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## 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

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.

Categories

## 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

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.

Categories

## 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

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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