MathDB
2013 BmMT Team Round - Berkley mini Math Tournament Fall

Source:

January 9, 2022
algebrageometrycombinatoricsnumber theorybmmt

Problem Statement

p1. If Bob takes 66 hours to build 44 houses, how many hours will he take to build 12 12 houses?
p2. Compute the value of 12+16+112+120\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}.
p3. Given a line 2x+5y=1702x + 5y = 170, find the sum of its xx- and yy-intercepts.
p4. In some future year, BmMT will be held on Saturday, November 1919th. In that year, what day of the week will April Fool’s Day (April 11st) be?
p5. We distribute 7878 penguins among 1010 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 1010 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 11 through 480480. How many times does the digit 88 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 22, what is the area of the flower?
p10. There are two non-consecutive positive integers a,ba, b such that a2b2=291a^2 - b^2 = 291. Find aa and bb.
p11. Let ABCABC be an equilateral triangle. Let P,Q,RP, Q, R be the midpoints of the sides BCBC, CACA and ABAB respectively. Suppose the area of triangle PQRPQR is 11. Among the 66 points A,B,C,P,Q,RA, B, C, P, Q, R, how many distinct triangles with area 11 have vertices from that set of 66 points?
p12. A positive integer is said to be binary-emulating if its base three representation consists of only 00s and 11s. Determine the sum of the first 1515 binary-emulating numbers.
p13. Professor XX can choose to assign homework problems from a set of problems labeled 1 1 to 3030, inclusive. No two problems in his assignment can share a common divisor greater than 1 1. What is the maximum number of problems that Professor XX can assign?
p14. Trapezoid ABCDABCD has legs (non-parallel sides) BCBC and DADA of length 55 and 66 respectively, and there exists a point XX on CDCD such that XBC=XAD=AXB=90o\angle XBC = \angle XAD = \angle AXB = 90^o . Find the area of trapezoid ABCDABCD.
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 ABCABC be a triangle and M be the midpoint of BCBC. If AB=AM=5AB = AM = 5 and BC=12BC = 12, what is the area of triangle ABCABC?
p17. A positive integer nn is called good if it can be written as 5x+8y=n5x+ 8y = n for positive integers x,yx, y. Given that 4242, 4343, 4444, 4545 and 4646 are good, what is the largest n that is not good?
p18. Below is a 7×7 7 \times 7 square with each of its unit squares labeled 11 to 4949 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 12251225. 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.png
p19. A circle of integer radius r r has a chord PQPQ of length 88. There is a point XX on chord PQPQ such that PX=2\overline{PX} = 2 and XQ=6\overline{XQ} = 6. Call a chord ABAB euphonic if it contains XX and both AX\overline{AX} and XB\overline{XB} are integers. What is the minimal possible integer r r such that there exist 66 euphonic chords for XX?
p20. On planet Silly-Math, two individuals may play a game where they write the number 324000324000 on a whiteboard and take turns dividing the number by prime powers – numbers of the form pkp^k for some prime pp and positive integer kk. 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 324000324000 by in order to ensure a win.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.