MATHCOUNTS 2015 ██ Mock Chapter Competition ██ Sprint Round Problems 130 Name __________________________________________________________________ State ___________________________________________________________________ DO NOT BEGIN UNTIL YOU HAVE SET YOUR TIMER TO FORTY MINUTES. This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the lefthand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted. Total Score Scorer’s Initials 1. chocolates A chocolate box consists of only macadamia and cashew chocolates. Every box has 100 chocolates, and the ratio of macadamia chocolates to cashew chocolates in each box is equal to 1 : 4 . How many macadamia chocolates are in the box? 2. Find the sum of the numbers below which are rational. Express your answer as a common fraction. 3. Angelica and Brae play a game, with each person having one fair, standard, sixsided dice, with faces labeled 1 6. They both roll their die once. If the numbers on the top faces match, Angelica wins. Otherwise, Brae will win. What is the probability that Angelica wins, in any given match? Express your answer as a common fraction. 4. players A football team labels every one of their jerseys’ backs with different numbers. Every number ranges from 20 to 40, inclusive, and must be even. Everybody on the team receives a labeled jersey, and every jersey has a distinct number on its back. What is the maximum number of players the team can have? 5. If z−5 2 = 0, what is z? 6. A 3 x 3 x 3 cube is composed of 27 unit cubes glued together. All the outer faces of the 3 x 3 x 3 cube are painted. What fraction of the unit cubes have at least one painted side? Express your answer as a common fraction. 7. The positive integer 6 is the smallest positive integer to have the property that its prime factorization contains at least two distinct primes, each with an exponent of 1. This is because its prime factorization is 2 × 3 . Notice that the number 12 does not follow this property, as its prime factorization is 22 × 3 . What is the value of the second smallest positive integer with this property? 8. th A class of 100 students are standing in line. Three of the students are friends: Randy, Andy, and Nancy. Nancy stands 25th in line, and Andy stands 35th in line. The number of people between Randy and Andy is equal to the number of people between Randy and Nancy. What place does Randy stand in line? 9. Concentric circles X and Y are drawn, such that circle X has a radius of 4 units. The annulus created by the two circles (a region that is contained in Y but not in X ) has an area of 65 π. What is the radius of Y ? 10. Two machines make a sound of a click at constant rates. One of the machines click every 20 seconds, when turned on. The other machine clicks every 25 seconds, when turned on. When both machines are turned on at the same time, they immediately start clicking. After how many seconds do they both click a second time? 11. miles Janine and Jasmine are on two different locations on a straight road with their scooters. They agree (by cell phone) that, at 11 AM sharp, they would begin scootering toward each other and meet somewhere on the road. At 12 PM, Janine had traveled exactly 5 miles. Jamie rode a bit slower, so at 12 PM, she only traveled 3 miles. The length of the road in between them is 19 of the whole entire road at that time. How far apart are they at 12 PM, in miles? 12. integers An integer is a palindrome if the number reads the same backwards. For example, 141 and 13631 are both palindromes. How many three digit positive integer palindromes are odd? 13. elements Set A has 12 elements. Set B has 18 elements. Set A and set B share 8 elements in common. How many elements are in the union of set A and set B ? 14. th Mikkal is on an elevator, on a certain floor of a building. He then goes down six floors to grab his suitcase, and then up eight floors to dine at the Hakka Ren Restaurant. He then goes down eleven floors to pick up his clothes from the laundry. Finally, he goes down three floors, to give the clerk his room card on the first floor. What floor did Mikkal start on, before he grabbed his suitcase? 