Difference between revisions of "2010 AMC 12B Problems/Problem 21"

(Solution 2)
Line 16: Line 16:
 
Then, plugging in values of <math>2,4,6,8,</math> we get  
 
Then, plugging in values of <math>2,4,6,8,</math> we get  
  
<math>P(2)-a=(2-1)(2-3)(2-5)(2-7)Q(2) = -15Q(2) = -2a</math>
+
<cmath>P(2)-a=(2-1)(2-3)(2-5)(2-7)Q(2) = -15Q(2) = -2a</cmath>
 +
<cmath>P(4)-a=(4-1)(4-3)(4-5)(4-7)Q(4) = 9Q(4) = -2a</cmath>
 +
<cmath>P(6)-a=(6-1)(6-3)(6-5)(6-7)Q(6) = -15Q(6) = -2a</cmath>
 +
<cmath>P(8)-a=(8-1)(8-3)(8-5)(8-7)Q(8) = 105Q(8) = -2a</cmath>
 +
<cmath>-2a=-15Q(2)=9Q(4)=-15Q(6)=105Q(8).</cmath>
 +
Thus, <math>a</math> must be a multiple of <math>\text{lcm}(15,9,15,105)=315</math>.
  
<math>P(4)-a=(4-1)(4-3)(4-5)(4-7)Q(4) = 9Q(4) = -2a</math>
+
Now we show that there exists <math>Q(x)</math> such that <math>a=315.</math> We have
 +
<cmath>Q(2)=42, Q(4)=-70, Q(6)=42, Q(8)=-6</cmath>
 +
Thus, <math>Q(x)=R(x)(x-2)(x-6)+42</math> for some <math>R(x).</math> From here it is clear that <math>Q(x)</math> exists, since we can take <math>R(x)=-8x+60.</math> <math>\blacksquare</math>
  
<math>P(6)-a=(6-1)(6-3)(6-5)(6-7)Q(6) = -15Q(6) = -2a</math>
+
Therefore, our answer is <math> \boxed{\textbf{(B)}\ 315} </math>.  
 
 
<math>P(8)-a=(8-1)(8-3)(8-5)(8-7)Q(8) = 105Q(8) = -2a</math>
 
 
 
<math>-2a=-15Q(2)=9Q(4)=-15Q(6)=105Q(8).</math>
 
Thus, the least value of <math>a</math> must be the <math>\text{lcm}(15,9,15,105)</math>.
 
Solving, we receive <math>315</math>, so our answer is <math> \boxed{\textbf{(B)}\ 315} </math>.
 
 
 
(This solution appears to be incomplete in that it only shows that <math>a</math> being a multiple of <math>315</math> is a <em>necessary</em> condition for <math>P(x)</math> to have integer coefficients.  However, it's not clear at this point that <math>a=315</math> is <em>sufficient</em> to guarantee that <math>P(x)</math> has only integer coefficients. For example, at this point how do we know that for <math>a=315</math> the equations above for <math>Q(2)</math>, <math>Q(4)</math>, <math>Q(6)</math>, and <math>Q(8)</math> don't require that, say, the second coefficient of <math>Q(x)</math> is <math>1/97</math>?)
 
  
 
== Solution 2 ==
 
== Solution 2 ==

Revision as of 10:03, 15 August 2017

Problem 21

Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that

$P(1) = P(3) = P(5) = P(7) = a$, and
$P(2) = P(4) = P(6) = P(8) = -a$.

What is the smallest possible value of $a$?

