we reach $49$ .

$49^2 = 2401$ making $ab = 390 = 2 * 3 * 5* 13$ .

$3 * 13 + 2* 5 = 49$ , so one root is $10$ and another is $39$ . The roots sum to zero, so the last root must be $-49$ .

$|-49|+10+39 = \boxed{098}$ .

Solution 3

Let us first note the obvious that is derived from Vieta's formulas: $a+b+c=0, ab+bc+ac=-2011$ . Now, due to the first equation, let us say that $a+b=-c$ , meaning that $a,b>0$ and $c<0$ . Now, since both $a$ and $b$ are greater than 0, their absolute values are both equal to $a$ and $b$ , respectively. Since $c$ is less than 0, it equals $-a-b$ . Therefore, $|c|=|-a-b|=a+b$ , meaning $|a|+|b|+|c|=2(a+b)$ . We now apply Newton's sums to get that $a^2+b^2+ab=2011$ ,or $(a+b)^2-ab=2011$ . Solving, we find that $49^2-390$ satisfies this, meaning $a+b=49$ , so $2(a+b)=\boxed{098}$ .

Solution 4

We have

As a result, we have

$a+b+c=0$

$ab+bc+ac=-2011$

$abc=-m$

So, $a=-b-c$

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

Solve $b=\frac {-c+\sqrt{c^2-4(c^2-2011)}}{2}$ and $\Delta =8044-3c^2=k^2$ , where $k$ is an integer

Cause $89<\sqrt{8044}<90$

So, after we tried for $2$ times, we get $k=88$ and $c=10$

then $b=39$ , $a=-b-c=-49$

As a result, $|a|+|b|+|c|=10+39+49=\boxed{098}$

Solution 5 (mod to help bash)

First, derive the equations $a=-b-c$ and $ab+bc+ca=-2011\implies b^2+bc+c^2=2011$ . Since the product is negative, $a$ is negative, and $b$ and $c$ positive. Now, a simple mod 3 testing of all cases shows that $b\equiv \{1,2\} \pmod{3}$ , and $c$ has the repective value. We can choose $b$ not congruent to 0, make sure you see why. Now, we bash on values of $b$ , testing the quadratic function to see if $c$ is positive. You can also use a delta argument like solution 4, but this is simpler. We get that for $b=10$ , $c=39, -49$ . Choosing $c$ positive we get $a=-49$ , so $|a|+|b|+|c|=10+29+39=\boxed{098}$ ~firebolt360

Solution 6

Note that $-c=b+a$ , so $c^2=a^2+2ab+b^2$ , or $-c^2+ab=-a^2-ab-b^2$ . Also, $ab+bc+ca=-2011$ , so $(a+b)c+ab=-c^2+ab=-2011$ . Substituting $-c^2+ab=-a^2-ab-b^2$ , we can obtain $a^2+ab+b^2=2011$ , or $\frac{a^3-b^3}{a-b}=2011$ . If it is not known that $2011$ is prime, it may be proved in $5$ minutes or so by checking all primes up to $43$ . If $2011$ divided either of $a, b$ , then in order for $a^3-b^3$ to contain an extra copy of $2011$ , both $a, b$ would need to be divisible by $2021$ . But then $c$ would also be divisible by $2011$ , and the sum $ab+bc+ca$ would clearly be divisible by $2011^2$ .

By LTE, $v_{2011}(a^3-b^3)=v_{2011}(a-b)$ if $a-b$ is divisible by $2011$ and neither $a,b$ are divisible by $2011$ . Thus, the only possibility remaining is if $a-b$ did not divide $2011$ . Let $a=k+b$ . Then, we have $(b+k)^3-b^3=2011k$ . Rearranging gives $3b(b+k)=2011-k^2$ . As in the above solutions, we may eliminate certain values of $k$ by using mods. Then, we may test values until we obtain $k=29$ , and $a=10$ . Thus, $b=39$ , $c=-49$ , and our answer is $49+39+10=098$ .

