6. Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.
10. Using the letters A, B, C and D it is possible to write 64 “words” of three letters. Three of these words are ABB, BAD and CCC. When the 64 words are placed in alphabetical order, what is the 19th word?
14. The degree measure of the sum of the interior angles of a convex polygon with n sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with n + 2 sides?
24. An 8-by-8 square is divided into 64 unit squares and each unit square is painted one of four possible colors. The ratio of the numbers of squares of each color is 1:2:3: x . How many integer values could x have?
28. Point ( x , y ) is randomly picked from the rectangular region with vertices at (0, 0), (2008, 0), (2008, 2009) and (0, 2009). What is the probability that x > 2 y ? Express your answer as a common fraction.
34. A book’s pages are numbered with integers 1 through n . When the book is open to the exact middle, the product of the two page numbers showing is 1980. What is the value of n , assuming that all pages of the book are numbered?
40. Owen has exactly a 50% chance of throwing a winner on each turn in ring toss. If he took three turns in ring toss and at least one of the turns was a winner, what was the probability that all three tosses were winners? Express your answer as a common fraction.
44. Each edge length of a rectangular solid is a prime number. If the volume of the rectangular solid is 385 cubic units, what is the total surface area, in square units, of the rectangular solid?
52. What is the ratio of the sum of the odd integers between 0 and 100, to the sum of the odd integers between 100 and 200? Express your answer as a common fraction.
56. The Corner Deli has four types of bread, five types of meat, and three types of cheese. Sandwiches come in three kinds: one bread and one meat; one bread and one cheese; or one bread with one meat and one cheese. How many sandwich combinations are possible at The Corner Deli?
62. In the set of numbers { , , , 30%, 0.37 } , what is the sum of the smallest and largest values? Express your answer as a decimal to the nearest tenth. 3 8 0.25 1 5
66. What is the largest possible product you can obtain from two fractions between 0 and 1 that are formed by using each of the digits 2, 3, 5 and 6 once and only once? Express your answer as a common fraction.
68. A four-digit positive integer (leading digit not 0) is chosen at random. What is the probability that at least two digits are the same? Express your answer as a common fraction.
72. An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction.
78. A regular tetrahedron has edges of length 14 cm each. What is the total surface area, in square centimeters, of the tetrahedron? Express your answer in simplest radical form.
80. The sum of two numbers is 12. The product of the same two numbers is 24. What is the sum of the reciprocals of the two numbers? Express your answer as a common fraction.
82. The 27 members of the P.J. Running Club who ran in the fall marathon represented 45% of the Club’s total membership. If the number of Club members has remained the same since just before the fall marathon and 15 of the Club’s members signed up for this spring’s marathon, what percent of the Club’s members signed up for this spring’s marathon?
86. A standard six-faced die and a standard four-faced die are both rolled once. The cube’s faces are numbered 1− 6, and the tetrahedron’s faces are numbered 1− 4. What is the probability that the sum of the downward-facing faces is greater than 6? Express your answer as a common fraction.
88. A positive five-digit integer is in the form AB,CBA; where A, B and C are each distinct digits. What is the greatest possible value of AB,CBA that is divisible by eleven?
90. Two sides of a triangle have lengths of 18 and 29 units. If the third side has an integer length, what is the positive difference between the maximum and minimum length of the third side?
100. Use each of the five digits 2, 4, 6, 7 and 9 only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product?
102. Two supplementary angles have measures, in degrees, of 2 x + 5 and 3 x + 15. What is the positive difference, in degrees, between these two angles?
108. The original price of an item is increased by 50%, then reduced by , and then increased by 25%. The new price is what percent of the original price? 1 3
110. An equilateral triangle and a regular hexagon have the same perimeter. What is the ratio of the area of the hexagon to the area of the triangle? Express your answer as a common fraction.
116. Cone L has a volume of 24 π cubic inches. If a plane cuts cone L parallel to its base at the midpoint of its altitude, what is volume, in cubic inches, of the small cone formed when the frustum is removed? Express your answer in terms of π . (A frustum of a cone is the part that is left when the cone is cut by a plane parallel to the base and the part containing the vertex is removed.)
120. Eleven students use 44 pencils and 22 erasers in a year. In half a year at that rate, 20 students use p pencils and e erasers. What is the value of p + e ?
122. Two segments tangent to a circle intersect at point B, as shown. Angle ABC forms a 60-degree angle, and AB and BC each measure units. What is the value of k if the area of the circle is k π square units? A B C
128. If each of three standard six-sided dice is rolled once, what is the probability that the sum of the three rolls is greater than 16? Express your answer as a common fraction.
130. The measure of each interior angle in a regular n -gon is exactly half the measure of each interior angle in a regular 2 n -gon. What is the value of n ?
136. When dribbling a basketball up the court, Gloria dribbles at a rate of two dribbles for every three steps she takes. At this rate, how many dribbles will she complete during her 51 steps to the other end of the court?
138. A rectangle’s sides are 2 x and 3 x units. A 4-by-4 square is cut out of one corner, as shown. The area of the gray region is 100 square units. The flaps are folded up to form two sides and the bottom of a box. If the other two sides and top of the box were added, what would be the volume of the box, in cubic units? fold f o l d 3 x 2 x 100 4 4
140. When x 7 – 2 x 4 + 5 x 3 + x – 9 is multiplied by – 3 x 6 – 3 x 4 + 4 x 3 – 5 x 2 + 1 and the like terms are combined, what is the coefficient of x 4 ?
