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# NMTC 2018 Stage II - KAPREKAR (Class 7, 8) - Problems and Solutions ###### Problem 1

A lucky year is one in which at least one date, when written in the form day / month / year, has the following property. The product of the month times the day equals the last two digits of the year. For example, is a lucky year because it has the date where , but 1962 is not a lucky year as or , where is not a valid date. From 1900 to 2018 how many years are not lucky (not including and ) ? Give proper explanation for your answer.

###### Problem 2

In the figure given, and are right angles. If and and , then find .

###### Problem 3

(a) is a hexagon in which and . Its interior angle is between and and is greater than . The rest of the angles are each. What is its area?
(b) A convex polygon with ' ' sides has all angles equal to , except one angle. List all possible values of .

###### Problem 4 are distinct non-zero reals such that Find all possible values of ###### Problem 5

Find the smallest positive integer such that it has exactly different positive integer divisors including and the number itself.

###### Problem 6

(a) What is the sum of the digits of the smallest positive integer which is divisible by and has all of its digits equal to ?
(b) When is divided by the odd number , the quotient is a prime number and the remainder is . What is ?

###### Problem 7

Consider the sums Express as an irreducible fraction.

###### Problem 8

Let be real numbers, not all of them are equal. Prove that if , then .

Prove the converse, if , then .

###### Problem 1

A lucky year is one in which at least one date, when written in the form day / month / year, has the following property. The product of the month times the day equals the last two digits of the year. For example, is a lucky year because it has the date where , but 1962 is not a lucky year as or , where is not a valid date. From 1900 to 2018 how many years are not lucky (not including and ) ? Give proper explanation for your answer.

###### Problem 2

In the figure given, and are right angles. If and and , then find .

###### Problem 3

(a) is a hexagon in which and . Its interior angle is between and and is greater than . The rest of the angles are each. What is its area?
(b) A convex polygon with ' ' sides has all angles equal to , except one angle. List all possible values of .

###### Problem 4 are distinct non-zero reals such that Find all possible values of ###### Problem 5

Find the smallest positive integer such that it has exactly different positive integer divisors including and the number itself.

###### Problem 6

(a) What is the sum of the digits of the smallest positive integer which is divisible by and has all of its digits equal to ?
(b) When is divided by the odd number , the quotient is a prime number and the remainder is . What is ?

###### Problem 7

Consider the sums Express as an irreducible fraction.

###### Problem 8

Let be real numbers, not all of them are equal. Prove that if , then .

Prove the converse, if , then .

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