$\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

Solution 1

There must be some polynomial $Q(x)$ such that $P(x)-a=(x-1)(x-3)(x-5)(x-7)Q(x)$

Then, plugging in values of $2,4,6,8,$ we get

\[P(2)-a=(2-1)(2-3)(2-5)(2-7)Q(2) = -15Q(2) = -2a\] \[P(4)-a=(4-1)(4-3)(4-5)(4-7)Q(4) = 9Q(4) = -2a\] \[P(6)-a=(6-1)(6-3)(6-5)(6-7)Q(6) = -15Q(6) = -2a\] \[P(8)-a=(8-1)(8-3)(8-5)(8-7)Q(8) = 105Q(8) = -2a\] \[-2a=-15Q(2)=9Q(4)=-15Q(6)=105Q(8).\] Thus, $a$ must be a multiple of $\text{lcm}(15,9,15,105)=315$.

Now we show that there exists $Q(x)$ such that $a=315.$ We have \[Q(2)=42, Q(4)=-70, Q(6)=42, Q(8)=-6\] Thus, $Q(x)=R(x)(x-2)(x-6)+42$ for some $R(x).$ From here it is clear that $Q(x)$ exists, since we can take $R(x)=-8x+60.$ $\blacksquare$

Therefore, our answer is $\boxed{\textbf{(B)}\ 315}$.

Solution 2

The evenly-spaced data suggests using discrete derivatives to tackle this problem. First, note that any polynomial of degree $n$

$P(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n$

can also be written as

$P(x) = b_0 + b_1 (x-1) + b_2 (x-1)(x-2) + \ldots + b_n (x-1)(x-1) \cdots (x-n)$.

Moreover, the coefficients $a_i$ are integers for $i=0, 1, 2, \ldots n$ iff the coefficients $b_i$ are integers for $i=0, 1, 2, \ldots n$. This latter form is convenient for calculating discrete derivatives of $P(x)$.

The discrete derivative of a function $f(x)$ is the related function $\Delta f(x)$ defined as

$\Delta f(x) = f(x+1) - f(x)$.

With this definition, it's easy to see that for any positive integer $k$ we have

$\Delta [(x-1)(x-2)\cdots(x-k)] = k(x-1)(x-2)\cdots(x-[k-1])$.

This in turn allows us to use successive discrete derivatives evaluated at $x=1$ to calculate all of the coefficients $b_i$ using

$P(1)=b_0$, $\Delta P(1) = b_1$, $\Delta^2 P(1) = 2 b_2$, $\ldots$, $\Delta^7 P(1) = 7! b_7$.

We can also calculate the following table of discrete derivatives based on the data points given in the problem statement:

$x$
$1$$2$$3$$4$$5$$6$$7$$8$
$P(x)$$a$$-a$$a$$-a$$a$$-a$$a$$-a$
$\Delta P(x)$$-2a$$2a$$-2a$$2a$$-2a$$2a$$-2a$
$\Delta^2 P(x)$$4a$$-4a$$4a$$-4a$$4a$$-4a$
$\vdots$
$\Delta^7 P(x)$$-2^7 a$

Thus we can read down the column for $x=1$ to find that $k! b_k = (-2)^k a$ for $k = 0, 1, \ldots, 7$. Interestingly, even if we choose $P(x)$ to have degree greater than $7$, the $8$ coefficients of lowest order always satisfy these conditions. Moreover, it's straightforward to show that the $P(x)$ of degree $7$ with $b_k$ satisfying these conditions will fit the data given in the problem statement. Thus, to find minimal necessary and sufficient conditions on the value of $a$, we need only consider these $8$ equations. As a result, $P(x)$ with integer coefficients fitting the given data exists iff $k!$ divides $2^k a$ for $k = 0, 1, \ldots, 7$. In other words, it's necessary and sufficient that

$0! | a$,

$1! | 2a$,

$2! | 2^2 a$,

$3! | 2^3 a$,

$4! | 2^4 a$,

$5! | 2^5 a$,

$6! | 2^6 a$, and

$7! | 2^7 a$.

The last condition holds iff $7 \cdot 3 \cdot 5 \cdot 3 = 315$ divides evenly into $a$. Since such $a$ will also satisfy the first $7$ conditions, our answer is $\boxed{\textbf{(B)}\ 315}$.

See also

2010 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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