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# NMTC 2018 Stage II - GAUSS (Class 5, 6) - Problems and Solutions

###### Problem 1

Write down all the ten digit numbers whose digital sum is . (The digital sum of a number is the sum of the digits of the number. The digital sum of is is ). Find the sum of all the digit numbers with digital sum .

###### Problem 2

The sum of the -digit numbers and is divisible by . Find all possible pairs .

###### Problem 3

Three congruent circles with centres and , are tangent to the sides of rectangle as shown. The circle with centre at has diameter and passes through the points and . Find the area of the rectangle .

###### Problem 4

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, 1944 is a lucky year because it has the date where . From to how many years are not lucky ? Give proper explanation for your answer.

###### Problem 5

The area of each of the four congruent L-shaped regions of this by square is of the total area. How many centimeters long is the side of the centre square?

###### Problem 6

For any positive integer is the sum of the digits of . What is the minimum value of when (1) and (2) .

###### Problem 7

A digit number is obtained by writing the digit numbers to i.e., . You have to remove digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in if we remove the four 's we get the number , but if we remove and the after , we get . This will be the largest number possible in this case.) What will be the first 10 digits of the largest number obtained?

###### Problem 8

Given the numbers and is defined as the remainder when the ordinary product is divided by . Find the product of every pair of these numbers including the product of number with itself. Fill in the table given below.
(1) Find , where we find the product of fifteen .
(2) Find where we have ten

###### Problem 1

Write down all the ten digit numbers whose digital sum is . (The digital sum of a number is the sum of the digits of the number. The digital sum of is is ). Find the sum of all the digit numbers with digital sum .

###### Problem 2

The sum of the -digit numbers and is divisible by . Find all possible pairs .

###### Problem 3

Three congruent circles with centres and , are tangent to the sides of rectangle as shown. The circle with centre at has diameter and passes through the points and . Find the area of the rectangle .

###### Problem 4

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, 1944 is a lucky year because it has the date where . From to how many years are not lucky ? Give proper explanation for your answer.

###### Problem 5

The area of each of the four congruent L-shaped regions of this by square is of the total area. How many centimeters long is the side of the centre square?

###### Problem 6

For any positive integer is the sum of the digits of . What is the minimum value of when (1) and (2) .

###### Problem 7

A digit number is obtained by writing the digit numbers to i.e., . You have to remove digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in if we remove the four 's we get the number , but if we remove and the after , we get . This will be the largest number possible in this case.) What will be the first 10 digits of the largest number obtained?

###### Problem 8

Given the numbers and is defined as the remainder when the ordinary product is divided by . Find the product of every pair of these numbers including the product of number with itself. Fill in the table given below.
(1) Find , where we find the product of fifteen .
(2) Find where we have ten

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