2024 AMC 10A Problems/Problem 11
Contents
[hide]Problem
How many ordered pairs of integers satisfy ?
Infinitely many
Solution 1
Note that is a nonnegative integer.
We square, rearrange, and apply the difference of squares formula to the given equation: It is clear that so Each ordered pair gives one ordered pair so there are such ordered pairs
Remark
From we get respectively.
~MRENTHUSIASM
Solution 2
Squaring both sides of the given equation gives Splitting into its factors (keep in mind it doesn't ask for positive integers, so the factors can be double negative, too) gives six cases:
.
Note that the square root in the problem doesn't have with it. Therefore, if there are two solutions, and , then these together are to be counted as one solution. The solutions expressed as are:
.
and are to be counted as one, same for and . Therefore, the solution is ~Tacos_are_yummy_1
Solution 3 (Crazy Rush)
From looking at the problem, it's obvious that are already solutions. From squaring and rearranging, we get We know that the difference between two consecutive squares is always odd, and for each pair of increasing consecutive squares, the difference starts from and increases by each time. This means that there is an existing pair, of consecutive squares that will satisfy this equation.
Also note that the answer cannot be infinity because the difference between two squares will increase as the two squares get higher, consecutive or not.
Therefore, the solutions and where and are consecutive that have a square difference of , give the answer of ~Tacos_are_yummy_1
Solution 4 (Quick)
Clearly, and are solutions. Notice that . Remembering our Pythagorean triples, we realize is a solution as well, and by extension, so is . Since 7 is not part of any other Pythagorean triple, we can conclude the answer is . ~i_am_suk_at_math_2
Solution 5 (idk why someone would ever use this)
Since in there are no restrictions of the sign of , we know that the solution will come in pairs, with negative and positive . Therefore, the answer will not be or . We can further confirm that it is not , since the problem would probably ask for an operation of and instead of the number of solutions. We can also guess it is not , since then the problem would ask for the sum of one of the variables in the solutions. It seems improbable that over the integers there would be infinite answers. So the answer is
~megaboy6679
(do not ever use this)
Video Solution by Pi Academy
https://youtu.be/ABkKz0gS1MU?si=ZQBgDMRaJmMPSSMM
Video Solution 1 by Power Solve
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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