2005 AIME II Problems/Problem 13

Problem

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2.$

Solution

We define $Q(x)=P(x)-x+7$, noting that it has roots at $17$ and $24$. Hence $P(x)-x+7=A(x-17)(x-24)$. In particular, this means that $P(x)-x-3=A(x-17)(x-24)-10$. Therefore, $x=n_1,n_2$ satisfy $A(x-17)(x-24)=10$, where $A$, $(x-17)$, and $(x-24)$ are integers. This cannot occur if $x\le 17$ or $x\ge 24$ because the product $(x-17)(x-24)$ will either be too large or not be a divisor of $10$. We find that $x=19$ and $x=22$ are the only values that allow $(x-17)(x-24)$ to be a factor of $10$. Hence the answer is $19\cdot 22=\boxed{418}$.

Solution 2

We know that $P(n)-(n+3)=0$ so $P(n)$ has two distinct solutions so $P(x)$ is at least quadratic. Let us first try this problem out as if $P(x)$ is a quadratic polynomial. Thus $P(n)-(n+3)= an^2+(b-1)n+(c-3)=0$ because $P(n)=an^2+bn+c$ where $a,b,c$ are all integers. Thus $P(x)=ax^2+bx+c$ where $a,b,c$ are all integers. We know that $P(17)$ or $289a+17b+c=10$ and $P(24)$ or $576a+24b+c=17$. By doing $P(24)-P(17)$ we obtain that $287a+7b=7$ or $41a+b=1$ or $-41a=b-1$. Thus $P(n)=an^2- (41a)n+(c-3)=0$. Now we know that $b=-41a+1$, we have $289a+17(-41a+1)+c=10$ or $408a=7+c$ which makes $408a-10=c-3$. Thus $P(n)=an^2-(41a)n+(408a-10)=0$. By Vieta's formulas, we know that the sum of the roots($n$) is equal to 41 and the product of the roots($n$) is equal to $408-\frac{10}{a}$. Because the roots are integers $\frac{10}{a}$ has to be an integer, so $a=1,2,5,10,-1,-2,-5,-10$. Thus the product of the roots is equal to one of the following: $398,403,406,407,409,410,413,418$. Testing every potential product of the roots, we find out that the only product that can have divisors that sum up to $41$ is $\boxed{418}$.

-David Camacho

Solution 3

We have $P(n_1) = n_1+3$. Using the property that $a - b \mid P(a) - P(b)$ whenever $a$ and $b$ are distinct integers, we get $$n_1 - 17 \mid P(n_1) - P(17) = (n_1+3) - 10 = n_1 - 7,$$and $$n_1 - 24 \mid P(n_1) - P(24) = (n_1+3)-17=n_1-14.$$Since $n_1 - 7 = 10 + (n_1-17)$ and $n_1-14 = 10 + (n_1-24)$, we must have $$n_1 - 17 \mid 10 \; \text{and} \; n_1-24 \mid 10.$$We look for two divisors of $10$ that differ by $7$; we find that $\{2, -5\}$ and $\{5, -2\}$ satisfies these conditions. Therefore, either $n_1 - 24 = -5$, giving $n_1 = 19$, or $n_1 - 24 = -2$, giving $n_1 = 22$. From this, we conclude that $n_1, n_2 = \boxed{19, 22}$ and that $n_1\cdot n_2=\boxed{418}$.

~ Alcumus (Solution)