2014 AMC 12A Problem 25

Below is the professionally curated solution for Problem 25 of the 2014 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 12A solutions, or check the answer key.

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Concepts:parabola lattice point Diophantine Equation

Difficulty rating: 2650

25.

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3).$ For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le1000?$

$38$

$40$

$42$

$44$

$46$

Solution:

Since $(0,0)$ is the midpoint of $A=(4,3)$ and $B=(-4,-3),$ the segment $AB$ is the latus rectum, so the directrix is parallel to $AB$ at distance $5$ on the far side, namely $4y-3x+25=0.$

Equating distances to focus and directrix gives $(4x+3y)^2=25\big(25+2(4y-3x)\big).$ Writing $4x+3y=5s$ forces $5\mid s,$ and $s=5t$ forces $t$ odd; with $t=2u+1$ the integer points are $x=-6u^2+2u+4,\qquad y=8u^2+14u+3.$

Then $|4x+3y|=|50u+25|\le1000$ iff $|2u+1|\le39,$ i.e. $-20\le u\le19.$ That gives $40$ lattice points.

Thus, the correct answer is B.

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