Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Coordinate Geometry.
Consider the set of all triangles OPQ where O is the origin and P and Q are distinct points in the plane with non negative integer coordinates (x,y) such that 41x+y=2009 . Find the number of such distinct triangles whose area is a positive integer.
Algebra
Equations
Geometry
Answer: is 600.
AIME, 2009, Question 11
Geometry Revisited by Coxeter
let P and Q be defined with coordinates; P=(\(x_1,y_1)\) and Q(\(x_2,y_2)\). Let the line 41x+y=2009 intersect the x-axis at X and the y-axis at Y . X (49,0) , and Y(0,2009). such that there are 50 points.
here [OPQ]=[OYX]-[OXQ] OY=2009 OX=49 such that [OYX]=\(\frac{1}{2}\)OY.OX=\(\frac{1}{2}\)2009.49 And [OYP]=\(\frac{1}{2}\)\(2009x_1\) and [OXQ]=\(\frac{1}{2}\)(49)\(y_2\).
2009.49 is odd, area OYX not integer of form k+\(\frac{1}{2}\) where k is an integer
41x+y=2009 taking both 25 \(\frac{25!}{2!23!}+\frac{25!}{2!23!}\)=300+300=600.
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