MathDB

2019

Part of MBMT Guts Rounds

Problems(3)

2019 MBMT Guts Round D16-30/ L10-15 Montgomery Blair Math Tournament

Source:

2/27/2022
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
Set 4
D.16 / L.6 Alex has 100100 Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most 11 spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them?
D.17 I start with the number 11 in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket?
D.18 / L.12 Point PP inside square ABCDABCD is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area 2525, 2525, and 1515, what are all the possible values for the area of the fourth triangle?
D.19 Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at 100100 or more decibels, then they lose the game. Mr. Stein yells first, at 8888 decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win?
D.20 / L.15 A semicircle of radius 11 has line ℓ\ell along its base and is tangent to line mm. Let rr be the radius of the largest circle tangent to ℓ\ell, mm, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of rr is the interval [a,b][a, b]. Find b−ab - a.
Set 5
D.21 / L.14 Hungryman starts at the tile labeled “SS”. On each move, he moves 11 unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats.
https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png
D.22 / L.11 How many triples of nonnegative integers (x,y,z)(x, y, z) satisfy the equation 6x+10y+15z=3006x + 10y +15z = 300?
D.23 / L.16 Anson, Billiam, and Connor are looking at a 3D3D figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a 5×55 \times 5 square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
D.24 / L.13 Tse and Cho are playing a game. Cho chooses a number x∈[0,1]x \in [0, 1] uniformly at random, and Tse guesses the value of x(1−x)x(1 - x). Tse wins if his guess is at most 150\frac{1}{50} away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins?
D.25 / L.20 Find the largest solution to the equation 2019(x2019x2019−20192+2019)2019)=2019x2019+1.2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.
Set 6
This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.
D.26 / L.26 What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were 4747 volunteers; this is the fifth MBMT.
D.27 / L.27 William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than 10%10\% of the entire chocolate bar.
D.28 / L.28 Let xx be the positive solution to the equation xxxx=1.1x^{x^{x^x}}= 1.1. Estimate 1x−1\frac{1}{x-1}.
D.29 / L.29 Estimate the number of dots in the following box: https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png It may be useful to know that this image was produced by plotting (4x,y)(4\sqrt{x}, y) some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval [0,1][0, 1].
D.30 / L.30 For a positive integer nn, let f(n)f(n) be the smallest prime greater than or equal to nn. Estimate (f(1)−1)+(f(2)−2)+(f(3)−3)+...+(f(10000)−10000).(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).
For 26≤i≤3026 \le i \le 30, let EiE_i be your team’s answer to problem ii and let AiA_i be the actual answer to problem ii. Your score SiS_i for problem ii is given by S26=max⁡(0,12−∣E26−A26∣/5)S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5) S27=max⁡(0,12−100∣E27−A27∣)S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|) S28=max⁡(0,12−5∣E28−A28∣))S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|)) S29=12max⁡(0,1−3∣E29−A29∣A29)S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right) S30=max⁡(0,12−∣E30−A30∣/2000)S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theoryMBMT
2019 MBMT Guts Round D1-15/ L1-9 Montgomery Blair Math Tournament

Source:

