Fiji Mathematics Team Competition – National Final
Form 7 - 2010
F7/1 The sum of the first five terms of an arithmetic sequence is 40 and the sum of the first ten
terms of the sequence is 155. What is the sum of the first fifteen terms of the sequence?
F7/2 In trapezium ABCD, side AB is parallel to side DC and diagonals AC and BD intersect at P. If the area of triangle APB is 4 cm2 and the area of triangle DPC is 9 cm2. Determine the area of the trapezium ABCD.
F7/3 Half the books on a teacher’s bookshelf are Mathematics books, a third of them are Physics books and th of them are History books. The remainder of the books is romantic novels. If 2 of the Mathematics books and 4 of the Physics books are replaced by romantic novels, then romance novels will comprise 15% of the books on the bookshelf. How many books in total are there on the bookshelf?
F7/4 A rectangle is inscribed in a triangle whose sides have lengths 30,40 and 50 cm. One edge of the rectangle lies on the hypotenuse of the triangle. What is the largest possible area the rectangle can have?
F7/5 Triangle ABC has sides that measure 12, 9 and 7 cm. Three circles centered at one of the triangle’s vertices are mutually tangent as shown. (Note that the diagram is not to scale). Calculate the radius of the largest of the three circles.
F7/6 The difference between the squares of two consecutive odd integers is 128. What is the product of the two integers?
F7/7 During the Christmas season, many postal workers have to work overtime, so a supervisor at one large post office planned a late-night snack for the employee. She ordered 1 extra large pizza for every two workers, 1 large bag of potato chips for every three workers and 1 two-liter bottle of cola for every four workers. When the order arrived, 26 items were delivered. How many employees were working that evening?
F7/8 Each orange tree on a farm produces 600 orange per year if no more than 20 trees are planted per acre. For each additional tree planted per acre, the yield decreases by 15 oranges. How many trees per acre should be planted to obtain the greatest number of oranges?
F7/9 Find the sum of all positive integers which are less than 2010 and are not divisible by 3.
F7/10 Marika and Daniel decide to ride their bicycles along a predetermined route. Marika can ride at a steady 18 mph and Daniel can ride at 19.5 mph. When Daniel finishes the ride he has to wait for 20 minutes for Marika to finish his ride. If they both rode the same distance, how far did they ride?
F7/11 Five consecutive integers have the property that the sum of the first 4 is exactly three times the fifth. Find the sum of the next five consecutive integers.
F7/12 Simplify
F7/13 There are twice as many liters of water in one container as in another. If 8 liters of water are removed from each of the two containers, there will be three times as many liters of water in one container as in the other. How many liters of water did the smaller container have to begin with?
F7/14 A Pythagorean Triple is a set of three positive integers that satisfy the equation
a2 + b2 = c2. If the smallest of the three is 21 and the difference of the other two is 3, find the area of a triangle that can be constructed with these lengths.
F7/15 Mike has a bicycle repair shop and needs to know how much to charge for labor. If he charges too much he will loose customers, if he charges too little he won’t make much money. At the present, he charges $40 per hour and has 15 hours of work a week. He knows that for every $5 increase in the hourly rate his workload drops by 3 hours, and for every $5 decrease his workload goes up by 3 hours. How much should Mike charge per hour to maximize his profits?
F7/16 An open box is to be created from a nine cm by twelve cm piece of poster board by cutting congruent squares from each corner and folding up the sides. The goal is to cut squares of the size that will produce the box with maximum volume. Determine that volume to the nearest cm.
F7/17 The remote control for Maggie’s TV picks randomly one of the first ten channels 1,2,3……,10 in such a way that each channel is equally likely to be picked. Maggie’s favorite channel is channel 8. She decides to enter it repeatedly using the remote until she gets channel 8.What is the probability (in fraction) that she will have to try at most three times.
F7/18 One step has 4 walls, two steps have 10 walls and three steps have 18 walls as shown below:
If you have a set of steps with 340 walls, following the same pattern as above, how many walls must be added to construct the next set of steps in the sequence?
F7/19 In the Granny Awards there were 103 different categories and five artists were nominated in each category. Some artists were nominated in more than one category. There were 50 artists nominated in at least two categories, 35 in at least three categories, 24 in at least four categories and 12 nominated in five categories. No artist was nominated in more than five categories. What is the number of distinct artists nominated for at least one Granny Award?
F7/20 The Preliminary Round of the AH High School Maths Contest is composed of 15 multiple choice questions. If each question is awarded 0 or 1, what is the number of students that must be in the contest to be sure that at least 30 students get the same final score in the contest?
F7/21 Several people started with $400 each and played a game with the following unusual rule: Each player pays $10 to the house at the beginning of each round. During each round, one active player is declared the loser and he distributes all of his money in equal amounts to the remaining players. The loser must then leave, but all of the other players go to the next round. The game is over as soon as one player remains. At the end of the game, the surviving player was surprised to discover that he had exactly $400, equaling his starting amount. How many players were there at the beginning of the game?
F7/22 Frankie and Johnny are of the way across a bridge when they hear a train coming behind them. Frankie runs toward the train and Johnny away from it. Luckily, both just made it off the bridge in the nick of time. Afterwards they learn that the train was traveling at 100 mph. If Frankie and Johnny ran at the same rate, how fast did they run?