15. Luths Two types of people are sitting around a circular table: the Triars, and the Luths. Triars always tell blatant lies, and Luths always tell the truth. Everyone sitting at the table says that neither one of the two people sitting adjacent to them are Luths. Given that there are exactly 100 people sitting around the table, how many of them are Luths? Assume that if someone is lying about "neither one of the two people sitting adjacent to them are Luths," then both of the people sitting adjacent to them are Luths. (If such a seating is not possible, write DNE.) 16. shrubberies The Knights Who Say “ Icky icky icky icky kapang zoop boing!” have sent King Arthur on a quest for shrubberies. He must buy a certain number of shrubberies depending on how many knights are part of the group, though. He must buy 1 shrubbery for the first knight, and then he sums the number of shrubberies he has previously brought, and gives the next knight that number of shrubberies. For example, if there are 4 knights, he buys 8 shrubberies (1+1+2+4). How many shrubberies does he buy if there are 10 knights? 17. Simplify √ 852−752 Express your answer in simplest radical form. 10 . 18. Four different people place each of their distinct purses into a bin. Then, one by one, each person grabs a random purse without looking, and without replacement, such that everyone ends up with one purse. What is the chance that everyone received a wrong purse? Express your answer as a common fraction. 19. A cylindrical can has radius 2 and height 5. There is a label on the side of the can that takes up 40% of the lateral surface area of the can. When the can is opened, half of the top base is removed. What fraction of the entire external surface area of the can does the label take up now? Express your answer as a common fraction. 20. versions Mini Mario Diner sells different versions of ice cream. You can choose to receive one, two, or three indistinguishable scoops on your cone. Also, you get to choose the flavor that every one of your scoops have: chocolate, strawberry, and vanilla. Finally, you get to choose any combination of five different types of sprinkles (you can choose not to get any sprinkles) on your ice cream. How many distinct versions of ice cream can you receive from Mini Mario Diner? Assume that the order of the scoops matter, but the sprinkles do not. 21. A cylinder and a cone have equal volumes. Given that the cone has twice the height of the cylinder’s, what is the ratio of the area of the cone’s base to the area of the cylinder’s base? Express your answer as a common fraction. 22. feet Two snowmen are built, such that they are relatively similar. The smaller of the snowmen took exactly 5 pounds of snow, and the other took 320 pounds of snow. Given that the smaller of the two snowmen measures 18 inches, how many feet tall is the larger of the two snowmen? 23. cents Madeline goes to the store to buy carrots, cabbages, and cilantro for her annual Thanksgiving dinner. She notices that you can buy two carrots and two cabbages for exactly 76 cents, three cabbages and three cilantros for exactly 108 cents, and four carrots and four cilantros for 104 cents. Given that information, how many cents is one cabbage, one cilantro, and one cabbage worth? 24. An equilateral triangle is inscribed in a circle O , as shown. The ratio of the area of the region enclosed by △AOC to the area of the region enclosed by and BC can be expressed in √ the form aπdπeb−f+c , where all radicals are simplified, and all variables are integers. Find a + b + c + d + e + f . 25. Given that the least common multiple of 5346 and 2314 is 6185322 , what is the greatest common factor of these two numbers? 26. The sum of all the positive integer divisors of a positive integer n is equal to 2n − 1. Find the sum of the 8 smallest possible values of n . 27. In the diagram, ⊙O1 and ⊙O2 are both congruent, and share common tangent AC . Quadrilateral ABCO is a rectangle, and AO = 6 cm. The area of pentagon 2 OO1O2BC is x cm , and OB = y cm. Given that the radius of measures 2 centimeters, compute x + y. 28. A sphere is inscribed inside a right circular cone with base radius 5, and slant angle 60° . Then, another sphere is put inside this cone. What is the maximum possible ratio of the radius of the smaller sphere to the radius of the larger sphere? 29. A right circular cone with radius 7 and slant angle 60° is moulded from clay. A right circular cone with radius 3 and the same slant angle is taken out of it to form the crosssection shown in the figure. If a sphere has the same volume as this figure, what is the radius of this sphere? 3 Express your answer in the form √a√b , where a and b are integers. 1850 30. ( , ) Given that a + b = 650 a = b is true for positive real numbers a and b , find (a, b) .
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