Video Solution

https://www.youtube.com/watch?v=QNbfAu5rdJI&t=26s ~ MathEx

@@ Line 2: / Line 2: @@
 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 <math>|a| + |b| + |c|</math>.
-==Solution==
+==Solution 1==
-With Vieta's formulas, we know that <math>a+b+c = 0</math>, and <math>ab+bc+ac = -2011</math>.
+From Vieta's formulas, we know that <math>a+b+c = 0</math>, and <math>ab+bc+ac = -2011</math>. Thus <math>a = -(b+c)</math>. All three of <math>a</math>, <math>b</math>, and <math>c</math> are non-zero: say, if <math>a=0</math>, then <math>b=-c=\pm\sqrt{2011}</math> (which is not an integer). <math>\textsc{wlog}</math>, let <math>|a| \ge |b| \ge |c|</math>. If <math>a > 0</math>, then <math>b,c < 0</math> and if <math>a < 0</math>, then <math>b,c > 0,</math> from the fact that <math>a+b+c=0</math>. We have <cmath>-2011=ab+bc+ac = a(b+c)+bc = -a^2+bc</cmath>
+Thus <math>a^2 = 2011 + bc</math>. We know that <math>b</math>, <math>c</math> have the same sign, so product <math>bc</math> is always positive. So <math>|a| \ge 45 = \lceil \sqrt{2011} \rceil</math>.
+Also, if we fix <math>a</math>, <math>b+c</math> is fixed, so <math>bc</math> is maximized when <math>b = c</math> . Hence, <cmath>2011 = a^2 - bc > \tfrac{3}{4}a^2 \qquad \Longrightarrow \qquad a ^2 < \tfrac{4}{3}\cdot 2011 = 2681+\tfrac{1}{3}</cmath>
-<br />
+So <math>|a| \le 51</math>. Thus we have bounded <math>a</math> as <math>45\le |a| \le 51</math>, i.e. <math>45\le |b+c| \le 51</math> since <math>a=-(b+c)</math>. Let's analyze <math>bc=(b+c)^2-2011</math>. Here is a table:
-<math>a,b,c\neq 0</math> since any one being zero will make the other two <math> \pm \sqrt{2011}</math>.
-<math>a = -(b+c)</math>. WLOG, let <math>|a| \ge |b| \ge |c|</math>.
-Then if <math>a > 0</math>, then <math>b,c < 0</math> and if <math>a < 0</math>, <math>b,c > 0</math>.
-<br />
-<math>ab+bc+ac = -2011 = a(b+c)+bc = -a^2+bc</math>
-<br />
-<math>a^2 = 2011 + bc</math>
-We know that <math>b</math>, <math>c</math> have the same sign. So <math>|a| \ge 45</math>. (<math>44^2<2011</math> and <math>45^2 = 2025</math>)
-Also, <math>bc</math> maximize when <math>b = c</math> if we fixed <math>a</math>. Hence, <math>2011 = a^2 - bc > \frac{3}{4}a^2</math>.
-So <math>a ^2 < \frac{(4)2011}{3} = 2681+\frac{1}{3}</math>.
-<math>52^2 = 2704</math> so <math>|a| \le 51</math>.
-<br />
-Now we have limited <math>a</math> to <math>45\le |a| \le 51</math>.
-Let's us analyze <math>a^2 = 2011 + bc</math>.
-<br />
-Here is a table:
 <table border = 1 cellspacing = 0 cellpadding = 5 style = "text-align:center;">
-<tr><th><math>|a|</math></th><th><math>a^2 = 2011 + bc</math></th></tr>
+<tr><th><math>|a|</math></th><th><math>bc=a^2-2011</math></th></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>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>198</math></td></tr>
-<tr><td><math>48</math></td><td><math>198 + 95 = 293</math></td></tr>
+<tr><td><math>48</math></td><td><math>293</math></td></tr>
-<tr><td><math>49</math></td><td><math>293 + 97 = 390</math></td></tr>
+<tr><td><math>49</math></td><td><math>390</math></td></tr>
 </table>
 <br />
@@ Line 99: / Line 71: @@
 <math>|-49|+10+39 = \boxed{098}</math>.
+== Solution 3 ==
+Let us first note the obvious that is derived from Vieta's formulas: <math>a+b+c=0, ab+bc+ac=-2011</math>. Now, due to the first equation, let us say that <math>a+b=-c</math>, meaning that <math>a,b>0</math> and <math>c<0</math>. Now, since both <math>a</math> and <math>b</math> are greater than 0, their absolute values are both equal to <math>a</math> and <math>b</math>, respectively. Since <math>c</math> is less than 0, it equals <math>-a-b</math>. Therefore, <math>|c|=|-a-b|=a+b</math>, meaning <math>|a|+|b|+|c|=2(a+b)</math>. We now apply Newton's sums to get that <math>a^2+b^2+ab=2011</math>,or <math>(a+b)^2-ab=2011</math>. Solving, we find that <math>49^2-390</math> satisfies this, meaning <math>a+b=49</math>, so <math>2(a+b)=\boxed{098}</math>.
