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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>(\pm1,\pm7)</math>, <math>(\pm5,\pm5)</math>, and <math>(\pm7,\pm1)</math>.
| + | bro whoever came up with this problem must've been drunk tbh, there are literally infinite solutions... |
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| − | <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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| − | So, we are looking for the number of 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.
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| − | 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.
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| − | There are <math>\binom{12}{2}=66</math> ways to pick two distinct lattice points, and thus <math>66</math> distinct lines which pass through two lattice points on the circle. However, <math>\frac{12}{2}=6</math> of these lines pass through the origin.
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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.
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| − | Thus, there are a total of <math>66-6+12=\boxed{72}</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.
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| | == See also == | | == See also == |
Revision as of 22:46, 15 September 2023
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
bro whoever came up with this problem must've been drunk tbh, there are literally infinite solutions...
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
of real numbers, the system of equations
of integers. How many such ordered pairs 
