Difference between revisions of "1998 AIME Problems/Problem 2"
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It should be noted that the cases for <math>x\le15</math> and <math>x>15</math> should be considered separately in order to ensure that <math>2y < 60</math>. | It should be noted that the cases for <math>x\le15</math> and <math>x>15</math> should be considered separately in order to ensure that <math>2y < 60</math>. | ||
− | ==Solution 4== | + | ==Solution 4 - Unrigorous engineers induction solution== |
We will try out small cases. | We will try out small cases. |
Revision as of 00:27, 3 June 2020
Problem
Find the number of ordered pairs of positive integers that satisfy
and
.
Contents
Solution
Solution 1
Pick's theorem states that:
![$A = I + \frac B2 - 1$](http://latex.artofproblemsolving.com/7/c/b/7cba92d5eeba4957d9f6127c3a7243cf3cbe15ae.png)
The conditions give us four inequalities: ,
,
,
. These create a quadrilateral, whose area is
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 .
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
corresponds with an integer value of
as
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.
![$450 = I + \frac {60}2 - 1$](http://latex.artofproblemsolving.com/e/b/b/ebb7fcc51788e8575f7924fc60da042ab8f4ade4.png)
![$I = 421$](http://latex.artofproblemsolving.com/e/3/8/e385581c8c77f9dcb0b15e14772efc79d72970e8.png)
Since the inequalities also include the equals case, we include the boundaries, which gives us ordered pairs. However, the question asks us for positive integers, so
doesn't count; hence, the answer is
.
Solution 2
First, note that all pairs of the form ,
work.
Now, considered the ordered pairs with , so that
is automatically satisfied. Since
, there are
possible values of
. Hence, given
, there are
values of possible
for which
and the above conditions are satisfied. But
, so this only works for
. Thus, there are
ordered pairs. For ,
must follow
. Hence, there are
possibilities for
, and there are
ordered pairs.
By symmetry, there are also ordered pairs with
and the above criteria satisfied.
Hence, the total is
Solution 3
Multiplying both sides by 2 yields:
Then the two inequalities can be merged to form the following inequality:
Additionally, we must ensure that
Therefore we must find pairs that satisfy the inequality above.
A bit of trial and error and observing patterns leads to the answer
.
It should be noted that the cases for and
should be considered separately in order to ensure that
.
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 (Problems • Answer Key • Resources) | ||
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.