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1 The Advantage Testing Foundation 013 Problem 1 The figure below shows two equilateral triangles each with area 1. The intersection of the two triangles is a regular hexagon. What is the area of the union of the two triangles? Express your answer as a fraction in simplest form. Answer: 4 3 Solution: The exterior angles of the regular hexagon are each 60. So the 6 little triangles sticking out are equilateral triangles. The side length of these little equilateral triangles is 1 that of the big equilateral triangles. Hence 3 the area of each little equilateral triangle is 1. 9 The union of the big triangles is the disjoint union of 1 big triangle and 3 little triangles. So the total area is = = 4 3. Problem When the binomial coefficient ( ) is written out in base 10, how many zeros are at the rightmost end? Answer: 0 1

2 Solution: The number of zeros at the end of ( ) is the number of factors of 10 in ( ) In turn, that number is the smaller of the number of factors of in ( ) ( and the number of factors of 5 in 15 ) 64. The number of factors of in ( ) can be found by looking at the base- (binary) representation of 64 and = 61. The binary representation of 64 is ( ). The binary representation of 61 is ( ). By Kummer s theorem, the number of factors of in ( ) is the number of carries when ( ) and ( ) are added. But ( ) and ( ) have no 1 s in common, so there are no carries in their binary addition. Hence the number of factors of in ( ) is 0. Thus the number of factors of 10 in ( ) is also 0. Problem 3 Let S 1, S,..., S 15 be 15 sets of 5 numbers each, comprising 65 distinct numbers. Let m i be the median of S i. Let M be the median of m 1, m,..., m 15. What is the greatest possible number of the 65 numbers that are less than M? Answer: 436 Solution: Without loss of generality, assume that m 1 > m > > m 15. So M = m 63. Let the set S i be {k i, l i, m i, n i, o i }, where k i > l i > m i > n i > o i. The numbers k i, l i, and m i for i 63 are each greater than or equal to M. All of the other numbers could be less than M. So the least possible number of the 65 numbers that are greater than or equal to M is 3 63 = 189. Hence the greatest possible number of the 65 numbers that are less than M is , which is 436. Problem 4 The MathMatters competition consists of 10 players P 1, P,..., P 10 competing in a ladder-style tournament. Player P 10 plays a game with P 9 : the loser is ranked 10th, while the winner plays P 8. The loser of that game is ranked 9th, while the winner plays P 7. They keep repeating this process until someone plays P 1 : the loser of that final game is ranked nd, while the winner is ranked 1st. How many different rankings of the players are possible? Answer: 51 Solution: The MathMatters tournament has 9 games. Each game has possible winners. So the tournament has 9 possible sequences of winners. Each sequence of winners leads to a unique ranking. So the number of possible rankings is 9, which is 51.

3 Problem 5 Say that a 4-digit positive integer is mixed if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 013 is mixed. How many 4-digit positive integers are mixed? Answer: 1680 Solution: The number of sets of 4 digits is ( 10 4 ), which is ! = 10. Let s fix one choice for the set of 4 digits. The leftmost digit is neither the biggest nor the smallest, so there are choices for it. (We don t have to worry about starting with a leading zero, because the leftmost digit is not the smallest.) Given that choice, the rightmost digit is neither the smallest nor the leftmost, so there are choices for it. Given those choices, the second digit from the left is neither the leftmost nor the rightmost, so there are choices for it. Finally, given those choices, the third digit from the left has only 1 choice, the remaining digit. Hence, given a set of 4 digits, there are 1 = 8 ways to order the digits to form a mixed integer. So the number of mixed integers is ( 10 4 ) 8, which is 10 8, or Note: This problem was proposed by Oleg Kryzhanovsky. Problem 6 Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. Express your answer as a fraction in simplest form. Answer: 1 or 1 Solution: The geometric sequence has the form a, ar, ar, where a and r are real numbers. The three terms are distinct, so a 0 and r 1. Because the three numbers form an arithmetic sequence (in some order), one of them is the average of the other two. There are three cases, depending on which number is the average. Case 1: a is the average of ar and ar. In equation form, a = ar + ar. So r + r =. Hence r is or 1. But r can t be 1, so r is. 3

