Let . Given distinct ordered pairs of non-negative integers , we wish to maximize the number of pairs with such that . Find the maximum possible number of such pairs.
The maximum possible number of such pairs is 198. Geometrically, the condition is equivalent to the triangle formed by the origin and points being a primitive lattice triangle of area . By Pick's Theorem and Euler's formula for planar graphs, the maximum number of such primitive triangles that can be formed by points is . For , this yields , which is achieved by the points and for .
The maximum possible number of such pairs is 197. Geometrically, the condition is equivalent to the triangle formed by the origin and points being a primitive lattice triangle of area . By Pick's Theorem and Euler's formula for planar graphs, the maximum number of such primitive triangles that can be formed by points is . For , this yields , which is achieved by the points and for .
The maximum possible number of such pairs is 196. Geometrically, the condition is equivalent to the triangle formed by the origin and points being a primitive lattice triangle of area . By Pick's Theorem and Euler's formula for planar graphs, the maximum number of such primitive triangles that can be formed by points is . For , this yields , which is achieved by the points and for .
The maximum possible number of such pairs is 199. Geometrically, the condition is equivalent to the triangle formed by the origin and points being a primitive lattice triangle of area . By Pick's Theorem and Euler's formula for planar graphs, the maximum number of such primitive triangles that can be formed by points is . For , this yields , which is achieved by the points and for .
Hint 1: Interpret the expression geometrically. Note that it is exactly twice the area of the triangle .
Hint 2: Use Pick's Theorem: the minimum area of a non-degenerate lattice triangle is . Thus, means the triangle has no other lattice points.
Hint 3: Formulate this as a triangulation problem of points in a coordinate plane. Use Euler's formula to bound the number of triangles by , and construct a case achieving this.
Step 1 (General Formula): For distinct points, the maximum number of such pairs is .
For , the maximum number of pairs is:
Step 2 (Geometric Interpretation): Let be points in the Cartesian plane. The condition is exactly twice the area of the triangle formed by the origin and the two points and . Thus, the condition is equivalent to .
By Pick's Theorem, a lattice triangle has area at least , and exactly if and only if it contains no other lattice points in its interior or boundary (a fundamental or primitive triangle).
Step 3 (Bounding): The set of segments divides the plane into sectors. The maximum number of primitive triangles sharing the origin that can be formed by points corresponds to a planar triangulation in coordinate space. By Euler's formula for planar graphs, the maximum number of such triangles is bounded by .
Step 4 (Construction): We can achieve exactly pairs using the following set of 100 points:
The origin , the point , and any point for form triangles of area .
The origin and any two consecutive points and for form triangles of area .
This gives a total of pairs.
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