2010 AIME I Problems/Problem 6

Problem

Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers $x$ , and suppose $P(11) = 181$ . Find $P(16)$ .

Solution

$[asy] import graph; real min = -0.5, max = 2.5; pen dark = linewidth(1); real P(real x) { return 8*(x-1)^2/5+1; } real Q(real x) { return (x-1)^2+1; } real R(real x) { return 2*(x-1)^2+1; } draw(graph(P,min,max),dark); draw(graph(Q,min,max),linetype("6 2")+linewidth(0.7)); draw(graph(R,min,max),linetype("6 2")+linewidth(0.7)); dot((1,1)); label("$P(x)$",(max,P(max)),E,fontsize(10)); label("$Q(x)$",(max,Q(max)),E,fontsize(10)); label("$R(x)$",(max,R(max)),E,fontsize(10)); /* axes */ Label f; f.p=fontsize(8); xaxis(-2, 3, Ticks(f, 5, 1)); yaxis(-1, 5, Ticks(f, 6, 1)); [/asy]$

Let $Q(x) = x^2 - 2x + 2$ , $R(x) = 2x^2 - 4x + 3$ . Completing the square, we have $Q(x) = (x-1)^2 + 1$ , and $R(x) = 2(x-1)^2 + 1$ , so it follows that $P(x) \ge Q(x) \ge 1$ for all $x$ (by the Trivial Inequality).

Also, $1 = Q(1) \le P(1) \le R(1) = 1$ , so $P(1) = 1$ , and $P$ obtains its minimum at the point $(1,1)$ . Then $P(x)$ must be of the form $c(x-1)^2 + 1$ for some constant $c$ ; substituting $P(11) = 181$ yields $c = \frac 95$ . Finally, $P(16) = \frac 95 \cdot (16 - 1)^2 + 1 = \boxed{406}$ .

2010 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2010 AIME I Problems/Problem 6

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