Difference between revisions of "2021 AIME I Problems/Problem 15"

Revision as of 10:14, 12 March 2021

Problem

Let $S$ be the set of positive integers $k$ such that the two parabolas $y=x^2-k~~\text{and}~~x=2(y-20)^2-k$ intersect in four distinct points, and these four points lie on a circle with radius at most $21$ . Find the sum of the least element of $S$ and the greatest element of $S$ .

Solution

Solution 1

With binary search you can narrow down the k value. Newton raphson method let you narrow down the x and y solution for that specific k value. With 3 (x,y) pairs you can find radius of the circle.

You end up finding the bounds of 5 and 280. The sum is 285.

~Lopkiloinm

Solution 2

Make the translation $x \rightarrow x+20$ to obtain $20+x=y^2-k , y=2x^2-k$ . Multiply the first equation by 2 and sum, we see that $2(x^2+y^2)=3k+40+2x+y$ . Completing the square gives us $(x- \frac{1}{2})^2+(y - \frac{1}{4})^2 = \frac{325+24k}{16}$ ; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that $LHS \leq 21^2=441 \rightarrow 24k \leq 6731$ , so $k \leq 280$ .

For the lower bound, we need to ensure there are 4 intersections to begin with. A quick check shows k=5 works while k=4 does not. Therefore, the answer is 5+280=285.

In general, this problem tells us that the intersection points of two conics without xy terms usually lie on a circle. When is this true/false will be left to the reader.

-Ross Gao

@@ Line 12: / Line 12: @@
 ===Solution 2===
-Make the translation <math>x \rightarrow x+20</math> to obtain <math>20+x=y^2-k , y=2x^2-k</math>. Multiply the second equation by 2 and sum, we see that <math>2(x^2+y^2)=3k+40+2x+y</math>. Completing the square gives us <math>(x- \frac{1}{2})^2+(y - \frac{1}{4})^2 = \frac{325+24k}{16}</math>; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that <math>LHS \leq 21^2=441 \rightarrow 24k \leq 6731</math>, so <math>k \leq 280</math>.
+Make the translation <math>x \rightarrow x+20</math> to obtain <math>20+x=y^2-k , y=2x^2-k</math>. Multiply the first equation by 2 and sum, we see that <math>2(x^2+y^2)=3k+40+2x+y</math>. Completing the square gives us <math>(x- \frac{1}{2})^2+(y - \frac{1}{4})^2 = \frac{325+24k}{16}</math>; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that <math>LHS \leq 21^2=441 \rightarrow 24k \leq 6731</math>, so <math>k \leq 280</math>.
 For the lower bound, we need to ensure there are 4 intersections to begin with. A quick check shows k=5 works while k=4 does not. Therefore, the answer is 5+280=285.

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