all are pui ching math invitational competition past questions. I hv the ans but i dun hv the steps. plz work out these questions for me. thank you. 1.peter wrote down all positive integers not exceeding 2003 which have at least one ‘2’ in their digits, i.e. 2, 12, 20, 21,…,2002, 2003. how many integers did peter write down? 2. P is a point in regular hexagon. given that each side of the regular hexagon is 2rt3. if we denote the perpendicular distances of P from the sixe sides of the hexagon 3. find the largest prime factor of 9991. 4. on the rectangular coordinate plane, points whose x-coordinates and y-coordinates are both integers are called ‘lattice points’. if a circle is drawn with the origin as centre and rt1000 as radius, how many lattice points will the circumference of this circle pass through? 5. given a 7*7 grid, we remove the 5*5 grid at the upper right hand corner to obtain an L-shaped grid as shown. we want to travel from the upper left hand corner to the lower right hand corner of this L-shaped grid, but in teach step we can only move rightward or downward along the gridlines. how many different paths are there? 6. if n is a positive integer and n^2 =11112222^2+3333^2+3334^2, find n. 7. alan and betty play a game. first, alan puts 5 coins on a 13*13 square grid such that the 5 coins are in 5 neighbouring cells lined up in a row or a column. they betty starts to guess where the coins are. every time when she guesses, she chooses a cell. if the cell is occupied by a coinm she wins. what is the minimum number of guesses she has to make to ensure that she wins? 8. 15 students, three from each of secondary 1 to secondary 5, take part in a workshop. they are to be divided into groups of three, and it is required that any two members in the same group differ by at most one form. in how many different ways can we divide the students into groups?