1994 AIME Problems/Problem 7
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... The vertices are chosen from the points in a grid as shown below. Since there are points in the grid and triangles consist of distinct vertices, we get an initial count of [asy] size(100); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1));
dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2));
dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3));
dot((1,4)); dot((2,4)); dot((3,4)); dot((4,4));
[/asy]
However, degenerate triangles (those with area) are precisely those formed by collinear points, so we have to remove these from the count. In each column or row, there are sets of collinear points. There are total columns and rows, so this gives us invalid triangles.
There are also such sets of collinear points in each of the two main diagonals and for each of the shorter diagonals. This means that we counted a total ofinvalid triangles in our initial computation of triangles, so the answer is .
Source: 1993 AHSME #28 Hint(s): There are probably more degenerate triangles than you think. Your Response(s): (a,b)$of real numbers, the system of equations\begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*}has at least one solution, and each solution is an ordered pair$ (Error compiling LaTeX. Unknown error_msg)(x,y)(a,b)x^2+y^2=50\sqrt{50}(\pm1,\pm7)(\pm5,\pm5)(\pm7,\pm1)3\cdot 2\cdot 2=12$lattice points. They are indicated by the blue dots below.
[asy] size(150); draw(circle((0,0),sqrt(50)));
draw((1,7)--(-1,-7),red); draw((7,1)--(5,-5), green);
dot((0,0));
dot((1,7),blue); dot((1,-7),blue); dot((-1,7),blue); dot((-1,-7),blue);
dot((5,5),blue); dot((5,-5),blue); dot((-5,5),blue); dot((-5,-5),blue);
dot((7,1),blue); dot((7,-1),blue); dot((-7,1),blue); dot((-7,-1),blue); [/asy]
Since$ (Error compiling LaTeX. Unknown error_msg)(x,y)=(0,0)a\cdot 0+b\cdot 0=0 \neq 1ax+by=112(p,q)(-p,-q)$through which it passes. And example is the red line above.
There are$ (Error compiling LaTeX. Unknown error_msg)\binom{12}{2}=6666(p,q)(-p,-q)\frac{12}{2}=612$ unique tangent lines to the circle at each of the lattice points.
Therefore, our final count of distinct lines which pass through one or two of the lattice points on the circle, but do not pass through the origin, is
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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