4 Case : ar is the average of a and ar. In equation form, ar = a + ar. So r + 1 = r. Hence r is 1. But r can t be 1, so Case is impossible. Case 3: ar is the average of a and ar. In equation form, ar = a + ar. So r = 1 + r. Hence r is 1 or 1. But r can t be 1, so r is 1. Looking at all three cases, we conclude that r is or 1. We can achieve r = 1, since 1, 1, 1 is a geometric sequence with common ratio 4 1 and 1, 1, 1 is an arithmetic sequence. So the greatest possible value 4 of r is 1. Note: This problem was proposed by Oleg Kryzhanovsky. Problem 7 In the figure below, ABC is an equilateral triangle. y B A(1, 1) C Point A has coordinates (1, 1), point B is on the positive y-axis, and point C is on the positive x-axis. What is the area of ABC? Express your answer in the form a n b, where a and b are positive integers and n is a square-free positive integer. Answer: 3 3 Solution: We will use complex numbers. For example, the point A(1, 1) in complex form is 1 + i. The vector AC in complex form is AC = C A = C (1 + i) = C 1 i. The vector AB is a 60 clockwise rotation of AC. So we have AB = cis( 60 ) ( 1 3 ) AC = i (C 1 i). The complex multiplication on the right expands as follows: AB = C x (C 1) i.

5 The vector AB has real part 1. So we get the equation C 1 3 = 1. Solving for C gives C = 3 1. We can calculate the squared length of AC as follows: AC = C 1 i = 3 i = ( 3 ) + 1 = So the squared side length of equilateral triangle ABC is Hence the area of ABC is (8 4 3) 3 4 = ( 3) 3 = 3 3. Problem 8 Let R be the set of points (x, y) such that x and y are positive, x + y is at most 013, and x y = x y. Compute the area of set R. Express your answer as a fraction in simplest form. Recall that a is the greatest integer that is less than or equal to a, and a is the least integer that is greater than or equal to a. Answer: 013 Solution: We may ignore the points when x or y is an integer, since the set of such points has area 0. Because x and y are non-integers, x = x + 1 and y = y + 1. So the given equation x y = x y becomes ( x + 1) y = x ( y + 1). The displayed equation simplifies to y = x. Let n be the floor of x (and y). Because x + y 013, the value of n is either 0, 1,..., 1005, or If n 1005, then (x, y) is any point in the unit square n < x < n+1 and n < y < n+1. If n = 1006, then (x, y) is any point in the right isosceles triangle given by 1006 < x < 1007, 1006 < y < 1007, and x + y

6 The cases n = 0, 1,..., 1005 give 1006 unit squares, for an area of The case n = 1006 gives a triangle of area 1. So the total area is , which is 013. Note: This problem was proposed by Oleg Kryzhanovsky. Problem 9 Let A and B be distinct positive integers such that each has the same number of positive divisors that 013 has. Compute the least possible value of A B. Answer: 1 Solution: The prime factorization of 013 is So 013 has 8 positive divisors. If A and B are distinct integers, then A B 1. Can we find two consecutive integers with exactly 8 positive divisors each? A positive integer with exactly 8 positive divisors has the form p 7 or p 3 q or pqr, where p, q, and r are distinct primes. In particular, if q is a prime bigger than, then every number of the form 8q has exactly 8 positive divisors. Numbers of that form include 4, 40, 56, 88, 104, 136. Let s factorize the numbers that are 1 away from these numbers. We see that 105 = So both 104 and 105 have exactly 8 positive divisors. Hence, we can set A = 104 and B = 105 to get A B = 1. Note: This problem was proposed by Oleg Kryzhanovsky. Problem 10 The following figure shows a walk of length 6: This walk has three interesting properties: O 6

7 1. It starts at the origin, labelled O.. Each step is 1 unit north, east, or west. There are no south steps. 3. The walk never comes back to a point it has been to. Let s call a walk with these three properties a northern walk. There are 3 northern walks of length 1 and 7 northern walks of length. How many northern walks of length 6 are there? Answer: 39 or 36 Solution: If k is a nonnegative integer, let N(k) be the number of northern walks of length k that end in a north step. (For convenience, say that the empty walk of length 0 ends in a north step.) Let E(k) be the number of northern walks of length k that end in an east step. By symmetry, E(k) is also the number of northern walks of length k that end in a west step. For initial conditions, we have N(0) = 1 and E(0) = 0. We claim that E(k + 1) = N(k) + E(k). That s because every northern walk of length k + 1 that ends in an east step consists of a northern walk of length k that ends in a north or west step followed by a final east step. Similarly, we claim that N(k + 1) = N(k) + E(k). That s because every northern walk of length k + 1 that ends in a north step consists of a northern walk of length k (which ends with a north, east, or west step) followed by a final north step. Using these recurrence relations, we can compute the first few values of N and E. k N(k) E(k) From the table above, we see that N(6) = 99 and E(6) = 70. The number of northern walks of length 6 is N(6) + E(6), because every northern walk of length 6 ends in a north, east, or west step. Hence the number of northern walks of length 6 is N(6) + E(6) = = = 39. Some students interpreted the problem as excluding walks that went beyond the displayed grid. Under that interpretation, three walks counted 7

