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2018 MBMT Team Round - Montgomery Blair Math Tournament

Source:

February 13, 2022
algebrageometryMBMTcombinatoricsnumber theory

Problem Statement

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
C1. Mr. Pham flips 20182018 coins. What is the difference between the maximum and minimum number of heads that can appear?
C2 / G1. Brandon wants to maximize □□+□\frac{\Box}{\Box} +\Box by placing the numbers 11, 22, and 33 in the boxes. If each number may only be used once, what is the maximum value attainable?
C3. Guang has 1010 cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have?
C4. The ninth edition of Campbell Biology has 14641464 pages. If Chris reads from the beginning of page 426426 to the end of page449449, what fraction of the book has he read?
C5 / G2. The planet Vriky is a sphere with radius 5050 meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey?
C6 / G3. Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from 00 to 100100 inclusive. However, according to school policy, if the quarter grade is less than or equal to 5050, then it is bumped up to 5050. What is the probability that Stan’s final quarter grade is 5050?
C7 / G5. What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length 11 and a square of side length 2020?
C8. You enter the MBMT lottery, where contestants select three different integers from 11 to 55 (inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win?
C9 / G7. Find a possible solution (B,E,T)(B, E, T) to the equation THE+MBMT=2018THE + MBMT = 2018, where T,H,E,M,BT, H, E, M, B represent distinct digits from 00 to 99.
C10. ABCDABCD is a unit square. Let EE be the midpoint of ABAB and FF be the midpoint of ADAD. DEDE and CFCF meet at GG. Find the area of △EFG\vartriangle EFG.
C11. The eight numbers 20152015, 20162016, 20172017, 20182018, 20192019, 20202020, 20212021, and 20222022 are split into four groups of two such that the two numbers in each pair differ by a power of 22. In how many different ways can this be done?
C12 / G4. We define a function f such that for all integers n,k,xn, k, x, we have that f(n,kx)=knf(n,x)andf(n+1,x)=xf(n,x).f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x). If f(1,k)=2kf(1, k) = 2k for all integers kk, then what is f(3,7)f(3, 7)?
C13 / G8. A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be 44, 7979, 10351035. How long is the longest possible sequence that satisfies these rules?
C14 / G11. ABCABC is an equilateral triangle of side length 88. PP is a point on side AB. If AC+CP=5⋅APAC +CP = 5 \cdot AP, find APAP.
C15. What is the value of (1)+(1+2)+(1+2+3)+...+(1+2+...+49+50)(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)?
G6. An ant is on a coordinate plane. It starts at (0,0)(0, 0) and takes one step each second in the North, South, East, or West direction. After 55 steps, what is the probability that the ant is at the point (2,1)(2, 1)?
G10. Find the set of real numbers SS so that ∏c∈S(x2+cxy+y2)=(x2−y2)(x12−y12).\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).
G12. Given a function f(x)f(x) such that f(a+b)=f(a)+f(b)+2abf(a + b) = f(a) + f(b) + 2ab and f(3)=0f(3) = 0, find f(12)f\left( \frac12 \right).
G13. Badville is a city on the infinite Cartesian plane. It has 2424 roads emanating from the origin, with an angle of 1515 degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from (10,0)(10, 0) to (3,3)(3, 3). What is the minimum distance he can take, only going on roads?
G14. Team AA and Team BB are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a 50%50\% chance of scoring 11 point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team AA will score 55 points before Team BB scores any?
G15. The twelve-digit integer A58B3602C91D‾,\overline{A58B3602C91D}, where A,B,C,DA, B, C, D are digits with A>0A > 0, is divisible by 1010110101. Find ABCD‾\overline{ABCD}.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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