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MATHCOUNTS ∎
2015 Mock Chapter Competition
Sprint Round Problems 1­30 ∎ ________________________________________________________ Name ________________________________________________________________ School ________________________________________________________________ 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 must set a timer to 40 minutes before you flip to the next page, beginning the test questions. You are not allowed to use calculators, books or other aids during this round. 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 Correct Scorer’s Initials 1. Solve for x ​
in the equation 5x + 7 = 12. 2. A bag has 3 red balls, 2 green balls, and 1 blue ball. What is the probability that you draw a red ball? Express your answer as a common fraction. 3. Diagonal AC is drawn from the vertices of square ABCD with side length 2. What is the perimeter of triangle ABC? Express your answer in simplest radical form. 4. Martin buys a sweater for 20% off its original price of $15. He then sells back the shirt for 20% profit. What was his profit? 5. How many square inches are in one square foot? 6. A popular music video on YouTube has no views. After 3 days, the video has 450 more views. After another 2 days, the video has 500 more views. What is the average number of views per day for this 5­day period? 7. Find the product of the solutions of the equation x2 + 9x + 11 =− 9. 8. What is the sum of the reciprocals of 43 and 54 ? Express your answer as a fraction in simplest form. 9. If I pick an integer from 1­20, what is the probability that it is prime? Express your answer as a common fraction in simplest form. 10. The total bill at a restaurant is 50 dollars. Adam, Ben, and James pay the bill in the ratio 7:2:1. How much more money does Adam pay than Ben? 11. At Joe’s Pizza Parlor, a large pizza costs $12.50 each and delivery costs $5 . On the other hand, pizzas at Bob’s Pizza Parlor cost $13 each and delivery costs $2. For how many pizzas would both options cost the same? 12. Line A has equation 3x + 2y = 12 . Line B, which is perpendicular to line A, has equation y = mx + b and passes through (3, 6). Find m + b, and express your answer as a fraction in simplest form. 13. On four tests, Joey’s scores were 81, 83, 87 and x. The mean of his test scores was 81. What was x ? 14. In a rectangular garden with area 750 square feet, the length is 5 more than the width. In feet, what is the perimeter? 15. How many more ways can you arrange the letters in “MAMMOGRAM” than the word “ANAGRAM”? 16. What is the sum of the coefficients of (x + y)4? 17. How many positive divisors does 224 have? 18. Equilateral ΔABC has perimeter 6√3 inches and is inscribed in circle O. What is the area of circle O? Express your answer in terms of π. 19. If I pick two distinct integers from 1­10, inclusive, what is the probability that their sum is a multiple of three? Express your answer as a fraction in simplest form. 20. If 5 workers can build 4 houses in 3 days, how long will it take 4 workers to build 3 houses? Assume that workers build at a constant rate, and express your answer as a fraction in simplest form. 21. a is a positive integer less than 200. When a is divided by 5 , there is a remainder of 2. When divided by 11, there is a remainder of 7. When divided by 3, there is a remainder of 2. What is the value of a? 22. A car driving from New York City to Philadelphia at a rate of 50 mph. After traveling 176 yards, a second car leaves New York City at a rate of 56 mph. After how many minutes will the two cars meet? 23. If x + y = 5 and xy = 10, then what is x4 + y4? 24. How many palindromes between 100­1000 are multiples of three? 25. When rolling two dice, what is the probability that the two numbers rolled are relatively prime? Express your answer as a fraction in simplest form. 26. In the figure below, right ΔABC has AC as the diameter of the circle. The length of EC is 2 more than the length of EB. If the radius of the circle is 2, then what is the length of BC? 27. Solve for x if x + 1 = 2+ 1 1 . Express your answer 1
2+2+...
as a radical in simplest form. 28. x, y, and z are positive integers less than 18 such that x2 − y2 = n . Find all the sum of all possible values of n. 29. In the diagram below, BD = 12, < BDC = 45 and < BCA = 75 . If ΔABC is isosceles, then what is the area of ΔBCD ? 30. a, b, c, and d are distinct, non­negative integers where a, b, and c form an arithmetic sequence and b, c, and d form a geometric sequence. If the common difference and common ratio are the same, what is the smallest possible value of a + b + c + d? MATHCOUNTS ∎
2015 Mock Chapter Competition
Target Round Problems 1 ­ 8 ∎ ___________________________________________________________ Name ________________________________________________________________ School ________________________________________________________________ DO NOT BEGIN UNTIL YOU HAVE SET YOUR TIMER TO SIX MINUTES. This section of the competition consists of eight problems, which will be presented in pairs. Work on one pair of problems will be completed and answers will be collected before the next pair is distributed. The time limit for each pair of problems is six minutes. The first pair of problems is on the other side of this sheet. When told to do so, turn the page over and begin working. This round assumes the use of calculators, and calculations also may be done on scratch paper, but no other aids are allowed. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left­hand column of the problem sheets. If you complete the problems before time is called, use the time remaining to check your answers. __________________________________________________________________________ Total Correct Scorer’s Initials 1. Find the sum of the mean and median of the set, {4, 1, 5, 6, 10, 7, 7} . Express your answer as a decimal to the nearest tenth. 2. In Ray’s state, he has to be ready to pay his taxes by April. 15% of his salary goes towards tax, in which 43% is for property tax and 57% is for income tax. How much more does Ray pay towards income tax than property tax, if his salary is $50,000? 3. If I pick two values from the set of integers from 1 to 19, what is the probability that their sum is greater than 12? Express your answer as a common fraction in simplest form. 4. Using the figure below, in how many ways can Johnny walk from school to home, assuming he doesn’t backtrack? 5. How many four­digit numbers between 8000 and 9000 are there for which the thousands digit is equal the sum of the other three digits? 6. How many arrangements of the letters in the word “ARRANGE” contain no double­letters? 7. Regular pentagon P QRST is inscribed in regular pentagon ABCDE with side length 6. The vertices of pentagon P QRST are the midpoints of the sides of ABCDE . What is the length of P R ? Express your answer as a decimal to the nearest tenth. 8. A cone with height 6 and radius 4 has the same volume as a rectangular prism with dimensions 12, r , and r + 1 . What is the value of r ? Express your answer as a decimal to the nearest tenth.