8 above should be excluded: the straight walks of length 6 going in the purely north, east, or west direction. So, under that interpretation, the answer is 39 3 = 36. Our intended answer was 39, but we accepted either 39 or 36. Problem 11 Alice throws two standard dice, with A being the number on her first die and B being the number on her second die. She then draws the line Ax + By = 013. Boris also throws two standard dice, with C being the number on his first die and D being the number on his second die. He then draws the line Cx + Dy = 014. Compute the probability that these two lines are parallel. Express your answer as a fraction in simplest form. 43 Answer: 648 Solution: The slope of the line Ax + By = 013 is A. The slope of the B line Cx + Dy = 014 is C. The lines are parallel if and only if their slopes D A and C are the same, which is equivalent to AD = BC. B D Because A and D are from the set {1,, 3, 4, 5, 6}, the possible values of AD are given by the multiplication table below Looking at the table above, we see 5 numbers (1, 9, 16, 5, and 36) that each appear 1 time, 10 numbers (, 3, 5, 8, 10, 15, 18, 0, 4, and 30) that each appear times, 1 number (4) that appears 3 times, and numbers (6 and 1) that each appear 4 times. By symmetry, BC has the same distribution. So the probability that AD = BC is ) ) ( 1 36 which simplifies to ( = = ( ) ( ) 3 4 +, 36 36

9 Note: This problem was proposed by Oleg Kryzhanovsky. Problem 1 The rectangular parallelepiped (box) P has some special properties. If one dimension of P were doubled and another dimension were halved, then the surface area of P would stay the same. If instead one dimension of P were tripled and another dimension were divided by 3, then the surface area of P would still stay the same. If the middle (by length) dimension of P is 1, compute the least possible volume of P. Express your answer as a fraction in simplest form. Answer: 3 Solution: Let the dimensions of the parallelepiped P be a, b, and c. The surface area of P is (ab + ac + bc). If a were doubled and b were halved, then the surface area would be (ab + ac + bc/). Equating the old and new surface area, we get (ab + ac + bc) = (ab + ac + bc/). Simplifying gives b = a. So the preservation of surface area under a doubling/halving means that one dimension is double another. If a were tripled and b were divided by 3, then the surface area of P would be (ab + 3ac + bc/3). Equating the old and new surface area, we get (ab + bc + ac) = (ab + 3ac + bc/3). Simplifying gives b = 3a. So the preservation of surface area under a tripling/dividing by 3 means that one dimension is triple another. We may assume that a b c. Because the middle dimension is 1, we have b = 1. Because one dimension is triple another, we have c 3a. So the doubling involves b. Hence either a = 1 or c =. Suppose that a = 1. Because one dimension is triple another, c is either 3 or 3. Either way, the volume of P is at least 3. 4 Suppose that c =. Because one dimension is triple another, a is either 1 or. Either way, the volume of P is at least Combining the two cases, we see that the volume of P is at least. We 3 can achieve that bound by setting a = 1, b = 1, and c =. So the least 3 possible volume of P is 3. Note: This problem was proposed by Oleg Kryzhanovsky. 9

10 Problem 13 Each of n boys and n girls chooses a random number from the set {1,, 3, 4, 5}, uniformly and independently. Let p n be the probability that every boy chooses a different number than every girl. As n approaches infinity, what value does n p n approach? Express your answer as a fraction in simplest form. 6 Answer: 5 Solution: The probability that each of the n boys chooses a number from the set {1,, 3} and each of the n girls chooses a number from {4, 5} is ( ) n ( ) n ( ) n 3 6 = So p n (6/5) n. More generally, let S be a subset of {1,, 3, 4, 5}. Let A S be the event that each of the n boys chooses a number from S and each of the n girls chooses a number not from S. The probability of A S is ( ) n ( ) n ( ) n S 5 S S (5 S ) = The numerator S (5 S ) is at most 6, so the probability of A S is at most (6/5) n. The event every boy chooses a different number than every girl is the union of the events A S over all subsets S of {1,, 3, 4, 5}. In other words, p n = Pr[ S A S ]. Because the probability of a union is at most the sum of the individual probabilities, we have p n Pr[A S ]. S In the previous paragraph, we showed that Pr[A S ] is at most (6/5) n, so p n ( ) n 6. 5 S Because S ranges over the 3 subsets of {1,, 3, 4, 5}, we have ( ) n 6 p n

