## QBM027

Q1. In a plane a set of n points (n ≥ 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length d. The number of diameters of the given set is at most? Q2. An integer n has property P if there are integers p and q such that 0 < p < q < n and the sum p+(p+1)+. . . +q is divisible by n. then n has property P if and only if a. n is not a power of 2. b. n is a power of 2. c. n is not a power of 3. d. n is a power of 3. Q3. A triomino is an arrangement of three squares with a "corner" as sketched below -- note that the figure may be rotated through any multiple of 90 degrees _____ | | | |__|__| | | |__| What is the largest number of squares on an 8 X 8 checkerboard which can be colored green so that any triomino on the board has at least one of its squares not colored green? Q4. We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12391239 a doublestring number. Find the smallest doublestring number which is a cube. Q.5 A technician mixes a solution having water and oil in the ratio 3 : 4 with another solution having oil and turpentine in the ratio 5 : 6. How much solution should he mix so that the amount of oil in the new mixture is 9 litres? Q6. If n is a positive integer, then n^5 + 5 and (n+1)^5 + 5 are relatively prime. a. Always true b. True for specific cases c. Always false d.Cant be said Q7. In an examination, the average marks obtained by students who passed was x%, while the average of those who failed was y%. The average marks of all students taking the exam was z%. Find in terms of x, y and z, the percentage of students taking the exam who failed.? Q8. Two-fifths of the voters promise to vote for P and the rest promise to vote for Q. Of these, on the last day, 15% of the voters went back on their promise to vote for P and 25% of voters went back of their promise to vote for Q, and P lost by 2 votes. Then the total number of voters is: (1)100 (2)110 (3) 90 (4) 95 Q9. In the game of picking the parcel, 4 players stand at the corners of a square and a parcel is kept at the center of the square. As soon as the signal goes up, a player has to run and pick up the parcel and proceed towards the diagonally opposite corner. The parcel changes hand, and the third player now runs with the parcel and taking a quarter-circular pat h lands at the spot vacated by the 1st player. He then places the parcel at the center and second player takes a quarter-circular path and passes the parcel to the player on his right returns of his spot. If the distance between opposite corners is 14 m, what is the total distance traveled by the parcel? (a)36m (b)44m (c)45m (d)47m Q10. When (629)24 is divided by 21, find the remainder. (1)1 (2)2 (3)5 (4)11 Q11. Which of the following is the highest? (1)122 +92 (2) 132 +82 (3) 142 +72 (4) 152 +62 Q12. What is the largest integer x so that x3 divides 40,000? (a) 10 (b) 20 (c) 25 (d) 26 (e) None of the above Q13. In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B? Q14. In the game G(m; n), for given integers m; n with 2 ≤ m≤ n, players A and B alternately subtract any positive integer less than m from a running score which starts at n. Player A starts, and the winner is the player who brings the score to zero. For given m; n there is always one player who can force a win. Find who, and explain how. Q15. This past weekend I installed a new bi-fold door on a closet in the attic. The door consists of two panels, each one foot wide, hinged together in the middle. One side of one panel is fixed to the wall, while the other side of the other panel runs along track along the top of the door. As the door opens and closes, it rubs against a carpet on the attic floor, tracing out a path on the carpet pile. What is the equation of the path traced out by the door on the carpet? END