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    National2009 National2009 Presentation Transcript

    • MATH COUNTS 2009 National Competition Countdown Round 
    • What power of 4 is equal to 8? Express your answer as a common fraction.
    • Answer: 3 2
    • How many two-digit positive integers are even but not multiples of 3 and not multiples of 7?
    • Answer: 25 (integers)
    • 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.
    • Answer: 16 25
    • What is the ones digit of 1 2009 + 2 2009 + 3 2009 + ... + 2009 2009 ?
    • Answer: 5
    • 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?
    • Answer: BAC
    • When the two-digit integer “AB” is cubed the value is 912,673. What is A + B?
    • Answer: 16
    • 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?
    • Answer: 2160 (degrees)
    • If 15 bahs are equal in value to 24 rahs and 9 rahs are equal in value to 15 yahs, how many bahs are equal in value to 1000 yahs?
    • Answer: 375 (bahs)
    • What is the minimum possible value for y in the equation y = x 2 − 6 x + 13?
    • Answer: 4
    • If the elements within a set are always listed in ascending order, how many subsets of S = {1, 2, 3, 4, 5, 6, 7, 8} contain a 1, 2 or 5?
    • Answer: 224 (subsets)
    • What is the value of x + y + z when 6 x + 5 y − 2 z = −4 and 4 x + 5 y + 12 z = 54?
    • Answer: 5
    • 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?
    • Answer: 4 (values)
    • Given that , what is the value of ? 2 2 1 7 x x + = 4 4 1 x x +
    • Answer: 47
    • 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.
    • Answer: 502 2009
    • What is the sum of the digits of the first 50 positive odd integers?
    • Answer: 475
    • What is the value of K in the equation: (7)(6)(6!)(5!) = K! ?
    • Answer: 10
    • 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?
    • Answer: 88
    • What is the ordered pair of integers ( x , y ) for which 12 x + 21 y = 15 and 21 x + 12 y = 51?
    • Answer: (3, -1)
    • What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48?
    • Answer: 84 (degrees)
    • 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.
    • Answer: 1 7
    • If the sum of 1! + 2! + 3! + 4! + ... + 49! + 50! is divided by 15, what is the remainder?
    • Answer: 3
    • 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?
    • Answer: 334 (sq units)
    • If (7, 9) and (10, 2) are the coordinates of two opposite vertices of a square, what is the sum of the y -coordinates of the other two vertices?
    • Answer: 11
    • The sum of a five-term arithmetic sequence is 100. If all terms are positive integers, what is the smallest possible value for a term?
    • Answer: 2
    • What is the product of the two integer values of x for which │ x 2 – 16 │ is a prime number?
    • Answer: –9
    • 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.
    • Answer: 1 3
    • What is the area, in square units, of the interior region formed by the lines y = 2 x − 4, y = −3 x +16 and the y -axis?
    • Answer: 40 (sq units)
    • 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?
    • Answer: 92 (combinations)
    • How many positive four-digit integers of the form _ _ 45 are divisible by 45?
    • Answer: 10 (integers)
    • If , what is the value of (5 + x )(20 – x )? 5 20 7 x x + + ‒ =
    • Answer: 144
    • 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
    • Answer: 0.7
    • A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?
    • Answer: 90 (units)
    • 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.
    • Answer: 5 9
    • 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.
    • Answer: 62 125
    • What is the ordered pair ( b , c ) such that each solution of 2 x 2 + bx + c = 0 is 2 units larger than a solution of x 2 − 3 x + 2 = 0?
    • Answer: (–14, 24)
    • 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.
    • Answer: 1 3
    • If a is 4 percent of b and b is 25 percent of c , what percent of a is c ?
    • Answer: 10,000 (percent)
    • If 10 51 − 9 is written as an integer in standard form, what is the sum of the integer’s digits?
    • Answer: 451
    • 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.
    • Answer: (sq cm) 196 3
    • 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.
    • Answer: 1 2
    • 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?
    • Answer: 25 (percent)
    • The sum of 27 consecutive positive integers is 3 7 . What is their median?
    • Answer: 81
    • 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.
    • Answer: 5 12
    • 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?
    • Answer: 96,569
    • 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?
    • Answer: 34 (units)
    • What is the sum of all real values of x that satisfy the equation ( x − 3)( x 2 + 4 x + 4) = x − 3?
    • Answer: ‒1
    • What is the largest prime factor of 5! + 6! ?
    • Answer: 7
    • The difference between the squares of two positive integers is 2009. What is the maximum possible difference between these two integers?
    • Answer: 41
    • In a regular polygon, the measure of an interior angle is 6.5 times the measure of an exterior angle. How many sides does the polygon have?
    • Answer: 15 (sides)
    • 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?
    • Answer: 762
    • 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?
    • Answer: 42 (degrees)
    • What power of 16 is equal to 8? Express your answer as a common fraction.
    • Answer: 3 4
    • What is the least common multiple of 14, 15 and 16?
