General Contest Guidelines: Keep the contests secure. Discussion about contest questions is not permitted prior to giving the contest. Due to the detail of some problems, print the contests using a normal or high quality setting. Rules, such as grading guidelines and eligible students, can be viewed online at www.atpacmath.com. Results must be submitted at www.atpacmath.com prior to the next contest date. Results can be viewed at www.atpacmath.com immediately after they are submitted. Contest Dates Contest dates for the 010/011 Atlantic-Pacific Contest Season are listed below. Contests may be given anytime prior to the next contest date, as long as results are submitted on time. Contest # Contest Date Submit Results by 1 Wednesday, November 10 Tuesday, December 7 Wednesday, December 8 Tuesday, January 11 Wednesday, January 1 Tuesday, February 8 4 Wednesday, February 9 Tuesday, March 8 5 Wednesday, March 9 Tuesday, April 5 6 Wednesday, April 6 Sunday, April 17
Contest 1 (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes Math Teacher 1.1 Simplify the expression: 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 Write the answer in exponential form. Answer Column 1. The interior angles of a convex polygon, P, have an average measure of 150 o. How many sides does P have? 1. What is the greatest common factor of 1x y 4, 0x 8 y, 16x 4 y 4? 1.4 What is the area of the circle, in terms of π, that circumscribes an equilateral triangle with a side of 8 m? 1.5 Given that the number 6d6 is divisible by 1, what is the sum of all the digits that could replace d? 1.6 Determine the value of m, such that the three lines y = x + 1, x + y = 5, and y = mx + are concurrent.
Contest (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes Math Teacher Answer Column.1 If x o y = xy, find ( o 48) o 9. Answer in simplified radical form.. Find the acute angle between the minute hand and hour hand of a clock at 6:45.. Find the perimeter, in centimeters, of a square with a diagonal of length 6 cm. Give an exact answer..4 Solve for x: 5x 10 = 7 +. x x.5 If the line 6 x + y = 18 is reflected over the x-axis, then what is the equation of the new line in standard form?.6 The length of a rectangle is increased by 10 percent and its width is decreased by 0 percent. What percent more or less is the new area compared with the original area?
Contest (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes Math Teacher.1 The height of a triangle is 5 units less than the length of its base. If the area of this triangle is 4 square units, how many units is the length of its base? Answer Column. What is the distance between the x-intercept of the graph of y = x + 8x + 16 and the y-intercept of the graph 4 x + y = 1? Give an exact answer.. If b 4ac = 15, and a =, c = 1, then what is the value of b? Answer in simplified radical form..4 A businesswoman drives to work at an average of rate 0 km/h. She returns home on the road at an average rate of 40 km/h. What is her average rate of speed, in km/h, for the entire trip? Round to the nearest tenth..5 A circle is inscribed in an isosceles trapezoid with base lengths of 8 and 18. What is the diameter of the circle?.6 Solve x x 5 = 15 for x.
Contest 4 (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes Math Teacher Answer Column 4.1 The legs of a right triangle are, and 7. What is the length of the hypotenuse? 4. If the length and width of a rectangle are each increased by ten percent, then the perimeter of the rectangle is increased by what percent? 4. Simplify 16 + 64 4.4 What is the distance from the vertex of the graph of y = x 4x + and the y-intercept of the graph of y = 4x 4? Give an exact answer. 4.5 Seven of the interior angles of a decagon have measures whose sum is 10. Of the remaining three angles, exactly two are complementary and exactly two are supplementary. Find the measures, in degrees, of these three angles. 4.6 Given x + y = a and x + 5y = b, solve for x.
Contest 5 (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes Math Teacher 5.1 How many squares are contained in a 5 by 5 square grid? Answer Column 5. Solve for x: x 10 + 1 = 4 5. If x = 4, then what is the exact value of x? 5.4 Find the points of intersection of the graphs in the system. x 4y 4 = 0 y + x + = 0 5.5 The side of a square is the same length as the altitude of an equilateral triangle. Find k if the area of the square is k times the area of the triangle. Give an exact answer. 5.6 How many integers are there such that 7x + < and x 5 > 1?
