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AIME I Algebra Arithmetic Coordinate Geometry Math Olympiad USA Math Olympiad

Ordered pair Problem | AIME I, 1987 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Ordered pair.

Ordered pair Problem – AIME I, 1987


An ordered pair (m,n) of non-negative integers is called simple if the additive m+n in base 10 requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to 1492.

  • is 107
  • is 300
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Ordered pair

Algebra

Check the Answer


Answer: is 300.

AIME I, 1987, Question 1

Elementary Algebra by Hall and Knight

Try with Hints


for no carrying required

the range of possible values of any digit m is from 0 to 1492 where the value of n is fixed

Number of ordered pair (1+1)(4+1)(9+1)(2+1)

=(2)(5)(10)(3)

=300.

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Categories
AIME I Algebra Arithmetic Coordinate Geometry Math Olympiad USA Math Olympiad

GCD and Ordered pair | AIME I, 1995 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on GCD and Ordered pair.

GCD and Ordered pair – AIME I, 1995


Find number of ordered pairs of positive integers (x,y) with \(y \lt x \leq 100\) are both \(\frac{x}{y}\) and \(\frac{x+1}{y+1}\) integers.

  • is 107
  • is 85
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

GCD

Ordered pair

Check the Answer


Answer: is 85.

AIME I, 1995, Question 8

Elementary Number Theory by David Burton

Try with Hints


here y|x and (y+1)|(x+1) \(\Rightarrow gcd(y,x)=y, gcd(y+1,x+1)=y+1\)

\(\Rightarrow gcd(y,x-y)=y, gcd(y+1,x-y)=y+1\)

\(\Rightarrow y,y+1|(x-y) and gcd (y,y+1)=1\)

\(\Rightarrow y(y+1)|(x-y)\)

here number of multiples of y(y+1) from 0 to 100-y \((x \leq 100)\) are

[\(\frac{100-y}{y(y+1)}\)]

\(\Rightarrow \displaystyle\sum_{y=1}^{99}[\frac{100-y}{y(y+1)}\)]=49+16+8+4+3+2+1+1+1=85.

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Coordinate Geometry Geometry Math Olympiad USA Math Olympiad

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.

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

Check the Answer


Answer: is 600.

AIME, 2009, Question 11

Geometry Revisited by Coxeter

Try with Hints


 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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