Shippensburg Math & Computer Day 2013 Individual Math Contest Solutions

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1 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 the sum of these coefficients is greater than 100? Solution: (b). For any polynomial f(x), f(1) is the sum of the coefficients. Thus, the sum of the coefficients in row n of Pascal s Triangle is (1 + 1) n =2 n. Now we see that 2 6 = 64 and 2 7 = 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? Solution: (e). There are = 192 possible outcomes when rolling the three dice. There are = 120 possible outcomes for which all three resulting numbers will be different. Thus, the probability is = Find the exact value of sin 1 (sin (7 /8)) Solution: (a) The domain of sin 1 is [ 1, 1] and the range (of the usual principal value) is [ /2, /2]. When the domain of the sine function is restricted to [ /2, /2], we have sin 1 (sin(x)) = x. Since sin(7 /8) = sin( /8) and /2 < /8 < /2, we have sin 1 (sin (7 /8)) = /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 Solution: (e). We have 2bxc +3 = 3bx 2c +5 2bxc + 3 = 3(bxc 2) + 5 bxc = 4. Therefore, 4 < x < 5, and y = 2bxc + 3 = 11. Hence 15 < x + y < 16. (Alternatively, from the graphs of y =2bxc +3and y =3bx 2c +5, one can see they overlap when 4 <x<5). 7

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? Solution: (e). Each edge of a graph connects two vertices, and therefore gets counted twice when summing the degrees of the vertices. Hence, the total sum of the degrees of the vertices is twice the number of edges. 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? Solution: (d). The sum of the numbers from 1 to 10 is 55. Thus, if 5 was omitted, Bob would have obtained the result of 50. Furthermore, this is the only solution. If n<10, then the sum from 1 to n is less than 50. On the other hand, if n>10, then the sum from 1 to n is greater than 50 by an amount greater than n. 7. 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 Solution: (b). It is perhaps easiest to proceed in general. Let r = a + bi. Since r and s are the complex zeroes of the same quadratic function, s = a bi. Notice that 1 r s 2 = r2 + s 2 (rs) 2 = 2(a2 b 2 ) (a 2 + b 2 ) 2. From the quadratic formula applied to p(x), we have that a =5/6 and b = p 59/6. Therefore, by substituting, simplifying and factoring we have 1 r s 2 = =

3 8. Solve the equation log 2 (2x + 1) + log 2 (2x + 3) = 3 for x. Solution: (a). Using properties of logarithms, we have log 2 (2x + 1) + log 2 (2x + 3) = 3 log 2 ((2x + 1)(2x + 3)) = 3 (2x + 1)(2x + 3) = 8 4x 2 +8x 5=0. Solving this quadratic equation (via the quadratic formula, rational root test, guess and check, etc.) we have x =1/2 and x = 5/2. However, x = 5/2 is an extraneous solution that does not satisfy the original equation. 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 2: 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? 9

4 Solution: (b). We can introduce a new cut as long as we keep track of how the diagram should be glued together. We will choose the cut from the top left corner down to the bottom right corner, and then separate the two pieces. We can now flip the upper right corner, and glue the diagram together along the a sides. Lastly, reshaping this into a square and relableing c as a, we see this diagram represents a Klein bottle. (Alternatively, if one knows that the gluing diagram of a Klein bottle is the same as that of a torus, but with a twist for one pair of edges, it can be deduced that this diagram represents a Klein bottle. Yet another way is if one knows that a Klein bottle is a Möbius strip with the edges glued together. Gluing the a sides together forms a Möbius strip and then gluing the b sides glues the edges of the Möbius strip to form a Klein bottle). 10

5 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 k=0 Solution: (c). This can be determined directly as in the table below. The a k column is computed using the recursive formula, and the P k 0 a k column can be computed by successively adding the next item in the a k column. a k k a k P k0 a k 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. Solution: (d). First we find the equation of the line `. Since ` is tangent to the given circle, its slope is perpendicular to the slope of the line passing through the points ( 3, 4) and (0, 0). This line has slope 4/3, and so ` has slope 3/4. Using that m =3/4 and ( 3, 4) are the slope and a point on ` respectively, we find the equation of ` is y = 3 4 x The x- and y-intercepts of ` can now be found to be x = 25/3 and y = 25/4. It follows that the triangle we are considering is a right triangle with height 25/4 and base 25/3. Thus, the area is 1 2 bh =( )( 3 )( 4 )=

6 12. 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? Solution: (b). Let s index the rows starting at 0, i.e., row 0 consists of 1, row 1 consists of 1 1, etc. We make the following observations: (a) The number of elements in row k is equal to the number of elements in row k-1 plus k (b) The sum of the elements in row k is equal to (k + 1)!. (c) Each row is symmetric, i.e., the same forwards and backwards. The row that begins with 1, 4, 9, 15, 20 is row 4. Thus, this row has 11 elements and each of these five numbers appears twice. Let our missing number be x. Then the sum of the elements in row 4 is 98 + x = 5! = 120. Thus our missing number is x = 22. (Alternatively, one can notice that an element in row k is equal to the sum of k +1 consecutive elements in row k 1, with zeroes as placeholders. For example, if we think of row 3 as , then row 4 is given by 1 = = = = = = etc.) 12

7 13. Find the largest integer k for which 2 k evenly divides 100!. Solution: (b). We count the number of factors of 2 in the factorization 100! = We organize our counting in the following way: every number between 1 and 100 divisible by 2 will add a factor of 2 in the prime factorization of 100!, every number divisible by 4 will add another factor of 2, every number divisible by 8 will add yet another factor, etc. The following table summarizes these results. divisible by count Every number divisible by a power of 2 adds another factor of 2 in the factorization of 100!. Thus we need to sum the counts in the table to get 97 factors of 2 in the prime factorization of 100!. 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? Solution: (d). I can get 1/12 of the job done in an hour, and each professional painter can get 1/6 of the job done in an hour. So, together we get 5/12 of the job done in an hour. If t is the numbers of hours it takes for the three of us to complete the job, then 5 12 t =1, or t = 12 5 =2.4 hours. This is 2 hours and 24 minutes, or about 2.5 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? Solution: (c). There is a number n so that the five angles of the pentagon measure 3n, 4n, 5n, 6n, and 7n and the smallest angle is 3n. Now, the sum of the angles is 25n = 540. It follows that n = 21.6 and the smallest angle is A pizza with radius 6 inches is cut into 12 slices. What is the perimeter of one slice? Solution: (c). The perimeter of a slice (sector of a circle) is the sum of the two radii and the arc length: P =2r + L =2r + r, where is the angle of the slice. In our case r =6and =(2 )/12 = /6. Thus, the perimeter is 12 + inches. 13

8 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? Solution: (b). Since the average of the nine students scores is 73, the sum or their scores is 9 73 = 657. If the tenth student is x, then we have (657 + x)/10 = 70. Solving for x, we have x = If a bell is rung once per second, starting at midnight, for 2013 rings. What are the hour and minute during the last ring? Solution: (a). If the bell is rung once per second, then 2013 seconds have passed. Converting this to minutes, we have 2013/60 = minutes. Thus, it would be 12:33 AM. 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 Solution: (b). Each exterior unit square which is removed exposes 4 interior unit squares, so the entire surface area in square meters is =

9 20. 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. Solution: (b). x (y + z) = (x + 1)(y + z + 1) 1, (x y)+(x z) = [(x + 1)(y + 1) 1] + [(x + 1)(z + 1) 1] = (x + 1)(y + z + 2) 2. Therefore, x (y + z) 6= (x y) +(x z). The remaining choices can easily be shown to be true. 15