+==Solution 4==
+We have
+ <math>(x-a)\cdot (x-b)\cdot (x-c)=x^3-(a+b+c)x+(ab+ac+bc)x-abc</math>
+As a result, we have
+<math>a+b+c=0</math>
+<math>ab+bc+ac=-2011</math>
+<math>abc=-m</math>
+So, <math>a=-b-c</math>
+As a result, <math>ab+bc+ac=(-b-c)b+(-b-c)c+bc=-b^2-c^2-bc=-2011</math>
+Solve <math>b=\frac {-c+\sqrt{c^2-4(c^2-2011)}}{2}</math> and
+<math>\Delta =8044-3c^2=k^2</math>, where <math>k</math> is an integer
+Cause <math>89<\sqrt{8044}<90</math>
+So, after we tried for <math>2</math> times, we get <math>k=88</math> and <math>c=10</math>
+then <math>b=39</math>, <math>a=-b-c=-49</math>
+As a result, <math>|a|+|b|+|c|=10+39+49=\boxed{098}</math>
+==Solution 5 (mod to help bash)==
+First, derive the equations <math>a=-b-c</math> and <math>ab+bc+ca=-2011\implies b^2+bc+c^2=2011</math>. Since the product is negative, <math>a</math> is negative, and <math>b</math> and <math>c</math> positive. Now, a simple mod 3 testing of all cases shows that <math>b\equiv \{1,2\} \pmod{3}</math>, and <math>c</math> has the repective value. We can choose <math>b</math> not congruent to 0, make sure you see why. Now, we bash on values of <math>b</math>, testing the quadratic function to see if <math>c</math> is positive. You can also use a delta argument like solution 4, but this is simpler. We get that for <math>b=10</math>, <math>c=39, -49</math>. Choosing <math>c</math> positive we get <math>a=-49</math>, so <math>|a|+|b|+|c|=10+29+39=\boxed{098}</math>
+~firebolt360
+==Solution 6==
+Note that <math>-c=b+a</math>, so <math>c^2=a^2+2ab+b^2</math>, or <math>-c^2+ab=-a^2-ab-b^2</math>. Also, <math>ab+bc+ca=-2011</math>, so <math>(a+b)c+ab=-c^2+ab=-2011</math>. Substituting <math>-c^2+ab=-a^2-ab-b^2</math>, we can obtain <math>a^2+ab+b^2=2011</math>, or <math>\frac{a^3-b^3}{a-b}=2011</math>. If it is not known that <math>2011</math> is prime, it may be proved in <math>5</math> minutes or so by checking all primes up to <math>43</math>. If <math>2011</math> divided either of <math>a, b</math>, then in order for <math>a^3-b^3</math> to contain an extra copy of <math>2011</math>, both <math>a, b</math> would need to be divisible by <math>2021</math>. But then <math>c</math> would also be divisible by <math>2011</math>, and the sum <math>ab+bc+ca</math> would clearly be divisible by <math>2011^2</math>.
+By LTE, <math>v_{2011}(a^3-b^3)=v_{2011}(a-b)</math> if <math>a-b</math> is divisible by <math>2011</math> and neither <math>a,b</math> are divisible by <math>2011</math>. Thus, the only possibility remaining is if <math>a-b</math> did not divide <math>2011</math>. Let <math>a=k+b</math>. Then, we have <math>(b+k)^3-b^3=2011k</math>. Rearranging gives <math>3b(b+k)=2011-k^2</math>. As in the above solutions, we may eliminate certain values of <math>k</math> by using mods. Then, we may test values until we obtain <math>k=29</math>, and <math>a=10</math>. Thus, <math>b=39</math>, <math>c=-49</math>, and our answer is <math>49+39+10=098</math>.
+==Video Solution==
+https://www.youtube.com/watch?v=QNbfAu5rdJI&t=26s ~ MathEx
 == See also ==
 {{AIME box|year=2011|num-b=14|after=Last Problem|n=I}}
 [[Category:Intermediate Algebra Problems]]
 {{MAA Notice}}

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