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HMMT November 2015 November 14, 2015 Individual 1. Find the number of triples (a, b, c) of positive integers such that a + ab + abc = 11. Proposed by: Yang Liu Answer: 3 We can write a + ab + abc = a(1 + b + bc). Since 11 is prime, a = 11 or a = 1. But since b, c are both positive integers, we cannot have a = 11, and so a = 1. Then 1 + b + bc = 11 =⇒ b + bc = 10 =⇒ b(c + 1) = 10, and since c is a positive integer, only b = 1, 2, 5 are possible. This gives the 3 triples (a, b, c) = (1, 1, 9), (1, 2, 4), (1, 5, 1). 2. Let a and b be real numbers randomly (and independently) chosen from the range [0, 1]. Find the probability that a, b and 1 form the side lengths of an obtuse triangle. Proposed by: Alexander Katz Answer: π−2 4 We require a + b > 1 and a2 + b2 < 1. Geometrically, this is the area enclosed in the quarter-circle centered at the origin with radius 1, not including the area enclosed by a + b < 1 (an isosceles right triangle with side length 1). As a result, our desired probability is π−2 4 . 3. Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds? Proposed by: Alexander Katz Answer: 13 Suppose instead Neo started at a weight of 2015 pounds, instead had green pills, which halve his weight, and purple pills, which increase his weight by a pound, and he wished to reduce his weight to one pound. It is clear that, if Neo were able to find such a sequence of pills in the case where he goes from 2015 pounds to 1 pound, he can perform the sequence in reverse (replacing green pills with red pills and purple pills with blue pills) to achieve the desired weight, so this problem is equivalent to the original. Suppose at some point, Neo were to take two purple pills followed by a green pill; this changes his weight from 2k to k + 1. However, the same effect could be achieved using less pills by first taking a green pill and then taking a purple pill, so the optimal sequence will never contain consecutive purple pills. As a result, there is only one optimal sequence for Neo if he is trying to lose weight: take a purple pill when his weight is odd, and a green pill when his weight is even. His weight thus becomes 2015 → 2016 → 1008 → 504 → 252 → 126 → 63 → 64 → 32 → 16 → 8 → 4 → 2 → 1 which requires a total of 13 pills. Reversing this sequence solves the original problem directly. 4. Chords AB and CD of a circle are perpendicular and intersect at a point P . If AP = 6, BP = 12, and CD = 22, find the area of the circle. Proposed by: Sam Korsky Answer: 130π Let O be the center of the circle and let M be the midpoint of segment AB and let N be the midpoint of segment CD. Since quadrilateral OM P N is a rectangle we have that ON = M P = AM − AP = 3 so p √ √ OC = ON 2 + N C 2 = 9 + 121 = 130 Hence the desired area is 130π . 5. Let S be a subset of the set {1, 2, 3, . . . , 2015} such that for any two elements a, b ∈ S, the difference a − b does not divide the sum a + b. Find the maximum possible size of S. Proposed by: Sam Korsky Answer: 672 From each of the sets {1, 2, 3}, {4, 5, 6}, {7, 8, 9}, . . . at most 1 element can be in S. This leads to an upper bound of 2015 = 672 which we can obtain with the set {1, 4, 7, . . . , 2014}. 3 6. Consider all functions f : Z → Z satisfying f (f (x) + 2x + 20) = 15. Call an integer n good if f (n) can take any integer value. In other words, if we fix n, for any integer m, there exists a function f such that f (n) = m. Find the sum of all good integers x. Proposed by: Yang Liu Answer: -35 For almost all integers x, f (x) 6= −x − 20. If f (x) = −x − 20, then f (−x − 20 + 2x + 20) = 15 =⇒ −x − 20 = 15 =⇒ x = −35. Now it suffices to prove that the f (−35) can take any value. f (−35) = 15 in the function f (x) ≡ 15. Otherwise, set f (−35) = c, and f (x) = 15 for all other x. It is easy to check that these functions all work. 7. Let 4ABC be a right triangle with right angle C. Let I be the incenter of ABC, and let M lie on AC and N on BC, respectively, such that M, I, N are collinear and M N is parallel to AB. If AB = 36 and the perimeter of CM N is 48, find the area of ABC. Proposed by: Alexander Katz Answer: 252 Note that ∠M IA = ∠BAI = ∠CAI, so M I = M A. Similarly, N I = N B. As a result, CM + M N + N C = CM + M I + N I + N C = CM + M A + N B + N C = AC + BC = 48. Furthermore, AC 2 +BC 2 = 362 . As a result, we have AC 2 +2AC ·BC +BC 2 = 482 , so 2AC ·BC = 482 −362 = 12·84, and so AC·BC = 3 · 84 = 252 . 2 8. Let ABCD be a quadrilateral with an inscribed circle ω that has center I. If IA = 5, IB = 7, IC = AB 4, ID = 9, find the value of CD . Proposed by: Sam Korsky Answer: 35 36 The I-altitudes of triangles AIB and CID are both equal to the radius of ω, hence have equal length. [AIB] AB = CD . Also note that [AIB] = IA · IB · sin AIB and [CID] = IC · ID · sin CID, Therefore [CID] but since lines IA, IB, IC, ID bisect angles ∠DAB, ∠ABC, ∠BCD, ∠CDA respectively we have that ∠AIB + ∠CID = (180◦ − ∠IAB − ∠IBA) + (180◦ − ∠ICD − ∠IDC) = 180◦ . So, sin AIB = sin CID. [AIB] IA·IB Therefore [CID] = IC·ID . Hence AB IA · IB 35 = = . CD IC · ID 36 games won 9. Rosencrantz plays n ≤ 2015 games of question, and ends up with a win rate i.e. ##ofofgames of played k. Guildenstern has also played several games, and has a win rate less than k. He realizes that if, after playing some more games, his win rate becomes higher than k, then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of k. Proposed by: Alexander Katz Answer: 1 2015 Write k = m n , for relatively prime integers m, n. For the property not to hold, there must exist integers a and b for which m a+1 a < < b n b+1 (i.e. at some point, Guildenstern must ”jump over” k with a single win) ⇐⇒ an + n − m > bm > an hence there must exist a multiple of m strictly between an and an + n − m. If n−m = 1, then the property holds as there is no integer between an and an+n−m = an+1. We now show that if n−m 6= 1, then the property does not hold. By Bzout’s Theorem, as n and m are relatively prime, there exist a and x such that an = mx−1, where 0 < a < m. Then an+n−m ≥ an+2 = mx+1, n so b = x satisfies the conditions. As a result, the only possible k are those in the form n+1 . We know that Rosencrantz played at most 2015 games, so the largest non-perfect winrate he could 1 2014 possibly have is 2015 . Therefore, k ∈ { 21 , 23 , . . . , 2014 2015 }, the product of which is 2015 . 10. Let N be the number of functions f from {1, 2, . . . , 101} → {1, 2, . . . , 101} such that f 101 (1) = 2. Find the remainder when N is divided by 103. Proposed by: Yang Liu Answer: 43 For convenience, let n = 101. Compute the number of functions such that f n (1) = 1. Since n is a prime, there are 2 cases: the order of 1 is either 1 or n. The first case gives nn−1 functions, and the second case gives (n − 1)! functions. By symmetry, the number of ways for f n (1) = 2 is 1 · (nn − nn−1 − (n − 1)!) = nn−1 − (n − 2)!. n−1 Plugging in n = 101, we need to find 101100 − 99! ≡ (−2)−2 − = 1/4 − 1/6 = 1/12 = 43 101! 6 (mod 103).