Contest 6 (010/011 Season) ATLANTIC-PACIFIC MATHEMATICS LEAGUE High School League No Calculators Permitted Name Grade Level Score Time 0 Minutes 6.1 Solve for x: x( x c) = 1 c Math Teacher Answer Column 6. What is the coordinate of the center of the circle given the equation x + 10x + y = 0? 6. The sides of a right triangle are m, 4 m, and 5 m in length. A point is taken on the hypotenuse at a distance of m from the vertex adjacent to the 4m side. Find the distance from this point to the vertex of the right angle. Give an exact answer. 6.4 A tangent and a secant are drawn to a circle from an external point. The tangent is 14 cm long and the internal and external segments of the secant are in the ratio of :1. Find the length, in centimeters, of the secant. 6.5 A rectangular box measures 4 by 5 by meters. What is the length, in meters, of the longest broomstick that can fit into the box? Give an exact answer. 6.6 If x y x + y 5 =, then what is x y equal to?
Solutions to Contest #1 (010/011 Season) High School Question #1: Answer: 6 7 Explanation: 6 6 (1 + 1 + 1 + 1 + 1 + 1) = 6 6 (6) = 6 7. Questions #: Answer: 1 Explanation: 180(n-) = 150n. Therefore n = 1 Questions #: Answer: 4x y Explanation: Questions #4: Answer: 64π Explanation: The apothem of the triangle, half the base of the triangle, and the radius of the circle form a 0-60-90 right triangle with the shorter leg equal to 4. Therefore the hypotenuse is 8; which is also the radius of the circle. Question #5: Answer: 8 Explanation: The number needs to be divisible by and 4. The only numbers that satisfy the divisibility rules are 1 and 7. Their sum is 8. Question #6: Answer: 4 Explanation: Solving the system y = x + 1 and -x + y = 5 gives x = and y = 7. Substituting those values into y = mx + and solving for m gives m = 4/
Solutions to Contest # (010/011 Season) High School Question #1: Answer: 6 Explanation: o 48 = 1, so 1 o 9 = 108 = 6 Questions #: Answer: 67.5 Explanation: If the hour hand was on the six, then the angle would be 90 degrees, but the hour hand moves ½ degree per minute. The angle is 90 ½(45) = 67.5 Questions #: Answer: 7 Explanation: If the diagonal is 6, then the side is 6/ or 18. Multiplying by 4 gives the perimeter. Questions #4: Answer: 5.5 Explanation: Multiplying both sides of the equation by x leaves 5x = 7x 1 + 10. Solve for x to get 5.5. Question #5: Answer: x y = 6 or -x + y = -6 Explanation: The x and y intercepts of the original equation are (, 0) and (0, 6). The x and y-intercepts of the reflected line are (, 0) and (0, -6). The equation of the line through those two points is y = x 6. Question #6: Answer: 1 Explanation: The area of the original rectangle is LW. The other rectangle is.88lw. Therefore the area of the rectangle is decreased by 1%.
