1995 AHSME Problems/Problem 12

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Problem

Let $f$ be a linear function with the properties that $f(1) \leq f(2), f(3) \geq f(4),$ and $f(5) = 5$ . Which of the following is true?

$\mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 }$

Solution

A linear function has the property that $f(a)\leq f(b)$ either for all a<b, or for all b<a. Since $f(3)\geq f(4)$ , $f(1)\geq f(2)$ . Since $f(1)\leq f(2)$ , $f(1)=f(2)$ . And if $f(a)=f(b)$ for a≠b, then f(x) is a constant function. Since $f(5)=5$ , $f(0)=5\Rightarrow \mathrm{(D)}$

1995 AHSME (Problems)
Preceded by Problem 11	Followed by Problem 13
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1995 AHSME Problems/Problem 12

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