11 Our previous work has shown that (6/5) n p n 3(6/5) n. Taking the nth root shows that 6 n p 5 n n As n approaches infinity, n 3 approaches 1. So n 6 p n approaches 5. Problem 14 How many positive integers n satisfy the inequality n + 1 > n ? Recall that a is the least integer that is greater than or equal to a. Answer: 15,049 Solution: Every positive integer n can be written uniquely in the form 101q r, where q is a positive integer and r is a nonnegative integer such that r 100. Here q = n/101. The given inequality becomes q + 1 > 101q r. 100 Solving for q gives q < r. Because q is an integer, that inequality is equivalent to q 99 + r. So for any given r, there are 99 + r choices for q. Hence the total number of choices is (99 + r) = 101(99) + r = = r=0 r=0 Note: This problem was proposed by Oleg Kryzhanovsky. Problem 15 Let ABC be a triangle with AB = 7, BC = 8, and AC = 9. Point D is on side AC such that CBD has measure 45. What is the length of BD? Express your answer in the form m x n y, where m and n are positive integers and x and y are square-free positive integers. Answer: Solution: The diagram below shows the triangle ABC and the given information. A 7 9 D B 45 8 C 11

12 By the Law of Cosines, we have Solving for cosine gives Hence the sine is 7 = cos C. cos C = sin C = 1 cos C = = = 6 9 = 3. 1 ( ) = Because the angles of triangle BCD add to 180, we have By the sine subtraction law, we have BDC = C = 135 C. sin BDC = sin(135 C) = By the Law of Sines, we have = Solving for BD gives BD sin C = 8 sin BDC = 8 6 = BD = (sin C)( ) = 5 3 ( ) = Problem 16 If 3 x < 3 x3 and x 1, define C(x) =. The real 1 x root of the cubic x 3 + 3x 7 is of the form pc 1 (q), where p and q are rational numbers. What is the ordered pair (p, q)? Express your answer using fractions ( in simplest form. 7 Answer: 3, 7 ) 98 1

13 Solution: Let x be the real root of x + 3x 7. By the Rational Root theorem, we see that x is irrational. According to the statement of the problem, x = pc 1 (q), where p and q are rational. Solving for q, we find that q = C ( x p Clearing fractions gives ) = (x/p)3 1 x/p = x 3 p 3 p x = p 3 q p qx = 7 3x. Grouping the x terms together, we get (p q 3)x = p 3 q x p 3 p x. Because x is irrational, it follows that p q = 3 and p 3 q = 7. Dividing one equation by the other shows that Hence p = p q p 3 q = 7 3. q = 3 p = 3 (7/3) = 33 (7) = 7 49 = ( 7 Therefore, the pair (p, q) is 3, 7 ). 98 Note: The function C 1 is called the curly root function by Dan Kalman in his book Uncommon Mathematical Excursions: Polynomia and Related Realms. He uses curly roots as an alternative method for solving cubic equations. Problem 17 Let f be the function defined by f(x) = sin(πx). How many values of x such that x satisfy the equation f(f(f(x))) = f(x)? Answer: 61 Solution: The graph of f on the interval [, ] is shown below. 13

14 y x As we can see, the equation f(x) = 0 has 5 solutions in [, ]. If y is a fixed nonzero number such that < y <, then the equation f(x) = y has 4 solutions in [, ]. Let s enumerate the solutions of f(f(f(x))) = f(x) in [, ]. Let y = f(x). Then the equation becomes f(f(y)) = y. So we need to enumerate the solutions of f(f(y)) = y in [, ]. Let x = f(y). Then y = f(x) and x = f(y). The graph of y = f(x) is the graph of f, as shown above. The graph of x = f(y) is the inverse graph, the reflection of y = f(x) around the line y = x. The diagram below shows the two graphs, restricted to x and y in [, ]. y x As we can see, ignoring the origin, the two graphs intersect at 4 points in the first quadrant, 3 points in the second quadrant, 4 points in the third quadrant, and 3 points in the fourth quadrant. Altogether, there are 14 intersection points, not counting the origin. So there are 14 nonzero solutions to f(f(y)) = y in [, ]. Neither nor is a solution. 14

