Difference between revisions of "1994 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
| − | For certain ordered pairs <math>(a,b)\,</math> of real | + | For certain ordered pairs <math>(a,b)\,</math> of [[real number]]s, the system of equations |
<center><math>ax+by=1\,</math></center> | <center><math>ax+by=1\,</math></center> | ||
<center><math>x^2+y^2=50\,</math></center> | <center><math>x^2+y^2=50\,</math></center> | ||
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== Solution == | == Solution == | ||
| − | <math>x^2+y^2=50</math> is the equation of a circle of radius <math>\sqrt{50}</math>, centered at the origin. The [[lattice points]] on this circle are <math>(\ | + | <math>x^2+y^2=50</math> is the equation of a circle of radius <math>\sqrt{50}</math>, centered at the origin. The [[lattice points]] on this circle are <math>(\pm1,\pm7)</math>, <math>(\pm5,\pm5)</math>, and <math>(\pm7,\pm1)</math>. |
<math>ax+by=1</math> is the equation of a line that does not pass through the origin. (Since <math>(x,y)=(0,0)</math> yields <math>a(0)+b(0)=0 \neq 1</math>). | <math>ax+by=1</math> is the equation of a line that does not pass through the origin. (Since <math>(x,y)=(0,0)</math> yields <math>a(0)+b(0)=0 \neq 1</math>). | ||
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Since there is a unique tangent line to the circle at each of these lattice points, there are <math>12</math> distinct lines which pass through exactly one lattice point on the circle. | Since there is a unique tangent line to the circle at each of these lattice points, there are <math>12</math> distinct lines which pass through exactly one lattice point on the circle. | ||
| − | Thus, there are a total of <math>66-6+12=\boxed{ | + | Thus, there are a total of <math>66-6+12=\boxed{072}</math> distinct lines which pass through either one or two of the <math>12</math> lattice points on the circle, but do not pass through the origin. |
== See also == | == See also == | ||
{{AIME box|year=1994|num-b=6|num-a=8}} | {{AIME box|year=1994|num-b=6|num-a=8}} | ||
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| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 17:44, 16 October 2008
Problem
For certain ordered pairs 
of real numbers, the system of equations
has at least one solution, and each solution is an ordered pair 
of integers. How many such ordered pairs are there?
Solution
is the equation of a circle of radius 
, centered at the origin. The lattice points on this circle are 
, 
, and 
.
is the equation of a line that does not pass through the origin. (Since 
yields 
).
So, we are looking for the number of lines which pass through either one or two of the 
lattice points on the circle, but do not pass through the origin.
It is clear that if a line passes through two opposite points, then it passes through the origin, and if a line passes through two non-opposite points, the it does not pass through the origin.
There are 
ways to pick two distinct lattice points, and thus 
distinct lines which pass through two lattice points on the circle. However, 
of these lines pass through the origin.
Since there is a unique tangent line to the circle at each of these lattice points, there are distinct lines which pass through exactly one lattice point on the circle.
Thus, there are a total of 
distinct lines which pass through either one or two of the lattice points on the circle, but do not pass through the origin.
See also
| 1994 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
