1998 AIME Problems/Problem 2

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Problem

Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$.

Solution

Solution 1

AIME 1998-2.png

Pick's theorem states that:

$A = I + \frac B2 - 1$

The conditions give us four inequalities: $x \le 30$, $y\le 30$, $x \le 2y$, $y \le 2x$. These create a quadrilateral, whose area is $\frac 12$ of the 30 by 30 square it is in. A simple way to see this is to note that the two triangles outside of the quadrilateral form half of the area of the 30 by 30 square.

So $A = \frac 12 \cdot 30^2 = 450$. $B$ we can calculate by just counting. Ignoring the vertices, the top and right sides have 14 lattice points, and the two diagonals each have 14 lattice points (for the top diagonal, every value of $x$ corresponds with an integer value of $y$ as $y = 2x$ and vice versa for the bottom, so and there are 14 values for x not counting vertices). Adding the four vertices, there are 60 points on the borders.

$450 = I + \frac {60}2 - 1$
$I = 421$

Since the inequalities also include the equals case, we include the boundaries, which gives us $421 + 60 = 481$ ordered pairs. However, the question asks us for positive integers, so $(0,0)$ doesn't count; hence, the answer is $480$.

Solution 2

First, note that all pairs of the form $(a,a)$, $1\le a\le30$ work.

Now, considered the ordered pairs with $x < y$, so that $x < 2y$ is automatically satisfied. Since $x < y\le 2x$, there are $2x - x = x$ possible values of $y$. Hence, given $x$, there are $x$ values of possible $y$ for which $x < y$ and the above conditions are satisfied. But $2y\le60$, so this only works for $x\le15$. Thus, there are

$\sum_{i=1}^{15} i=\frac{(15)(16)}{2}$

ordered pairs. For $x > 15$, $y$ must follow $x < y\le 30$. Hence, there are $30 - x$ possibilities for $y$, and there are

$\sum_{i=16}^{30}(30-i)=\sum_{i=0}^{14}i=\frac{(14)(15)}{2}$

ordered pairs.

By symmetry, there are also $\frac {(15)(16)}{2} + \frac {(14)(15)}{2}$ ordered pairs with $x > y$ and the above criteria satisfied.

Hence, the total is

$\frac{(15)(16)}{2}+\frac{(14)(15)}{2}+\frac{(15)(16)}{2}+\frac{(14)(15)}{2}+30=480.$

Solution 3

$y\le2x\le60$

Multiplying both sides by 2 yields:

$2y\le4x\le120$

Then the two inequalities can be merged to form the following inequality:

$x\le2y\le4x\le120$

Additionally, we must ensure that $2y<60$

Therefore we must find pairs $(x,y)$ that satisfy the inequality above. A bit of trial and error and observing patterns leads to the answer $480$.

It should be noted that the cases for $x\le15$ and $x>15$ should be considered separately in order to ensure that $2y < 60$.

Solution 4 - Unrigorous engineers induction solution

We will try out small cases.

By replacing 60 in this problem with 2, we count only 1 ordered pair. By doing with 4, we count 4 ordered pairs. By continuing on, we suspect that if 60 was replaced with 2n, we will find 1+3+3+5+5+7+7 ...., where there will be n terms. Thus, our answer is 1+3+3+5+5.... 29+29+31 = 16*30 = 480.

See also

1998 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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