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# NMTC 2019 Stage II - Kaprekar (Class 7, 8) - Problems and Solutions

###### Problem 1

Let be the units place of . Prove that the decimal is a rational number and represent it as , where and are natural numbers.

###### Problem 2

(a) Find the positive integers such that .
(b) Find the positive integers such that .
(c) Using this idea, prove that we can find for any positive integer , distinct integers, such that .

###### Problem 3

Does there exist a positive integer which is a multiple of and whose sum of the digits is ? If no, prove it. If yes, give one such number.

###### Problem 4

In a triangle , the medians drawn through and are perpendicular. Then show that is the smallest side of .

###### Problem 5

Let be a triangle of area . Extend to such that to such that and to such that . Find the area of .

###### Problem 6

Find the real numbers and given that and .

###### Problem 7

The difference of the eight digit number and the eight digit number is divisible by . Prove that and .

###### Problem 8

is a parallelogram with area . is the intersection point of the diagonals of the parallelogram. is a point on . The intersection point of and is and the intersection point of and is . The sum of the areas of triangles and is . What is the area of the quadrilateral ?

###### Problem 1

Let be the units place of . Prove that the decimal is a rational number and represent it as , where and are natural numbers.

###### Problem 2

(a) Find the positive integers such that .
(b) Find the positive integers such that .
(c) Using this idea, prove that we can find for any positive integer , distinct integers, such that .

###### Problem 3

Does there exist a positive integer which is a multiple of and whose sum of the digits is ? If no, prove it. If yes, give one such number.

###### Problem 4

In a triangle , the medians drawn through and are perpendicular. Then show that is the smallest side of .

###### Problem 5

Let be a triangle of area . Extend to such that to such that and to such that . Find the area of .

###### Problem 6

Find the real numbers and given that and .

###### Problem 7

The difference of the eight digit number and the eight digit number is divisible by . Prove that and .

###### Problem 8

is a parallelogram with area . is the intersection point of the diagonals of the parallelogram. is a point on . The intersection point of and is and the intersection point of and is . The sum of the areas of triangles and is . What is the area of the quadrilateral ?

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