NMTC Stage 2 KAPREKAR Sub Junior Level 2018

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 1

Report

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, 1956 is a lucky year because it has the date 7 / 8 / 56 where $$7 \times 8 = 56$$, but 1962 is not a lucky year as $$62 = 62 \times 1$$ or $$31 \times 2$$, where 31/2/1962 is not a valid date. From 1900 to 2018 how many years are not lucky (not including 1900 and 2018)? Give proper explanation for your answer.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 2

Report

In the figure given, A, B and C are right angles. If $$\angle AEB = 40^\circ$$ and $$\angle BED = \angle BDE$$, then find $$\angle CDE$$.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 3

Report

(a) ABCDEF is a hexagon in which $$AB=BC=CD=DE=2$$ and $$EF=FA=1$$. Its interior angle C is between $$90^\circ$$ and $$180^\circ$$ and F is greater than $$180^\circ$$. The rest of the angles are $$90^\circ$$ each. What is its area?
(b) A convex polygon with n sides has all angles equal to $$150^\circ$$, except one angle. List all possible values of n.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 4

Report

a, b, c are distinct non-zero reals such that $$\frac{1+a^3}{a} = \frac{1+b^3}{b} = \frac{1+c^3}{c}$$. Find all possible values of $$a^3+b^3+c^3$$.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 5

Report

Find the smallest positive integer such that it has exactly 100 different positive integer divisors, including 1 and the number itself.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 6

Report

(a) What is the sum of the digits of the smallest positive integer which is divisible by 99 and has all of its digits equal to 2? (b) When 270 is divided by the odd number n, the quotient is a prime number and the remainder is 0. What is n?

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 7

Report

Consider the sums $$A = \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{99 \cdot 100}$$ and $$B = \frac{1}{51 \cdot 100} + \frac{1}{52 \cdot 99} + \cdots + \frac{1}{100 \cdot 51}$$. Express $$\frac{A}{B}$$ as an irreducible fraction.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution

NMTC Stage 2 KAPREKAR Sub Junior Level 2018 - Question 8

Report

Let a, b, c be real numbers, not all of them equal. Prove that if $$a+b+c=0$$, then $$a^2+ab+b^2 = b^2+bc+c^2 = c^2+ca+a^2$$. Prove the converse: if $$a^2+ab+b^2 = b^2+bc+c^2 = c^2+ca+a^2$$, then $$a+b+c=0$$.

Backspace
789
456
123
0.-
Clear All

Submit

View Solution