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2023 MBMT Guts Round B16-30 G11-30 Montgomery Blair Math Tournament

Source:

August 11, 2023
MBMTalgebrageometrycombinatoricsnumber theory

Problem Statement

[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.
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