MathDB

Ind. Round

Part of 2013 BmMT

Problems(1)

2013 BmMT Individual Round - Berkley mini Math Tournament

Source:

10/25/2023
p1. Ten math students take a test, and the average score on the test is 2828. If five students had an average of 1515, what was the average of the other five students' scores?
p2. If a⊗b=a2+b2+2aba\otimes b = a^2 + b^2 + 2ab, find (−5⊗7)⊗4(-5\otimes 7) \otimes 4.
p3. Below is a 3×43 \times 4 grid. Fill each square with either 11, 22 or 33. No two squares that share an edge can have the same number. After filling the grid, what is the 44-digit number formed by the bottom row? https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png
p4. What is the angle in degrees between the hour hand and the minute hand when the time is 6:306:30?
p5. In a small town, there are some cars, tricycles, and spaceships. (Cars have 44 wheels, tricycles have 33 wheels, and spaceships have 66 wheels.) Among the vehicles, there are 2424 total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town?
p6. You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails?
p7. In the below diagram, ∠ABC\angle ABC and ∠BCD\angle BCD are right angles. If AB‾=9\overline{AB} = 9, BD‾=13\overline{BD} = 13, and CD‾=5\overline{CD} = 5, calculate AC‾\overline{AC}. https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png
p8. Out of 100100 customers at a market, 8080 purchased oranges, 6060 purchased apples, and 7070 purchased bananas. What is the least possible number of customers who bought all three items?
p9. Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following: Francis: Fred ate the cookies. Fred: Ted did not eat the cookies. Ted: Francis is lying. If exactly one of them is telling the truth, who ate all the cookies?
p11. Let ABCABC be a triangle with a right angle at AA. Suppose AB‾=6\overline{AB} = 6 and AC‾=8\overline{AC} = 8. If ADAD is the perpendicular from AA to BCBC, what is the length of ADAD?
p12. How many three digit even numbers are there with an even number of even digits?
p13. Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line.
p14. A man 55 foot, 1010 inches tall casts a 1414 foot shadow. 2020 feet behind the man, a flagpole casts ashadow that has a 99 foot overlap with the man's shadow. How tall (in inches) is the flagpole?
p15. Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has nn balls and 3 divides nn, then he throws away a third of the balls. If 33 does not divide nn but 22 divides nn, then he throws away half of them. If neither 33 nor 22 divides nn, he stops throwing away the balls. If he began with 14581458 balls, after how many steps does he stop throwing away balls?
p16. Oski has 5050 coins that total to a value of 8282 cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter).
p17. Let ABCABC be a triangle. Let MM be the midpoint of BCBC. Suppose MA‾=MB‾=MC‾=2\overline{MA} = \overline{MB} = \overline{MC} = 2 and ∠ACB=30o\angle ACB = 30^o. Find the area of the triangle.
p18. A spirited integer is a positive number representable in the form 20n+13k20^n + 13k for some positive integer nn and any integer kk. Determine how many spirited integers are less than 20132013.
p19. Circles of radii 2020 and 1313 are externally tangent at TT. The common external tangent touches the circles at AA, and BB, respectively where A≠BA \ne B. The common internal tangent of the circles at TT intersects segment ABAB at XX. Find the length of AXAX.
p20. A finite set of distinct, nonnegative integers {a1,...,ak}\{a_1, ... , a_k\} is called admissible if the integer function f(n)=(n+a1)...(n+ak)f(n) = (n + a_1) ... (n + a_k) has no common divisor over all terms; that is, gcd(f(1),f(2),...f(n))=1gcd \left(f(1), f(2),... f(n)\right) = 1 for any integern n. How many admissible sets only have members of value less than 1010? {4}\{4\} and {0,2,6}\{0, 2, 6\} are such sets, but {4,9}\{4, 9\} and {1,3,5}\{1, 3, 5\} are not.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
bmmtalgebrageometrycombinatoricsnumber theory