MathDB

2023

Part of MBMT Guts Rounds

Problems(2)

2023 MBMT Guts Round B1-15, G1-10 Montgomery Blair Math Tournament

Source:

8/11/2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
Set 1
B1 / G1 Find 203+22+3120^3 + 2^2 + 3^1.
B2 A piece of string of length 1010 is cut 44 times into strings of equal length. What is the length of each small piece of string?
B3 / G2 What is the smallest perfect square that is also a perfect cube?
B4 What is the probability a 55-sided die with sides labeled from 11 through 55 rolls an odd number?
B5 / G3 Hanfei spent 1414 dollars on chicken nuggets at McDonalds. 44 nuggets cost 33 dollars, 66 nuggets cost 44 dollars, and 1212 nuggets cost 99 dollars. How many chicken nuggets did Hanfei buy?
Set 2
B6 What is the probability a randomly chosen positive integer less than or equal to 1515 is prime?
B7 Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled 11 through 66. What is the probability that the sum is greater than 55?
B8 / G4 What is the radius of a circle with area 44?
B9 What is the maximum number of equilateral triangles on a piece of paper that can share the same corner?
B10 / G5 Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in 2020 minutes. Bab can make a pizza in 3030 minutes. If Bob and Bab want to make 5050 pizzas in total, how many hours would that take them?
Set 3
B11 / G6 Find the area of an equilateral rectangle with perimeter 2020.
B12 / G7 What is the minimum possible number of divisors that the sum of two prime numbers greater than 22 can have?
B13 / G8 Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins?
B14 / G9 Aven has 44 distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table?
B15 / G10 Find the largest 77-digit palindrome that is divisible by 1111.
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMTalgebrageometrycombinatoricsnumber theory
2023 MBMT Guts Round B16-30 G11-30 Montgomery Blair Math Tournament

Source:

8/11/2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
Set 4
B16 / G11 Let triangle ABCABC be an equilateral triangle with side length 66. If point DD is on ABAB and point EE is on BCBC, find the minimum possible value of AD+DE+CEAD + DE + CE.
B17 / G12 Find the smallest positive integer nn with at least seven divisors.
B18 / G13 Square AA is inscribed in a circle. The circle is inscribed in Square BB. If the circle has a radius of 1010, what is the ratio between a side length of Square AA and a side length of Square BB?
B19 / G14 Billy Bob has 55 distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other?
B20 / G15 Six people make statements as follows: Person 11 says “At least one of us is lying.” Person 22 says “At least two of us are lying.” Person 33 says “At least three of us are lying.” Person 44 says “At least four of us are lying.” Person 55 says “At least five of us are lying.” Person 66 says “At least six of us are lying.” How many are lying?
Set 5
B21 / G16 If xx and yy are between 00 and 11, find the ordered pair (x,y)(x, y) which maximizes (xy)2(x2−y2)(xy)^2(x^2 - y^2).
B22 / G17 If I take all my money and divide it into 1212 piles, I have 1010 dollars left. If I take all my money and divide it into 1313 piles, I have 1111 dollars left. If I take all my money and divide it into 1414 piles, I have 1212 dollars left. What’s the least amount of money I could have?
B23 / G18 A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation.
B24 / G20 A regular 1212-sided polygon is inscribed in a circle. Gaz then chooses 33 vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right?
B25 / G22 A book has at most 77 chapters, and each chapter is either 33 pages long or has a length that is a power of 22 (including 11). What is the least positive integer nn for which the book cannot have nn pages?
Set 6
B26 / G26 What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers?
B27 / G27 Estimate 12345112312345^{\frac{1}{123}}.
B28 / G28 Let OO be the center of a circle ω\omega with radius 33. Let A,B,CA, B, C be randomly selected on ω\omega. Let MM, NN be the midpoints of sides BCBC, CACA, and let AMAM, BNBN intersect at GG. What is the probability that OG≤1OG \le 1?
B29 / G29 Let r(a,b)r(a, b) be the remainder when aa is divided by bb. What is ∑i=1100r(2i,i)\sum^{100}_{i=1} r(2^i , i)?
B30 / G30 Bongo flips 20232023 coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets HHHTTHTTHHHHTHHHHT T HT T HHHHT H, he’d have maximal runs of length 3,1,4,13, 1, 4, 1. Bongo squares the lengths of all his maximal runs and adds them to get a number MM. What is the expected value of MM?
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G19 Let ABCDABCD be a square of side length 22. Let MM be the midpoint of ABAB and NN be the midpoint of ADAD. Let the intersection of BNBN and CMCM be EE. Find the area of quadrilateral NECDNECD.
G21 Quadrilateral ABCDABCD has ∠A=∠D=60o\angle A = \angle D = 60^o. If AB=8AB = 8, CD=10CD = 10, and BC=3BC = 3, what is length ADAD?
G23 △ABC\vartriangle ABC is an equilateral triangle of side length xx. Three unit circles ωA\omega_A, ωB\omega_B, and ωC\omega_C lie in the plane such that ωA\omega_A passes through AA while ωB\omega_B and ωC\omega_C are centered at BB and CC, respectively. Given that ωA\omega_A is externally tangent to both ωB\omega_B and ωC\omega_C, and the center of ωA\omega_A is between point AA and line BC‾\overline{BC}, find xx.
G24 For some integers nn, the quadratic function f(x)=x2−(6n−6)x−(n2−12n+12)f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12) has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form 2k2^k for some nonnegative integer kk. What is the sum of all possible values of nn?
G25 In a triangle, let the altitudes concur at HH. Given that AH=30AH = 30, BH=14BH = 14, and the circumradius is 2525, calculate CHCH
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMTalgebrageometrycombinatoricsnumber theory