MathDB

2017 MMATHS

Part of MMATHS problems

Subcontests

(6)
Mixer Round
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2017 MMATHS Mixer Round - Math Majors of America Tournament for High Schools

p1. Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls 11, 22, 33, 44, 55, and then 66 in that order is pp. The probability that he rolls 22, 22, 44, 44, 66, and then 66 in that order is qq. What is p−qp - q?
p2. What is the smallest positive integer xx such that x≡2017x \equiv 2017 (mod 20162016) and x≡2016x \equiv 2016 (mod 20172017) ?
p3. The vertices of triangle ABCABC lie on a circle with center OO. Suppose the measure of angle ACBACB is 45o45^o. If ∣AB∣=10|AB| = 10, then what is the distance between OO and the line ABAB?
p4. A “word“ is a sequence of letters such as YALEYALE and AELYAELY. How many distinct 33-letter words can be made from the letters in BOOLABOOLABOOLABOOLA where each letter is used no more times than the number of times it appears in BOOLABOOLABOOLABOOLA?
p5. How many distinct complex roots does the polynomial p(x)=x12−x8−x4+1p(x) = x^{12} - x^8 - x^4 + 1 have?
p6. Notice that 1=12+13+161 = \frac12 + \frac13 + \frac16 , that is, 11 can be expressed as the sum of the three fractions 12\frac12 , 13\frac13 , and 16\frac16 , where each fraction is in the form 1n\frac{1}{n}, with each nn different. Give a 66-tuple of distinct positive integers (a,b,c,d,e,f)(a, b, c, d, e, f) where a<b<c<d<e<fa < b < c < d < e < f such that 1a+1b+1c+1d+1e+1f=1\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1 and explain how you arrived at your 66-tuple. Multiple answers will be accepted.
p7. You have a Monopoly board, an 11×1111 \times 11 square grid with the 9×99 \times 9 internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard 66-sided dice. Let SS be the set of squares on the board such that if you are initially on a square in SS, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in SS as your starting position. What is the probability that you land on Go?
p8. Using LL-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a 33-by-20172017 rectangle without any gaps?
p9. Does there exist a pair of positive integers (x,y)(x, y), where x<yx < y, such that x2+y2=10093x^2 + y^2 = 1009^3? If so, give a pair (x,y)(x, y) and explain how you found that pair. If not, explain why.
p10. Triangle ABCABC has inradius 88 and circumradius 2020. Let MM be the midpoint of side BCBC, and let NN be the midpoint of arc BCBC on the circumcircle not containing AA. Let sAs_A denote the length of segment MNMN, and define sBs_B and sCs_C similarly with respect to sides CACA and ABAB. Evaluate the product sAsBsCs_As_Bs_C.
p11. Julia and Dan want to divide up 256256 dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until 44 rejections have been made; once 44 rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round?
p12. A perfect partition of a positive integer NN is an unordered set of numbers (where numbers can be repeated) that sum to NN with the property that there is a unique way to express each positive integer less than NN as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of 33 are {1,1,1}\{1, 1, 1\} and {1,2}\{1, 2\}. {1,1,3,4}\{1, 1, 3, 4\} is NOT a perfect partition of 99 because the sum 44 can be achieved in two different ways: 44 and 1+31 + 3. How many integers 1≤N≤401 \le N \le 40 each have exactly one perfect partition?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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2017 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS

