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

Problem 1

Write down all the ten digit numbers whose digital sum is $2$ . (The digital sum of a number is the sum of the digits of the number. The digital sum of $4022$ is $4+0+2+2$ is $8$ ). Find the sum of all the $10$ digit numbers with digital sum $2$ .

Problem 2

The sum of the $3$ -digit numbers $35 a$ and $4 b 7$ is divisible by $36$ . Find all possible pairs $(a, b)$ .

Problem 3

Three congruent circles with centres $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ , are tangent to the sides of rectangle $\mathrm{ABCD}$ as shown. The circle with centre at $Q$ has diameter $5 \mathrm{~cm}$ and passes through the points $P$ and $R$ . Find the area of the rectangle $A B C D$ .

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 $11 / 4 / 44$ where $11 \times 4=44$ . From $1951$ to $2000$ 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 $100 \mathrm{~cm}$ by $100 \mathrm{~cm}$ square is $\frac{3}{16}$ of the total area. How many centimeters long is the side of the centre square?

Problem 6

For any positive integer $n, s(n)$ is the sum of the digits of $n$ . What is the minimum value of $\frac{n}{s(n)}$ when (1) $10 \leq n \leq 99$ and (2) $100 \leq n \leq 999$ .

Problem 7

A $122$ digit number is obtained by writing the $2$ digit numbers $39$ to $99$ i.e., $39404142434445 . . . . . .96979899$ . You have to remove $61$ digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in $15161718$ if we remove the four $1$ 's we get the number $5678$ , but if we remove $1,5,1$ and the $1$ after $6$ , we get $6718$ . 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 $2,4,8,10,14$ and $16: a \% b$ is defined as the remainder when the ordinary product $a \cdot b$ is divided by $18$ . Find the $\%$ product of every pair of these numbers including the product of $a$ number with itself. Fill in the table given below.
(1) Find $2 \% 2 \% 2 \% \ldots \% 2$ , where we find the $\%$ product of fifteen $2 's$ .
(2) Find $8 \% 8 \% 8 \% \ldots . \ldots 8$ where we have ten $8 's$

Problem 1

Write down all the ten digit numbers whose digital sum is $2$ . (The digital sum of a number is the sum of the digits of the number. The digital sum of $4022$ is $4+0+2+2$ is $8$ ). Find the sum of all the $10$ digit numbers with digital sum $2$ .

Problem 2

The sum of the $3$ -digit numbers $35 a$ and $4 b 7$ is divisible by $36$ . Find all possible pairs $(a, b)$ .

Problem 3

Three congruent circles with centres $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ , are tangent to the sides of rectangle $\mathrm{ABCD}$ as shown. The circle with centre at $Q$ has diameter $5 \mathrm{~cm}$ and passes through the points $P$ and $R$ . Find the area of the rectangle $A B C D$ .

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 $11 / 4 / 44$ where $11 \times 4=44$ . From $1951$ to $2000$ 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 $100 \mathrm{~cm}$ by $100 \mathrm{~cm}$ square is $\frac{3}{16}$ of the total area. How many centimeters long is the side of the centre square?

Problem 6

For any positive integer $n, s(n)$ is the sum of the digits of $n$ . What is the minimum value of $\frac{n}{s(n)}$ when (1) $10 \leq n \leq 99$ and (2) $100 \leq n \leq 999$ .

Problem 7

A $122$ digit number is obtained by writing the $2$ digit numbers $39$ to $99$ i.e., $39404142434445 . . . . . .96979899$ . You have to remove $61$ digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in $15161718$ if we remove the four $1$ 's we get the number $5678$ , but if we remove $1,5,1$ and the $1$ after $6$ , we get $6718$ . 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 $2,4,8,10,14$ and $16: a \% b$ is defined as the remainder when the ordinary product $a \cdot b$ is divided by $18$ . Find the $\%$ product of every pair of these numbers including the product of $a$ number with itself. Fill in the table given below.
(1) Find $2 \% 2 \% 2 \% \ldots \% 2$ , where we find the $\%$ product of fifteen $2 's$ .
(2) Find $8 \% 8 \% 8 \% \ldots . \ldots 8$ where we have ten $8 's$