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Explore the Back-StoryA 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.

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

(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 .

are distinct non-zero reals such that

Find all possible values of

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

(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 ?

Consider the sums

Express as an irreducible fraction.

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

Prove the converse, if , then .

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.

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

(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 .

are distinct non-zero reals such that

Find all possible values of

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

(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 ?

Consider the sums

Express as an irreducible fraction.

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

Prove the converse, if , then .

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