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Coordinate Geometry Problem | AIME I, 2009 Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Coordinate Geometry.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Coordinate Geometry.

Coordinate Geometry Problem – AIME 2009

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.

• is 107
• is 600
• is 840
• cannot be determined from the given information

Key Concepts

Algebra

Equations

Geometry

But try the problem first…

Source

AIME, 2009, Question 11

Geometry Revisited by Coxeter

Try with Hints

First hint

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)$$y_2$$.

Second Hint

2009.49 is odd, area OYX not integer of form k+$$\frac{1}{2}$$ where k is an integer

Final Step

41x+y=2009 taking both 25  $$\frac{25!}{2!23!}+\frac{25!}{2!23!}$$=300+300=600.

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