2013 BmMT Team Round - Berkley mini Math Tournament Fall
Source:
January 9, 2022
algebrageometrycombinatoricsnumber theorybmmt
Problem Statement
p1. If Bob takes hours to build houses, how many hours will he take to build houses?
p2. Compute the value of .
p3. Given a line , find the sum of its - and -intercepts.
p4. In some future year, BmMT will be held on Saturday, November th. In that year, what day of the week will April Fool’s Day (April st) be?
p5. We distribute penguins among people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take?
p6. A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry?
p7. Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct?
p8. The pages of a book are consecutively numbered from through . How many times does the digit appear in this numbering?
p9. A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length , what is the area of the flower?
p10. There are two non-consecutive positive integers such that . Find and .
p11. Let be an equilateral triangle. Let be the midpoints of the sides , and respectively. Suppose the area of triangle is . Among the points , how many distinct triangles with area have vertices from that set of points?
p12. A positive integer is said to be binary-emulating if its base three representation consists of only s and s. Determine the sum of the first binary-emulating numbers.
p13. Professor can choose to assign homework problems from a set of problems labeled to , inclusive. No two problems in his assignment can share a common divisor greater than . What is the maximum number of problems that Professor can assign?
p14. Trapezoid has legs (non-parallel sides) and of length and respectively, and there exists a point on such that . Find the area of trapezoid .
p15. Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.)
p16. Let be a triangle and M be the midpoint of . If and , what is the area of triangle ?
p17. A positive integer is called good if it can be written as for positive integers . Given that , , , and are good, what is the largest n that is not good?
p18. Below is a square with each of its unit squares labeled to in order. We call a square contained in the figure good if the sum of the numbers inside it is odd. For example, the entire square is good because it has an odd sum of . Determine the number of good squares in the figure.1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49
https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.pngp19. A circle of integer radius has a chord of length . There is a point on chord such that and . Call a chord euphonic if it contains and both and are integers. What is the minimal possible integer such that there exist euphonic chords for ?
p20. On planet Silly-Math, two individuals may play a game where they write the number on a whiteboard and take turns dividing the number by prime powers – numbers of the form for some prime and positive integer . Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide by in order to ensure a win.
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