142. The average age of three members of a quartet is 57 years. What is the age of the fourth member, in years, if the quartet’s overall average age is 62 years?
150. Segment AB has endpoints A(3, 5) and B( – 2, 4). This segment is reflected across the y – axis to segment A ’ B ’ . What is the sum of all four coordinates of A ’ and B ’ ?
152. The net of a standard die is shown to the left. Two standard dice are stacked on a wooden table, as shown to the right. There are five dots showing on the top face of the top die. What is the total number of dots on the three faces, including the bottom face of the stack, that can’t be seen from any perspective?
154. An 8 ½ ” by 11 ” rectangular piece of paper is cut parallel to the 11-inch side into one-half-inch-wide rectangular strips. These strips are taped to each other short end to short end without overlap. What is the length, in inches, of the longest strip formed by taping together all of the smaller strips?
162. In the following barter system, what is the value of x ? 4 grapefruits trade for 1 watermelon plus 3 oranges 2 watermelons trade for 5 grapefruits 1 grapefruit trades for x oranges
164. Place a different number from the set {1, 2, 3, 4} into each of the four empty circles such that the sum of the three numbers along each main diagonal of the hexagon is 12. What is the product of the three numbers that are in circles directly connected to the circle containing 2? 7 6 5
168. When investigating an accident, forensic analysts estimate the speed of a car by assuming that the coefficient of friction is f = s 2 ÷ (30 d ), where s is the speed of the vehicle, in miles per hour, and d is the length of skid marks, in feet. What is the value of the coefficient of friction for a vehicle initially traveling at 30 miles per hour that leaves skid marks 60 feet long? Express your answer as a common fraction.
170. Kia sold of her cakes in the morning. In the afternoon she sold of the remaining cakes. At the end of the day she had 12 cakes left. How many cakes did she have at the beginning of the day?
176. Two marbles will be drawn at random and without replacement from a jar of 5 red marbles and 5 white marbles. What is the probability that both marbles will be the same color? Express your answer as a common fraction.
182. Celeste ate a pizza with her friends. She ate exactly seven-sixteenths of the pizza. She paid for her portion with a $20 bill and received $12.44 in change. What was the total cost of the pizza?
190. Stock A and Stock B have the same initial price. Over a two-day period, Stock A rises 20% and then falls 15%. By what percent must Stock B rise over that two-day period to finish at the same price as Stock A?
192. On an archaeological dig, a shard of pottery was found that was a sector of a circle, as shown. The central angle of the shard was 30°. Assuming that the original piece of pottery was circular, what fraction of the whole piece of pottery was found? Express your answer as a common fraction.
196. A student government is comprised of four 8 th -graders and five 7 th -graders. If the president must be an 8 th -grader and the vice-president and treasurer must each be a different 7 th -grader, in how many ways can the three officers be assigned?
198. What is the slope of the line containing the origin and the midpoint of the segment with endpoints at (2, 5) and (4, 9)? Express your answer as a common fraction.
200. Two bits is equivalent to 25¢. What is the value, in cents, of the total amount of money mentioned in the first line of the following verse? Two bits, four bits, six bits, a dollar, All for MATHCOUNTS, stand up and holler!
210. What is the area, in square units, of a parallelogram having diagonals: • which measure 5 units and 8 units and • which form a 45 degree angle with each other? Express your answer in simplest radical form.
214. If m is removed at random from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and then n is chosen at random from the remaining numbers, what is the probability that x 2 + 2 mx + n 2 = 0 will have two real solutions? Express your answer as a common fraction.
216. The tens digit of a positive two-digit number is larger than the units digit and neither digit is zero. The new number formed by interchanging the two digits is 54 less than the original number. What is the sum of all possible original numbers?
218. Mary runs twice as fast as Bob and Billy runs at 80% of Bob’s pace. What fraction of a mile will Billy complete in the same time that Mary runs of a mile? Express your answer as a common fraction. 1 3
220. A rectangle with sides of 30- and 40-units is placed with its center at the origin of a Cartesian coordinate system. When the rectangle is rotated around the origin, what is the maximum y -value any vertex of the rectangle will achieve?
224. For integers a , b and c and products 7 a , 5 b and 8 c , we have 7 a − 5 b = 8 c . If 7 a > 0 and 8 c < 1000, what is the minimum possible value for b ?
226. What is the smallest positive integer that has a remainder of 1 when divided by 2, a remainder of 2 when divided by 3 and a remainder of 4 when divided by 5?
228. A bag contains only red marbles, white marbles and green marbles. With replacement after each selection, the probability of drawing a white marble followed by a white marble is . With replacement after each selection, the probability of drawing a green marble followed by a green marble is . What is the smallest possible number of red marbles in the bag? 1 16 9 100
230. An isosceles triangle is inscribed in a circle so that its base is the diameter of the circle. The diameter of the circle is one foot. What is the area of the triangle, in square inches?
232. A farmer just finished fencing off his rectangular field measuring 30 ft by 18 ft. Fence posts were placed at the corners of the field and along the sides so that the centers of adjacent posts are 6 feet apart. How many posts were used?