15 As we just discovered, there are 14 nonzero solutions to f(f(y)) = y in the open interval (, ). As we mentioned earlier, for every nonzero y in (, ), there are 4 solutions to f(x) = y in [, ]. So there are 14 4 = 56 solutions to f(f(f(x))) = f(x) and f(x) 0 in [, ]. Plus the 5 solutions to f(x) = 0 in [, ] give 5 solutions to f(f(f(x))) = f(x) and f(x) = 0 in [, ]. Altogether, the number of solutions to f(f(f(x))) = f(x) in [, ] is , which is 61. Problem 18 Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 013 such steps, let D be the number of dice on the table. What is the expected value (average value) of 6 D? Answer: 10,071 Solution: For every nonnegative integer n, let D n be the number of dice on the table after n steps. At the start, D 0 = 1. Given D n, we are told that D n+1 is D n + 1 with probability 6 Dn and is D n otherwise. Hence the expected value of 6 D n+1, given D n, is E(6 D n+1 D n ) = 6 Dn 6 Dn+1 + ( 1 6 Dn) 6 Dn = Dn 1 = 6 Dn + 5. Averaging over D n shows that E(6 D n+1 ) = E(E(6 D n+1 D n )) = E(6 Dn + 5) = E(6 Dn ) + 5. Each expected value is 5 more than the previous one So we have E(6 D 013 ) = E(6 D 0 ) = E(6 1 ) = = Note: This problem is related to the approximate counting randomized algorithm in computer science that is used to count a large number of events with a small amount of memory. Problem 19 If n is a positive integer, let φ(n) be the number of positive integers less than or equal to n that are relatively prime to n. Compute the value of the infinite sum φ(n) n 9 n. n n=1 Express your answer as a fraction in simplest form. Answer:

16 Solution: Let x =. Then the given sum can be rewritten as 9 By geometric series, we have x n φ(n) 1 x. n n=1 x n 1 x = n k=1 x kn. By setting m = kn, we can rewrite the last sum as x n 1 x = n where m ranges over all positive multiples of n. So our original sum becomes the double sum φ(n)x m. n=1 m=1 n m m=1 n m By switching the order of the sums, we get m=1 x m, φ(n)x m. n=1 n m For every fixed positive integer m, we claim that φ(n) = m. n=1 n m If d is a positive divisor of m, then the number of positive integers x less than or equal to m such that gcd(x, m) = d is φ(m/d). Summing over all d shows that φ(m/d) = m. d d m Setting n = m/d proves the claim. 16

17 We can use the claim of the last paragraph to simplify the double sum before that: φ(n)x m = mx m. m=1 n=1 n m m=1 The sum on the right can be evaluated in standard ways: mx m = m=1 m=1 k=1 Plugging in x =, we get 9 m x m = k=1 m=k x m = k=1 /9 (1 /9) = 9 = x k 1 x = x (1 x). Note: The infinite sum in the statement of the problem is an example of a Lambert series. Problem 0 Let a 0, a 1, a,... be an infinite sequence of real numbers such that a 0 = 4 5 and a n = a n 1 1 for every positive integer n. Let c be the smallest number such that for every positive integer n, the product of the first n terms satisfies the inequality a 0 a 1... a n 1 c n. What is the value of 100c, rounded to the nearest integer? Answer: 167 Solution: The recurrence a n = a n 1 1 resembles the double-angle identity cos θ = cos θ 1. With that in mind, we shall set Then we have In the same way, we find that θ = cos 1 a 0 = cos a 1 = a 0 1 = cos θ 1 = cos θ. a k = cos k θ. 17

18 We re interested in the product cos θ cos θ cos 4θ... cos n 1 θ. Using the double-angle identity sin θ = sin θ cos θ, we get the extended identity sin θ cos θ cos θ cos 4θ... cos n 1 θ = sin n θ. Because a k = cos k θ, we have n (sin θ)a 0 a 1 a... a n 1 = sin n θ. Because cos θ = 4 5, we have sin θ = 3 5. Therefore Because sin n θ 1, we have a 0 a 1 a... a n 1 = 5 sin n θ 3 n. a 0 a 1 a... a n n. Hence c 5 3. If sin n θ gets arbitrarily close to 1, then the analysis above would show that c = 5 3. Does sin n θ get arbitrarily close to 1? Probably, but I don t know how to prove it. (It s related to the binary representation of θ.) Nevertheless, we can check by computer that Then the analysis above shows that sin 36 θ > c 5 3 sin 36 θ > = So 100c is greater than and at most 500 = 166. Hence 100c rounded 3 3 to the nearest integer is

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