2/27/2022
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
Set 1
D.1 / L.1 Find the units digit of 313373^{1^{3^{3^7}}}.
D.2 Find the positive solution to the equation x3−x2=x−1x^3 - x^2 = x - 1.
D.3 Points AA and BB lie on a unit circle centered at O and are distance 11 apart. What is the degree measure of ∠AOB\angle AOB?
D.4 A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between 11 and 20192019, inclusive?
D.5 Ted has four children of ages 1010, 1212, 1515, and 1717. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now?
Set 2
D.6 Mr. Schwartz is on the show Wipeout, and is standing on the first of 55 balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is 1/21/2, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish.
D.7 / L. 5 Kevin has written 55 MBMT questions. The shortest question is 55 words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question?
D.8 / L. 3 Square ABCDABCD with side length 11 is rolled into a cylinder by attaching side ADAD to side BCBC. What is the volume of that cylinder?
D.9 / L.4 Haydn is selling pies to Grace. He has 44 pumpkin pies, 33 apple pies, and 11 blueberry pie. If Grace wants 33 pies, how many different pie orders can she have?
D.10 Daniel has enough dough to make 88 1212-inch pizzas and 1212 88-inch pizzas. However, he only wants to make 1010-inch pizzas. At most how many 1010-inch pizzas can he make?
Set 3
D.11 / L.2 A standard deck of cards contains 1313 cards of each suit (clubs, diamonds, hearts, and spades). After drawing 5151 cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs?
D.12 / L. 7 Let s(n)s(n) be the sum of the digits of nn. Let g(n)g(n) be the number of times s must be applied to n until it has only 11 digit. Find the smallest n greater than 20192019 such that g(n)≠g(n+1)g(n) \ne g(n + 1).
D.13 / L. 8 In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the 20192019 contestants, is secretly told that her score is SS. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of SS?
D.14 / L. 9 Let AA and BB be opposite vertices on a cube with side length 11, and let XX be a point on that cube. Given that the distance along the surface of the cube from AA to XX is 11, find the maximum possible distance along the surface of the cube from BB to XX.
D.15 A function ff with f(2)>0f(2) > 0 satisfies the identity f(ab)=f(a)+f(b)f(ab) = f(a) + f(b) for all a,b>0a, b > 0. Compute f(22019)f(23)\frac{f(2^{2019})}{f(23)}.

PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theoryMBMT
2019 MBMT Guts Round L10,16-30 Montgomery Blair Math Tournament

Source:

2/27/2022
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
L.10 Given the following system of equations where x,y,zx, y, z are nonzero, find x2+y2+z2x^2 + y^2 + z^2. x+2y=xyx + 2y = xy 3y+z=yz3y + z = yz 3x+2z=xz3x + 2z = xz
Set 4
L.16 / D.23 Anson, Billiam, and Connor are looking at a 3D3D figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a 5×55 \times 5 square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
L.17 The repeating decimal 0.MBMT‾0.\overline{MBMT} is equal to pq\frac{p}{q}, where pp and qq are relatively prime positive integers, and M,B,TM, B, T are distinct digits. Find the minimum value of qq.
L.18 Annie, Bob, and Claire have a bag containing the numbers 1,2,3,...,91, 2, 3, . . . , 9. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so 123123, 213213, and 321321 all count as arithmetic sequences.)
L.19 Consider a set SS of positive integers. Define the operation f(S)f(S) to be the smallest integer n>1n > 1 such that the base 2k2^k representation of nn consists only of ones and zeros for all k∈Sk \in S. Find the size of the largest set SS such that f(S)<22019f(S) < 2^{2019}.
L.20 / D.25 Find the largest solution to the equation 2019(x2019x2019−20192+2019)2019=2019x2019+1.2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.
Set 5
L.21 Steven is concerned about his artistic abilities. To make himself feel better, he creates a 100×100100 \times 100 square grid and randomly paints each square either white or black, each with probability 12\frac12. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png
L.22 Let x be chosen uniformly at random from [0,1][0, 1]. Let n be the smallest positive integer such that 3nx3^n x is at most 14\frac14 away from an integer. What is the expected value of nn?
L.23 Let AA and BB be two points in the plane with AB=1AB = 1. Let ℓ\ell be a variable line through AA. Let ℓ′\ell' be a line through BB perpendicular to ℓ\ell. Let X be on ℓ\ell and YY be on ℓ′\ell' with AX=BY=1AX = BY = 1. Find the length of the locus of the midpoint of XYXY .
L.24 Each of the numbers aia_i, where 1≤i≤n1 \le i \le n, is either −1-1 or 11. Also, a1a2a3a4+a2a3a4a5+...+an−3an−2an−1an+an−2an−1ana1+an−1ana1a2+ana1a2a3=0.a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0. Find the number of possible values for nn between 44 and 100100, inclusive.
L.25 Let SS be the set of positive integers less than 320193^{2019} that have only zeros and ones in their base 33 representation. Find the sum of the squares of the elements of SS. Express your answer in the form ab(cd−1)(ef−1)a^b(c^d - 1)(e^f - 1), where a,b,c,d,e,fa, b, c, d, e, f are positive integers and a,c,ea, c, e are not perfect powers.
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theoryMBMT