Round 5
p13. Points A,B,CA, B, C, and DD lie in a plane with AB=6AB = 6, BC=5BC = 5, and CD=5CD = 5, and ABAB is perpendicular to BCBC. Point E lies on line ADAD such that D≠ED \ne E, AE=3AE = 3 and CE=5CE = 5. Find DEDE.
p14. How many ordered pairs of integers (x,y)(x,y) are solutions to x2y=36+yx^2y = 36 + y?
p15. Chicken nuggets come in boxes of two sizes, aa nuggets per box and bb nuggets per box. We know that 899899 nuggets is the largest number of nuggets we cannot obtain with some combination of aa-sized boxes and bb-sized boxes. How many different pairs (a,b)(a, b) are there with a<ba < b?
Round 6
p16. You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value pp is chosen randomly and uniformly between 00 and 11. Alice then makes a biased coin that lands on heads with probability pp, and Bob makes a biased coin that lands on heads with probability 1−p1 -p. You design your own biased coin to maximize your chance of winning without knowing pp. What is the probability that you win?
p17. There are NN distinct students, numbered from 11 to NN. Each student has exactly one hat: yy students have yellow hats, bb have blue hats, and rr have red hats, where y+b+r=Ny + b + r = N and y,b,r>0y, b, r > 0. The students stand in a line such that all the rr people with red hats stand in front of all the bb people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The yy people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is 20162016. What is 100y+10b+r100y + 10b + r?
p18. Let P be a point in rectangle ABCDABCD such that ∠APC=135o\angle APC = 135^o and ∠BPD=150o\angle BPD = 150^o. Suppose furthermore that the distance from P to ACAC is 1818. Find the distance from PP to BDBD.
Round 7
p19. Let triangle ABCABC be an isosceles triangle with ∣AB∣=∣AC∣|AB| = |AC|. Let DD and EE lie on ABAB and ACAC, respectively. Suppose ∣AD∣=∣BC∣=∣EC∣|AD| = |BC| = |EC| and triangle ADEADE is isosceles. Find the sum of all possible values of ∠BAC\angle BAC in radians. Write your answer in the form 2arcsin(ab)+cdπ2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi, where ab\frac{a}{b} and cd\frac{c}{d} are in lowest terms, −1≤ab≤1-1 \le \frac{a}{b} \le 1, and −1≤cd≤1-1 \le \frac{c}{d} \le 1.
p20. Kevin is playing a game in which he aims to maximize his score. In the nthn^{th} round, for n≥1n \ge 1, a real number between 00 and 13n\frac{1}{3^n} is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is a+bca + b\sqrt{c} where a,ba, b, and cc are integers and cc is positive and not divisible by any positive perfect square other than 11. What is 100a+10b+c100a + 10b + c?
p21. Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at (2016,2017)(2016, 2017). Define the “area” of a path as the area below the path and above the xx-axis. The sum of areas over all paths that Lisa can take can be represented as as a⋅(40332016)a \cdot {{4033} \choose {2016}} . What is the remainder when aa is divided by 10001000?
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2017 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS

Round 1
p1. Jom and Terry both flip a fair coin. What is the probability both coins show the same side?
p2. Under the same standard air pressure, when measured in Fahrenheit, water boils at 212o212^o F and freezes at 32o32^o F. At thesame standard air pressure, when measured in Delisle, water boils at 00 D and freezes at 150150 D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have y=ax+by = ax + b. What is the value of a+ba + b? (Ignore units.)
p3. What are the last two digits of 51+52+53+⋅⋅⋅+510+5115^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}?
Round 2
p4. Compute the average of the magnitudes of the solutions to the equation 2x4+6x3+18x2+54x+162=02x^4 + 6x^3 + 18x^2 + 54x + 162 = 0.
p5. How many integers between 11 and 10000001000000 inclusive are both squares and cubes?
p6. Simon has a deck of 4848 cards. There are 1212 cards of each of the following 44 suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card?
Round 3
p7. Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability pp of shooting, and 1−p1 - p of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, pp is constant. Once the ball has been shot, the game is over. What is the maximum value of pp that makes Christine’s total probability of shooting the ball 320\frac{3}{20} ?
p8. If xx and yy are real numbers, then what is the minimum possible value of the expression 3x2−12xy+14y23x^2 - 12xy + 14y^2 given that x−y=3x - y = 3?
p9. Let ABCABC be an equilateral triangle, let DD be the reflection of the incenter of triangle ABCABC over segment ABAB, and let EE be the reflection of the incenter of triangle ABDABD over segment ADAD. Suppose the circumcircle Ω\Omega of triangle ADEADE intersects segment ABAB again at XX. If the length of ABAB is 11, find the length of AXAX.
Round 4
p10. Elaine has cc cats. If she divides cc by 55, she has a remainder of 33. If she divides cc by 77, she has a remainder of 55. If she divides cc by 99, she has a remainder of 77. What is the minimum value cc can be?
p11. Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a 1010-sided die (its faces are numbered from 11 to 1010) xx times. If, within those xx rolls, the number 1010 appears, then you win. Assuming that you like winning, what is the highest value of xx where you would prefer to play the coin-flipping game over the die-rolling game?
p12. Let be the set X={0,1,2,...,100}X = \{0, 1, 2, ..., 100\}. A subset of XX, called NN, is defined as the set that contains every element xx of XX such that for any positive integer nn, there exists a positive integer kk such that n can be expressed in the form n=xa1+xa2+...+xakn = x^{a_1}+x^{a_2}+...+x^{a_k} , for some integers a1,a2,...,aka_1, a_2, ..., a_k that satisfy 0≤a1≤a2≤...≤ak0 \le a_1 \le a_2 \le ... \le a_k. What is the sum of the elements in NN?
PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2017 MMATHS Individual Round - Math Majors of America Tournament for High School