Solutions to Contest # (010/011 Season) High School Question #1: Answer: 1 Explanation: The area of the triangle is ½(b)(b-5) = 4. Therefore, b 5b 84 = 0. Solving the quadratic equation gives x = 1 and -7. The base is 1. Questions #: Answer: 4 Explanation: The x-intercept of y = x + 8x + 16 is (-4, 0). The y-intercept of 4x + y = 1 is (0, 4). The distance between the two points is, simplified 4. Questions #: Answer: ± 41 Explanation: Squaring both sides after substituting gives b 4()(1) = 69. Isolating b gives b = 69 or b = ± 41 Questions #4: Answer: 4. Explanation: If the trip to work was 10km, then the roundtrip would be 40km in 7 hours. 40/7 = 4.km/hr. Question #5: Answer: 1 Explanation: Common external tangents gives the leg of the trapezoid to be 1. Dropping the perpendicular will give the smaller leg of right triangle CED to be 5. Using the Pythagorean Theorem yields CE to be 1. A 8 C 4 9 B 5 8 E D 5 Question #6: Answer: {-1, } Explanation: x x 5 = 5, therefore x x = or x x = 0. This gives (x - )(x + 1) = 0 or {-, 1}
Solutions to Contest #4 (010/011 Season) High School Question #1: Answer: 9 Explanation: Squaring both legs gives 18 and 6, which leaves The hypotenuse is 9. 81 for the hypotenuse of the right triangle. Questions #: Answer: 10 Explanation: The perimeter of the original rectangle is (l+w) and the perimeter of the new rectangle is.(l+w). The increase is 10% Questions #: Answer: 6400 Explanation: 16 = 64 and 6 = 16. (64 + 16) = 80 = 6400 Questions #4: Answer: Explanation: The vertex of x 4x + is (, -), and the y-intercept of y = 4x 4 is (0, -4). The distance between the two points is 8, simplified is. Question #5: Answer: 40, 50, 10 Explanation: The sum of the angles of a decagon is 1440. Seven of the angles have a sum of 10, so the remaining three angles have a sum of 0. Two angles are supplementary, x + y = 180, and two other angles are complementary, x + z = 90. Setting up a system gives x + y + z = 0 and x + y + z = 70. Solving the system for y gives 50. By substitution the other two angles are 10 and 40. Question #6: Answer: x = -5a + b Explanation: Using the multiplication/addition method to eliminate y gives x = -5a + b.
Solutions to Contest #5 (010/011 Season) High School Question #1: Answer: 55 Explanation: A 1x1 grid has 1 square, a x grid has 5 squares, a x grid has 14 squares. Following the pattern to 5x5 gives 55 squares. Questions #: Answer: {11, -1} Explanation: Isolating the absolute value gives x 10 = 1. Solving x 10 = 1 and x 10 = -1 gives x = 11 and x = -1 Questions #: Answer: 16 4 1 1 Explanation: x = 4, squaring both sides x = 4 ( ) = 16 4. Questions #4: Answer: (-, 0), (4, ), (4, - ) Explanation: Solving x 4y 4 = 0 for y gives y = 1/4x 1. Substituting this into the other equation and solving for x gives x = -, 4. Substituting x = - back into the equation gives y = 0. Substituting x = 4 gives y = +/- Question #5: Answer: Explanation: The area of the square is s. The area of the equilateral triangle is s /. The square is times larger than the triangle. Question #6: Answer: 1 Explanation: Solving both inequalities gives x < and x >. There is only one integer that satisfies both inequalities.
Question #1: Answer: {1, c 1} Solutions to Contest #6 (010/011 Season) High School Explanation: x cx = 1 c, then x 1 = cx c, which leads to (x + 1)(x 1) = c(x 1) or (x + 1)(x 1) - c(x 1) = 0. Factoring gives (x 1)[(x + 1) c ] = 0, so x 1 = 0 or x + 1 c = 0. Solving for x gives 1 and c - 1 Questions #: Answer: (-5, 0) Explanation: Completing the square gives x + 10x + 5 + y + 0 = 5. Rewriting in circle equation form gives (x + 5) + y = 5. Therefore the center is (-5,0). 6 Questions #: Answer: 5 5 Explanation: Solve Solve x = 5 y 4 = 5 to find the x coordinate of P. So, x = 5 6. 1 to find the y coordinate of P. So, y =. 5 Distance from P to origin gives Questions #4: Answer: 8 180 6 = 5 5 5 B: (0.00, 4.00) 4 B BP =.00 cm P PA =.00 cm A C A: (.00, 0.00) - 5 Explanation: Using the tangent-secant power theorem: x(x +y) = z ; where x is the external part of the secant, y is the internal part and z is the tangent leaves x(x + x) = 14 or 4x = 196. Solving gives x = 7 and by substitution the secant is 8 cm long. Question #5: Answer: 5 Explanation: Connecting the top right, rear point of the box to the front left bottom point, creates a right triangle with a leg of and 41. Using the Pythagorean Th. to find the hypotenuse gives + ( 41 ) = c. Solving for c gives 45 or 5. Question #6: Answer: 7 Explanation: Cross products give x y = 5x + 5y. Moving x and y to each side of the equation x 7 leaves -x = 7y or = y