Shippensburg Math & Computer Day 2013 Individual Math Contest

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Bethany Anthony
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1 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 (b) 7 (c) 8 (d) 9 (e) If a 4-sided die, a 6-sided die, and an 8-sided die are rolled together, what is the probability that all three resulting numbers will be different? (a) 3/8 (b) 5/6 (c) 7/8 (d) 3/5 (e) 5/8 3. Find the exact value of sin 1 (sin (7 /8)) (a) /8 (b) 3 /8 (c) 5 /8 (d) 7 /8 (e) 9 /8 4. Let bzc denote the greatest integer not exceeding z. Let x and y satisfy the simultaneous equations If x is not an integer, then x + y is y = 2bxc +3 y = 3bx 2c +5 (a) an integer (b) between -4 and 4 (c) 16.5 (d) between 4 and 5 (e) between 15 and 16 1

2 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. 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 (b) 2002 (c) 2013 (d) 3014 (e) 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 (b) 8 (c) 9 (d) 10 (e) Let r and s be the complex zeroes of the function p(x) =3x 2 5x +7. Find the exact value of 1 r s 2 (a) 17/18 (b) 17/49 (c) 5/6 (d) 3/7 (e) p 59/36 8. Solve the equation log 2 (2x + 1) + log 2 (2x + 3) = 3 for x. (a) 1/2 (b) 1/2 (c) 5/2 (d) 5/2 (e) The equation has no solution. 2

See the example below for the gluing diagram of a torus.

3 9. A gluing diagram of a square 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 1: How the gluing diagram of a torus glues together Here are gluing diagrams for four different surfaces: Which surface does the gluing diagram below represent? (a) Torus (b) Klein Bottle (c) Projective Plane (d) Sphere (e) None of the Above 3

4 10. The sequence {a n } is defined recursively by a 0 =3, a k =2 a k 1 2 for all k 1. Find the value of X10 (a) 1023 (d) 2083 (b) 1026 (e) 2089 (c) 2069 k=0 a k 11. The line ` is tangent to the circle x 2 + 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) (b) 25 (c) (d) (e) The triangle of Mahonian numbers begins as follows: The next row begins with 1, 4, 9, 15, 20. What is the following number in this row? (a) 20 (b) 22 (c) 24 (d) 25 (e) Find the largest integer k for which 2 k evenly divides 100!. (a) 96 (b) 97 (c) 98 (d) 99 (e) 100 4

5 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 (b) around 1.5 hours (c) around 2 hours (d) around 2.5 hours (e) around 3 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 (b) 21.6 (c) 64.8 (d) 108 (e) None of the Above 16. A pizza with radius 6 inches is cut into 12 slices. What is the perimeter of one slice? (a) 3+ inches (b) 6+ inches (c) 9+ inches (d) 12 + inches (e) None of the Above 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 (b) 43 (c) 47 (d) 53 (e) 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 (b) 12:43 AM (c) 12:53 AM (d) 1:03 AM (e) midnight (24 hours later) 5

6 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 (b) 72 (c) 76 (d) 84 (e) If the operation x y is defined by x y =(x + 1)(y + 1) 1, then which one of the 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

Shippensburg Math & Computer Day 2013 Individual Math Contest Solutions

Shippensburg Math & Computer Day 2013 Individual Math Contest Solutions 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