HMMT February, 2022

Algebra Difficulty 5.1 AIME, harder Prove it United States

Problem:

Positive integers $a$ , $b$ , and $c$ are all powers of $k$ for some positive integer $k$ . It is known that the equation $a x^{2} - b x + c = 0$ has exactly one real solution $r$ , and this value $r$ is less than $100$ . Compute the maximum possible value of $r$ .

Proposed by: Akash Das

Solution

Solution:

Note that for there to be exactly one solution, the discriminant must be $0$ , so $b^{2} - 4 a c = 0$ . Thus, $b$ is even, so $k = 2$ . Since $r = \frac{b}{2a}$ , then $r$ is also a power of $2$ , and the largest power of $2$ less than $100$ is $64$ . This is achieved by $(x - 64)^{2} = x^{2} - 128 x + 4096$ .

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