    • Answer: 1680
    • 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
    • Answer: 125 (percent)
    • 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.
    • Answer : 3 2
    • How many three-digit integers between 599 and 800 contain exactly one 7 and no 4s?
    • Answer: 80 (integers)
    • What percent of the reciprocal of h is k ? Express your answer in terms of h and k .
    • Answer: 100 hk or 100 kh (percent)
    • 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.)
    • Answer: 3 π (cubic inches)
    • What is the least positive integer that has no odd digits and is divisible by 9?
    • Answer: 288
    • 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 ?
    • Answer: 60
    • 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
    • Answer: 100
    • What is the smallest positive integer that is a product of two different primes and is one more than a multiple of 6?
    • Answer: 55
    • Define the operation # as a # b = a 2 + b . What is the value of (2 # 1) # (2 # 1)?
    • Answer: 30
    • 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.
    • Answer: 1 54
    • 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 ?
    • Answer: 3
    • The graph of 16 x – 2 y = 48 intersects the y -axis at the point ( a , b ). What is the sum of a and b ?
    • Answer: –24
    • A rhombus has sides of length 10 inches, and the lengths of its diagonals differ by 4 inches. What is the area of the rhombus, in square inches?
    • Answer: 96 (square inches)
    • 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?
    • Answer: 34 (dribbles)
    • 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
    • Answer: 400 (cubic units)
    • 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 ?
    • Answer: 29
    • 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?
    • Answer: 77 (years)
    • What is the arithmetic mean of and ? Express your answer in simplest radical form.
    • Answer:
    • The edge lengths of a rectangular box are 6 inches, 4 yards and 9 feet. What is the volume of the box in cubic yards?
    • Answer: 2 (cubic yards)
    • What is the result of alternately adding and subtracting the squares of the first 10 odd positive integers, 1 2 – 3 2 + 5 2 – 7 2 +…?
    • Answer: –200
    • 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 ’ ?
    • Answer: 8
    • 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?
    • Answer: 9 (dots)
    • 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?
    • Answer: 187 (inches)
    • The product of three distinct positive integers is 30. What is the smallest possible sum of these integers?
    • Answer: 10
    • When the sum 14 52 + 5 21 is simplified to an integer, what is the ones digit?
    • Answer: 1
    • Given that ( x – 2 y ) ÷ ( x + 2 y ) = 2, what is the value of ( x + y ) ÷ ( x – y )? Express your answer as a common fraction.
    • Answer: 5 7
    • 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
    • Answer: 2
    • 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
    • Answer: 140
    • What is the largest possible area, in square centimeters, of a quadrilateral inscribed in a circle of radius 10 cm?
    • Answer: 200 (square centimeters)
    • 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.
    • Answer: 1 2
    • 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?
    • Answer: 64 (cakes)
    • Two consecutive perfect squares differ by 101. What is the larger of the two squares?
    • Answer: 2601
    • How many square feet of carpet are unused when a 9-foot by 12-foot floor is covered with a carpet that is 3 yards by 5 yards?
    • Answer: 27 (square feet)
    • 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.
    • Answer:
    • If a circle is enlarged so that the area of the new circle is four times the area of the original circle, by what factor did the radius increase?
    • Answer: 2
    • In the set {1, 2, 3, 4, x } where x is an integer greater than 4, the mean is greater than the median. What is the least possible value of x ?
    • Answer: 6
    • 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?
    • Answer: $17.28
    • What is the greatest integer value of x that is not a solution to the inequality | 2 x – 3 | > 4?
    • Answer: 3
    • If $1.88 is divided into three parts proportional to 12:15:20, how many cents are in the smallest part?
    • Answer: 48 (cents)
    • How many positive three-digit integers contain three different digits?
    • Answer: 648 (integers)
    • 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?
    • Answer: 2 (percent)
    • 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.
    • Answer: 1 12
    • Define f ( x ) = 2 x + 3. If f ( f ( x )) = x , what is the value of x ?
    • Answer:  3
    • 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?
    • Answer: 80 (ways)
    • 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.
    • Answer:
    • 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!
    • Answer: 250 (cents)
    • What is the smallest positive six-digit multiple of 6 whose ones digit is 6?
    • Answer: 100,026
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    • 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.
    • Answer: (sq units) 10 2
    • How many positive integers n produce a remainder of 9 when 2009 is divided by n and n > 9?
    • Answer: 15 (integers)
    • 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.
    • Answer: 1 2
    • 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?
    • Answer: 246
    • 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
    • Answer: 2 15
    • 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?
    • Answer: 25
    • How many 2-by-2-by-2 cubes must be added to an 8-by-8-by-8 cube to make a 10-by-10-by-10 cube?
    • Answer: 61 (cubes)
    • 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 ?
    • Answer: –197
    • 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?
    • Answer: 29
    • 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
    • Answer: 9 (marbles)
    • 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?
    • Answer: 36 (square inches)
    • 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?
    • Answer: 16 (posts)