UNC Charlotte Super Competition - Comprehensive test March 2, 2015

Size: px

Start display at page:

Download "UNC Charlotte Super Competition - Comprehensive test March 2, 2015"

Bertram Kelley
5 years ago
Views:

1 March 2, triangle is inscribed in a semi-circle of radius r as shown in the figure: θ The area of the triangle is () r 2 sin 2θ () πr 2 sin θ () r sin θ cos θ () πr 2 /4 (E) πr 2 /2 2. triangle is formed by the coordinate axes and the tangent line at the point with coordinates (x 0, y 0 ) lying on the curve = {(x, y) : xy = k}, as shown in the figure. y (x 0, y 0 ) x The numbers x 0, y 0, and the slope m of the tangent line satisfy y 0 = mx 0. The area of the triangle is () 2k () 2k 2 () x 0 + 2k () k(x 0 + y 0 ) (E) k/(x 0 + y 0 )

2 3. There are three triangles of different sizes: small, medium and large. The small one is inscribed in the medium one such that its vertices are at the midpoints of the three edges of the medium one. The medium triangle is inscribed in the large triangle in the same way as shown in the figure. If the small triangle has area 1, what is the sum of the areas of the three triangles? () 14 () 16 () 19 () 21 (E) escribe the shape of the graph of f(x) = 6x2 + 7x 3. 2x + 3 () hyperbola () parabola () circle () full line (E) line with a hole 5. How many of elements of the set { 12.3, 9, 5, 1, 2.14, 3.6, 5.2, 7.8, 101} are not solutions of the equation 2x 2 9x + 6 = 2x 2 9x + 6? () 2 () 3 () 5 () 6 (E) 8 6. The number of elements in the intersection of the solution set of with the set { 7, 5, 2, 1, 1, 2, 107, 108} is () 0 () 1 () 2 () 3 (E) 4 2 x 1 2x x + 4 > 0 7. How many positive integers are in the range of the function f(x) = 2x 2 + 8x 3? () 0 () 2 () 3 () 5 (E) 6 8. The number of elements in the intersection of = { 1, 12, 0, 12, 1, 32 }, 2 with the solution set of 1 + x + 2 ) ( ( 1 + x 3 ) > 0 is x 1 2 () 01 () 1 () 2 () 3 (E) 4

3 9. The line segment divided the rectangle in the picture into two parts. The proportion of the areas of these parts is 2 : 9. 2t 9t x y In what proportion are the lengths x and y? () 2 : 9 () 2 : 7 () 4 : 7 () 4 : 9 (E) 5 : onsider a convex 4-gon. Reflect about to get E, about to get F, about to get G and about to get H. What percentage is the area of of the resulting 4-gon EF GH? G H F E () 10% () 20% () 25% () 30% (E) 33.5% 11. What is the number of real solutions of the equation 2015x x 3 +x 2 +2x+2016 = 0 if we count each different solution only once, without multiplicities? () 0 () 1 () 2 () 4 (E) How many four-digit positive integers are there with exactly two even and two odd, pairwise different digits? () 2008 () 2100 () 2160 () 2240 (E) What is the nth digit after the decimal point in the decimal representation of ( 26 5) n. () 0 () 1 () 6 () 5 (E) an not be determined

4 14. The diagonals of the convex four-gon divide the four-gon into four triangles, whose areas are a 1, a 2, a 3, and a 4 respectively, as shown in the figure. a 4 a 1 K a 3 a 2 Which of the following identities must hold for these areas? () a 1 + a 2 = a 3 + a 4 () a 1 + a 3 = a 2 + a 4 () a 1 a 2 = a 3 a 4 () a 1 a 3 = a 2 a 4 (E) a 1 a 3 = a 2 a The lengths of 3 edges, meeting in one vertex, of a rectangular box, measured in centimeters are distinct positive integers. One of the edges has length 6 cm. What is the sum of the lengths of the other two edges if the number expressing the volume of the box in cubic centimeters is the same as the number expressing its surface area in square centimeters. () 10 () 12 () 14 () 16 (E) The National ssembly of a country has 10 committees. Every member of the National ssembly works in exactly two committees and any pair of committees has exactly one member in common. How many members does the National ssembly have? () 25 () 30 () 45 () 50 (E) What is the sum of all integers n, for which Q(n) = 4n2 4n 24 n 3 3n 2 4n + 12 () 10 () 11 () 12 () 13 (E) infinity is also an integer? 18. We draw all diagonals of a convex 12-gon. Each intersection of diagonals is contained in exactly two diagonals. How many intersections of diagonals are there inside the 12-gon? () 216 () 480 () 495 () 500 (E) What is the exact value of expression is defined)? sin(3α) + sin α sin(2α) cos(α) () 2 () 1 () 0 () 1 (E) 2 (for all values of α such that the above

5 20. We subdivide the rectangle with two pairs of parallel lines into 9 parts, as shown in the figure. The areas of five of these parts, measured in square centimeters is provided. What is the area of the rectangle, measured in square centimeters? () 150 () 190 () 180 () 155 (E) What is the sum of the squares of the solutions of the equation log 3 (x) + log x (9) = 3? () 0 () 9 () 12 () 81 (E) In the language adspeak, the alphabet contains only the letters a, b, and c. How many 4-letter words in adspeak contain the letter c? () 81 () 108 () 108 () 65 (E) If log x = 8 and log y = 14, then log x 2 y 3 =? () 112 () 26 () () 6 (E) Let w be a real number. What is the sum of the (possibly complex) roots of the equation x x + w = 0? () w () w () 13 + w () 13 w (E) n a b c block of unit cubes has faces with surface areas 48, 60, and 80. Suppose the six faces are painted, after which the block is cut into unit cubes. Which of the following is the probability that a randomly selected cube is painted on exactly two faces? () 1/2 () 3/20 () 2/21 () 7/60 (E) 1/8

f(x + y) + f(x y) = 10.

f(x + y) + f(x y) = 10. Math Field Day 202 Mad Hatter A A Suppose that for all real numbers x and y, Then f(y x) =? f(x + y) + f(x y) = 0. A2 Find the sum + 2 + 4 + 5 + 7 + 8 + 0 + + +49 + 50 + 52 + 53 + 55 + 56 + 58 + 59. A3