Download Individual Mathematics Contest 2013

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
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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
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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
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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.
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1, then which one of the