Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Shippensburg Math & Computer Day 2013 Individual Math Contest 1. Row n of Pascal’s Triangle lists all the coefficients of the expansion of (1 + x)n . What is the smallest value of n for which the sum of these coefficients is greater than 100? (a) 6 (d) 9 (b) 7 (e) 10 (c) 8 2. If a standard 4-sided die, 6-sided die, and 8-sided die are rolled together, what is the probability that all three resulting numbers will be different? (a) 3/8 (d) 3/5 (b) 5/6 (e) 5/8 (c) 7/8 3. Find the exact value of sin 1 (sin (7⇡/8)) (a) ⇡/8 (d) 7⇡/8 (b) 3⇡/8 (e) 9⇡/8 (c) 5⇡/8 4. Let bzc denote the greatest integer not exceeding z. Let x and y satisfy the simultaneous equations y = 2bxc + 3 y = 3bx 2c + 5 If x is not an integer, then x + y is (a) an integer (d) between 4 and 5 (b) between -4 and 4 (e) between 15 and 16 (c) 16.5 1 5. A graph consists of points called vertices (the singular form of vertices is vertex) and lines connecting the points called edges. Below is an example of a graph. Figure: Example of a graph The number edges meeting at a given vertex is called the degree of the vertex. If a graph has 2013 edges and 1001 vertices, what is the total sum of the degrees of the vertices? (a) 1001 (d) 3014 (b) 2002 (e) 4026 (c) 2013 6. Bob tries to sum the numbers from 1 to n and gets a result of 50. But then he realizes he inadvertently omitted one number in finding this sum. What is n? (a) 7 (d) 10 (b) 8 (e) 11 (c) 9 7. Let r and s be the complex zeroes of the function p(x) = 3x2 value of 1 1 + 2 2 r s (a) 17/18 (b) 17/49 (c) 5/6 (d) 3/7 p (e) 59/36 8. Solve the equation log2 (2x + 1) + log2 (2x + 3) = 3 for x. (a) 1/2 (b) 1/2 (c) 5/2 (d) 5/2 (e) The equation has no solution. 2 5x + 7. Find the exact 9. A gluing diagram of a square is a square with sides paired up (one pair is labeled with a, the other pair with b) and each side is given a direction indicated by an arrow. The diagram represents a two-dimensional surface. Sides with the same label are glued together so that the arrows coincide. See the example below for the gluing diagram of a torus. Figure: How the gluing diagram of the square “glues together” to make a torus Two gluing diagrams of a square are considered equivalent if one can be turned into the other by any combination of reflecting the square, rotating the square, and reversing the direction of both arrows for a given pair of sides. How many nonequivalent gluing diagrams are there for a square? (a) 4 (d) 7 (b) 5 (e) 8 (c) 6 10. The sequence {an } is defined recursively by a0 = 3, ak = 2 · ak Find the value of 10 X 1 2 for all k 1. ak k=0 (a) 1023 (d) 2083 (b) 1026 (e) 2089 (c) 2069 11. The line ` is tangent to the circle x2 + y 2 = 25 at the point ( 3, 4). Find the area of the triangle created by the line `, the x-axis, and the y-axis. (a) 625 6 (b) 25 (c) (d) (e) 625 24 625 12 169 24 3 12. The triangle of Mahonian numbers begins as follows: 1 1 1 3 1 1 2 2 5 6 1 5 3 1 The next row begins with 1, 4, 9, 15, 20 and has eleven total elements. What is the next number in the row after 20? (a) 20 (d) 25 (b) 22 (e) 29 (c) 23 13. Find the largest integer k for which 2k evenly divides 100!. (a) 96 (d) 99 (b) 97 (e) 100 (c) 98 14. A professional painter can paint my bedroom in 6 hours. It would take me 12 hours to paint my bedroom alone. If I hire two professional painters and work alongside them, how long will it take all three of us to paint my bedroom together? (a) around 1 hour (d) around 2.5 hours (b) around 1.5 hours (e) around 3 hours (c) around 2 hours 15. If the ratios between adjacent angles of a pentagon are 3 : 4 : 5 : 6 : 7, what is the size of the smallest angle? (a) 14.4 (d) 108 (b) 21.6 (e) None of the Above (c) 64.8 4 16. A pizza with radius 6 inches is cut into 12 equal slices as in the picture below. What is the perimeter of one slice? Figure: The sliced pizza (a) 3 + ⇡ inches (d) 12 + ⇡ inches (b) 6 + ⇡ inches (e) None of the Above (c) 9 + ⇡ inches 17. The average of nine students’ test scores is 73. When one more student’s score is recorded, the average of the ten test scores is 70. What is the test score of the tenth student? (a) 37 (d) 53 (b) 43 (e) 57 (c) 47 18. If a bell is rung once per second, starting at midnight, for 2013 rings. What are the hour and minute during the last ring? (a) 12:33 AM (d) 1:03 AM (b) 12:43 AM (e) midnight (24 hours later) (c) 12:53 AM 5 19. The figure below depicts a wooden cube having edges of length 3 meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The edges of the holes are parllel to the edges of the cube. The entire surface area including the inside, in square meters, is (a) 54 (d) 84 (b) 72 (e) 86 (c) 76 20. If the operation x ⇤ y is defined by x ⇤ y = (x + 1)(y + 1) following is false? (a) x ⇤ y = y ⇤ x for all real x and y. (b) x ⇤ (y + z) = (x ⇤ y) + (x ⇤ z) for all real x, y, and z. (c) (x 1) ⇤ (x + 1) = (x ⇤ x) 1 for all real x. (d) x ⇤ 0 = x for all real x. (e) x ⇤ (y ⇤ z) = (x ⇤ y) ⇤ z for all real x, y, and z. 6 1, then which one of the