Subcontests
(2)2014 BmMT Individual Round - Berkley mini Math Tournament
p1. Compute 172+17⋅7+72.
p2. You have $1.17 in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $1.17. How many coins do you have?
p3. Suppose that there is a 40% chance it will rain today, and a 20% chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.)
p4. A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than 200.
p5. Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin?
Alice: I have the coin.
Bob: Carl has the coin.
Carl: Exactly one of us is telling the truth.
Dave: The person who has the coin is male.
p6. Vicky has a bag holding some blue and some red marbles. Originally 32 of the marbles are red. After Vicky adds 25 blue marbles, 43 of the marbles are blue. How many marbles were originally in the bag?
p7. Given pentagon ABCDE with BC=CD=DE=4, ∠BCD=90o and ∠CDE=135o, what is the length of BE?
p8. A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is 15 meters per second. From Stanford to San Jose, the train's average speed is 20 meters per second. What is the train's average speed for the entire trip?
p9. Find the area of the convex quadrilateral with vertices at the points (−1,5), (3,8), (3,−1), and (−1,−2).
p10. In an arithmetic sequence a1, a2, a3, ... , twice the sum of the first term and the third term is equal to the fourth term. Find a4/a1.
p11. Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this?
p12. Usually, spaceships have 6 wheels. However, there are more advanced spaceships that have 9 wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have 2 wheels). There are 10 vehicles and 49 wheels in total. How many bicycles are there?
p13. If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both (1,3,2) and (1,2,3) as arithmetic sequences.)
p14. Given regular hexagon ABCDEF with center O and side length 6, what is the area of pentagon ABODE?
p15. Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many di erent options are there for dinner if each person must have at least one dish that they can eat?
p16. Consider the graph of f(x)=x3+x+2014. A line intersects this cubic at three points, two of which have x-coordinates 20 and 14. Find the x-coordinate of the third intersection point.
p17. A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius 8, upper radius 4, and height 3?
p18. A quadrilateral ABCD is de ned by the points A=(2,−1), B=(3,6), C=(6,10) and D=(5,−2). Let ℓ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of ℓ?
p19. Consider the sequence 1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, ... where the elements are Fibonacci numbers and the Fibonacci number Fn appears Fn times. Find the 2014th element of this sequence. (The Fibonacci numbers are defined as F1=F2=1 and for n>2, Fn=Fn−1+Fn−2.)
p20. Call a positive integer top-heavy if at least half of its digits are in the set {7,8,9}. How many three digit top-heavy numbers exist? (No number can have a leading zero.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2014 BmMT Team Round - Berkley mini Math Tournament Fall
p1. Roll two dice. What is the probability that the sum of the rolls is prime?
p2. Compute the sum of the first 20 squares.
p3. How many integers between 0 and 999 are not divisible by 7,11, or 13?
p4. Compute the number of ways to make 50 cents using only pennies, nickels, dimes, and quarters.
p5. A rectangular prism has side lengths 1,1, and 2. What is the product of the lengths of all of the diagonals?p6. What is the last digit of 7654321 ?
p7. Given square ABCD with side length 3, we construct two regular hexagons on sides AB and CD such that the hexagons contain the square. What is the area of the intersection of the two hexagons?
https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png
p8. Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of 10 miles per hour per minute until she reaches the speed limit of 25 miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of 20 miles per hour per minute until he reaches the rate of 40 miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially?
p9. Find the ordered pair of positive integers (x,y) such that 144x−89y=1 and x is minimal.
p10. How many zeroes does the product of the positive factors of 10000 (including 1 and 10000) have?
p11. There is a square configuration of desks. It is known that one can rearrange these desks such that it has 7 fewer rows but 10 more columns, with 13 desks remaining. How many desks are there in the square configuration?
p12. Given that there are 168 primes with 3 digits or less, how many numbers between 1 and 1000 inclusive have a prime number of factors?
p13. In the diagram below, we can place the integers from 1 to 19 exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum.
https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png
p14. Let E be a random point inside rectangle ABCD with side lengths AB=2 and BC=1. What is the probability that angles ABE and CDE are both obtuse?
p15. Draw all of the diagonals of a regular 13-gon. Given that no three diagonals meet at points other than the vertices of the 13-gon, how many intersection points lie strictly inside the 13-gon?
p16. A box of pencils costs the same as 11 erasers and 7 pencils. A box of erasers costs the same as 6 erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined?
p17. In the following figure, all angles are right angles and all sides have length 1. Determine the area of the region in the same plane that is at most a distance of 1/2 away from the perimeter of the figure.
https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png
p18. Given that 468751=58+57+1 is a product of two primes, find both of them.
p19. Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color?
p20. Consider a cylinder with height 20 and radius 14. Inside the cylinder, we construct two right cones also with height 20 and radius 14, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.