p1. What is the smallest positive prime divisor of 1012−992101^2 -99^2?
p2. The product 81⋅1357981 \cdot 13579 equals 1099A991099A99 for some digit AA. What is AA?
p3. On a MMATHS team of 66 students, all students forget which of the 55 MMATHS sites (Yale, UVA, UF, UM, and Columbia) they are supposed to attend. On competition day, each student chooses a random site uniformly and independently of each other to attend, and they each take a flight to the one they picked. What is the probability that all of the sites end up with at least one person from that team?
p4. Square ABCDABCD has area 1212. Let EE be the midpoint of ADAD, and let FF be the point of intersection of BEBE and ACAC. Find the area of quadrilateral EFCDEFCD.
p5. Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a 23\frac23 chance that it is an end piece and a 13\frac13 chance that it is a middle piece. After placing a randomly selected olive slice on his plate, he checks to see if he can rearrange the slices on his plate so that he has at least one whole olive, which consists of any middle piece and any two end pieces. He can successfully rearrange the olive slices on his plate into at least one whole olive after randomly selecting nn olive slices with probability of at least 0.80.8. What is the minimum n for which this is true?
p6. What is the smallest positive integer nn whose digits multiply to 80008000?
p7. Mitchell has a bunch of 2×12\times 1 rectangular tiles. He needs to arrange them in such a way that they exactly cover a 2×102\times 10 grid of 1×11\times 1 squares. How many ways are there for him to do this? Two such “tilings“ are considered distinct from each other if not all of the 22-by-11 rectangular tiles in the two tilings are placed in exactly the same position and orientation.
p8. Trapezoid ABCDABCD is inscribed in a circle of radius 100100. If AB=200AB = 200 and BC=50BC = 50, what is the perimeter of ABCDABCD?
p9. How many ordered pairs of nonzero integers (a,b)(a, b) are there such that 1a+1b=124\frac{1}{a} + \frac{1}{b} = \frac{1}{24}?
p10. Suppose aa and bb are complex numbers such that ∣a∣=∣b∣=1|a|= |b| = 1 and a+b=13+14ia + b = \frac13 + \frac14 i. Find the product abab.
p11. Find the number of ordered triples of positive integers (a1,a2,a3)(a_1, a_2, a_3) such that a1+a2+a3=2017a_1 + a_2 + a_3 = 2017 and 22 does not divide a1a_1, 33 does not divide a2a_2, and 44 does not divide a3a_3.
p12. Circles Γ\Gamma and Ω\Omega have radii of 2020 and 1717, respectively, and their centers are separated by a distance of 55. Let XX be a fixed intersection point of these two circles, and let ZZ be some point on Γ\Gamma for which segment XZXZ intersects Ω\Omega at a second point YY . (If XZXZ is tangent to Ω\Omega, then define YY to be XX.) Find the maximum possible length of segment YZY Z.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.