2025 AIME I Problems/Problem 4: Difference between revisions

Revision as of 09:37, 31 July 2025

Problem

Find the number of ordered pairs $(x,y)$ , where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$ .

Video solution by grogg007

https://youtu.be/PNBxBvvjbcU?t=401

Solution 1

We begin by factoring, $12x^2-xy-6y^2=(3x+2y)(4x-3y)=0.$ Since the RHS is $0$ we have two options,

$\underline{\text{Case 1:}}\text{ } 3x+2y = 0$

In this case we have, $y=\frac{-3x}{2}.$ Using the bounding on $y$ we have, $-100\le\frac{-3x}{2}\le 100.$ $\frac{200}{3}\ge x \ge \frac{-200}{3}.$ In addition in order for $y$ to be integer $2 | x,$ so we substitute $x=2k.$ $\frac{200}{3}\ge 2k \ge \frac{-200}{3}.$ $\frac{100}{3}\ge k \ge \frac{-100}{3}.$ From this we have solutions starting from $-33$ to $33$ which is $67$ solutions.

$\underline{\text{Case 2: }}\text{ } 4x-3y = 0$

On the other hand, we have, $y=\frac{4x}{3}.$ From bounds we have, $-100\le\frac{4x}{3}\le 100.$ $-75 \le x \le 75.$ In this case, for $y$ to be integer $3 | x,$ so we substitute $x=3t.$ $-75 \le 3t \le 75.$ $-25 \le t \le 25.$ This gives us $51$ solutions.

Finally we overcount one case which is the intersection of the $2$ lines or the point $(0,0).$ Therefore our answer is $67+51-1=\boxed{117}$

~mathkiddus

Solution 2

First, notice that $(0,0)$ is a solution.

Divide the equation by $y^2$ , getting $12(\frac{x}{y})^2-\frac{x}{y}-6 = 0$ . (We can ignore the $y=0$ case for now.) Let $a = \frac{x}{y}$ . We now have $12a^2-a-6=0$ . Factoring, we get $(4a-3)(3a+2) = 0$ . Therefore, the graph is satisfied when $4a=3$ or $3a=-2$ . Substituting $\frac{x}{y} = a$ back into the equations, we get $4x=3y$ or $3x=-2y$ .

Remember that both $x$ and $y$ are bounded by $-100$ and $100$ , inclusive. For $4x=3y$ , the solutions are $(-75,-100), (-72,-96), (-69, -92), \dots, (72,96), (75,100)$ . Remember to not count the $x=y=0$ case for now. There are $25$ positive solutions and $25$ negative solutions for a total of $50$ .

For $3x-2y$ , we do something similar. The solutions are $(-66,99), (-64,96), \dots, (64, -96), (66, -99)$ . There are $33$ solutions when $x$ is positive and $33$ solutions when $x$ is negative, for a total of $66$ .

Now we can count the edge case of $(0,0)$ . The answer is therefore $50+66+1 = \boxed{117}$ .

~lprado

Solution 3

You can use the quadratic formula for this equation: $12x^2 - xy - 6y^2 = 0$ ; Although this solution may seem to be misleading, it works! What we are doing is considering the quadratic with respect to $1$ , where $12x^2$ is the coefficient of $1^2$ , $-xy$ is the coefficient of $1$ , and $-6y^2$ is the constant term.

You get: $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$\frac{xy \pm \sqrt{x^2 y^2 + (12\cdot6\cdot4\cdot x^2 \cdot y^2)}}{24x^2}$

$= \frac{xy \pm\sqrt{289x^2 y^2}}{24x^2}$

$= \frac{18xy}{24x^2}\text{, and }\frac{-16xy}{24x^2}$

Rather than putting this equation as zero, the numerators and denominators must be equal [EDIT: We set each value equal to one because the quadratic is taken with respect to the root $1$ . These two equations simplify to:

$3y = 4x;$ $-2y = 3x;$

As $x$ and $y$ are between $-100$ and $100$ , for the first equation, $x$ can be from $[-75,75]$ , but $x$ must be a multiple of $3$ , so there are:

$((75+75)/3) + 1 = 51$ solutions for this case.

For $-2y = 3x:$

$x$ can be between $[-66, 66]$ , but $x$ has to be a multiple of $2$ .

Therefore, there are $(66+66)/2 + 1 = 67$ solutions for this case.

However, the one overlap would be $x = 0$ , because y would be $0$ in both solutions.

Therefore, the answer is $51+67-1 = \boxed{117}.$

-U-King3.14Root -LaTeX corrected by Andrew2019 -clarified by golden_star_123

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=J-0BapU4Yuk

Video Solution by yjtest

https://www.youtube.com/watch?v=P7LEfKK1Vew

@@ Line 43: / Line 43: @@
 You can use the quadratic formula for this equation: <math>12x^2 - xy - 6y^2 = 0</math>;
-Although this solution may seem to be misleading, it works!
+Although this solution may seem to be misleading, it works! What we are doing is considering the quadratic with respect to <math>1</math>, where <math>12x^2</math> is the coefficient of <math>1^2</math>, <math>-xy</math> is the coefficient of <math>1</math>, and <math>-6y^2</math> is the constant term.
 You get: <cmath>\frac{-b \pm \sqrt{b^2-4ac}}{2a}</cmath>
@@ Line 55: / Line 55: @@
 <cmath>= \frac{18xy}{24x^2}\text{, and }\frac{-16xy}{24x^2}</cmath>
-Rather than putting this equation as zero, the numerators and denominators must be equal. These two equations simplify to:
+Rather than putting this equation as zero, the numerators and denominators must be equal [EDIT: We set each value equal to one because the quadratic is taken with respect to the root <math>1</math>. These two equations simplify to:
 <cmath>3y = 4x;</cmath> <cmath>-2y = 3x;</cmath>
@@ Line 72: / Line 72: @@
 Therefore, the answer is <math>51+67-1 = \boxed{117}.</math>
 -U-King3.14Root
 -LaTeX corrected by Andrew2019
+-clarified by golden_star_123
 ==Video Solution 1 by SpreadTheMathLove==

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