Q1.

What is the least positive integer that is not a factor of 25! and is not a prime number?

Q2.

Let n>3 be a positive integer. Equilateral triangle ABC is divided into n^2 (N to be power 2) smaller congruent equilateral triangles (with sides parallel to its sides). Let m be the number of rhombuses that contain two small equilateral triangles and d the number of rhombuses that contain eight small equilateral triangles. Find the difference m-d in terms of n.

Q3.

A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?

Q4.

Let r, s be two positive integers and P a 'chessboard' with r rows and s columns. Let M denote the maximum value of rooks placed on P such that no two of them attack each other. Determine M.

Q5.

In the figure below, there are 4 distinct dots A, B, C, and D, joined by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many completed colorings are possible? Fig. image link: https://wiki-images.artofproblemsolving.com//6/69/Usc93.19.PNG