NMTC Stage 2 GAUSS Primary Level 2015

A unit fraction is one of the form $$\frac{1}{a}$$ where a is a natural number ($$\neq $$1). Any proper fraction can be written as the sum of two or more unit fractions.
{Ex.$$\frac{1}{2}=\frac{1}{3}+\frac{1}{6}, \frac{5}{6}=\frac{1}{2}+ \frac{1}{3}, \frac{1}{24}= \frac{1}{54}+ \frac{1}{72}+ \frac{1}{108}$$}.
In the case of $$\frac{1}{2}$$,the factors of 2 are 1 and 2 only
$$\frac{1}{2}= \frac{1+2}{2(1+2)} =\frac{1}{2(1+2)} + \frac{2}{2(1+2)} = \frac{1}{6} + \frac{1}{3}$$
Express $$\frac{1}{15}$$ as the sum of two different unit fractions in 4 different ways.

(a) The numbers 11284 and 7655, when divided by a certain 3-digit number, leave the same remainder. Find the number and the remainder. (b) What is the least number to be subtracted from 1936 so that when divided by 9, 10 and 15 it will leave the same remainder in each case?

A, M, T, I represent different non-zero digits. It is given that
$$A+M+T+I=11$$,
$$A+M+I=10$$,
$$A+M=1$$.
Further in the following addition only one digit is given.

ABC is a three digit number in which the digit A is greater than the digits B and C. If the difference between ABC and CBA is 297 and the difference between ABC and BAC is 450, find all possible three digit numbers ABC and find their sum also.

take a triangle. Three straight lines are drawn through the vertices of the triangle as shown.The maximum number of points of intersection is 3.
Draw two lines through each vertex of a triangle to meet the opposite sides. What is the maximumnumber of points of intersection.

Find again drawing the maximum number of points of intersection when three lines are drawn through each vertex.
Without drawing can you guess the maximum points of intersection for 4 lines ?