MATHCOUNTS ∎
2015 Mock State 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. What is (30 + 101)2 − (29 + 8)2? 2. What is the absolute difference between the product of all real numbers and the sum of all real numbers? 3. What is the total number of combinations of four switches if a switch can only have two states and switches next to each other cannot have the same state? 4. If a + b = 5 b + c = 6 c + a = 7 Find a + b + c. 5. Kinetic energy is measured as 12 mv2 , where m is the mass in kilograms and v is the velocity in meters per second. What is the kinetic energy of a bowling ball with mass 10,000 grams traveling at a rate of 360 meters per minute? 6. ​
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ABCD is a square with vertex A at the center of a circle. The opposite vertex to A lies on that circle. The radius of the circle is 16 in. What is the number of square inches in the area of the triangle BCD? 7. What is the area of an octagon circumscribing a unit square? Express your answer in simplest radical form. 8. In how many points do the diagonals of a regular hexagon intersect? 9. Solve the following equation: √x + 5 = √x + 1. Express your answer in simplest radical form. 10.In the Fibonnacci sequence, where the first two terms are both equal to 1 and each term after is the sum of the two terms before it, ​
how many of the first 20 terms of the Fibonnacci sequence are multiples of 5? 11. ​
Uwotm8 ​
collects action figures. When he tries to organize them in groups of 3, 4, and 5, he has 2 left over each time. If he has less than 100 action figures, find the greatest number of rows he can organizes his action figures such that each row has more than one action figure. 12. The diagonals of a rectangle intersect at a point P . Point P is 5 cm further from the shortest side than the longer side. The perimeter of the rectangle is 88 cm. What is the number of the square centimeters in the area of the rectangle 13. What is the sum of the first 10 terms of an arithmetic sequence with first term ­500 and common difference 95? 14. In 2015, 100 students enroll in the AoPS classes. If 70 students take the introductory classes, 50 take the intermediate series, 10 take WOOT, and no one takes all three classes, how many students take exactly two classes? 15. In Mathcounts, the top 10 competitors participate in a countdown round. The final two competitors, A and B, must answer three questions correctly before his opponent to win. if person A has a 35 chance of answering a question correctly before person B and each question is answered correctly by either person A or person B, what is the probability that person B wins the round? Express your answer as a common fraction. 16. ​
Uwotm8​
is gambling again. He is playing a game where a fair coin is flipped and a fair die is rolled. If the coin shows heads, he loses one dollar no matter what the dice shows. If the coin shows tails and the number shown on the die is prime, he earns five dollars, and if the coin shows tails and the number shown on the die is not prime he loses two dollars. What is his expected balance after 10 rounds of this game, given that he starts with 0 dollars? 17. Segment AB is a common external tangent segment to two externally tangent circles with radii 34 and 14. Find the length of AB, and express your answer in simplest radical form. 18. How many digits are in the number 532 × 242 ? 19. Genevieve biked 441 miles on a 6­day trip. On each day n of the trip, she biked n times as many miles as she biked on the first day. (For example, on day 3, she biked three times as many miles as she had on the first day.) How many miles did she bike on the first day? 20. Find the area of a triangle with side lengths 26, 28, and 30. 21. A point S is chosen inside the square MNPQ. What is the probability that the angle MSN is acute? Express your answer as a common fraction in terms of π and 3in simplest form. 22. Hexagon ABCDEF with side length 6 has diagonal AC constructed such that the center of circle O is the midpoint of AC and that it is tangent to AB and BC. Find the radius of circle O, given that . Express your answer in simplest radical form. 23. What is 101101​
+ 132​
? Express your answer as a base­three numeral. 2​
5​
24. In ΔABC , the angle bisector of angle A intersects BC at point D. If AB = 6, AC = 7 , and AD = 4, find the area of △ABC . 25. Circle O is located in the interior of regular pentagon ABCDE with side length 6, such that circle O is only tangent to the midpoint of one side and the midpoints of two others. What is the radius of circle O? Express your answer in simplest radical form. 26. ​
Let ABCD be a square. We draw a circle of center C and of radius CB . A point P is on the arc BD that is situated inside the square and is such that the distance from P to the side AB is 4 inches and the distance from P to the side AD is 8 inches. Find the side of the square in inches. 27. What is the largest 4­digit positive integer abcd , such that a, b, c, and d are distinct positive integers and √ad ≥ b√c ? 28. The intersection points of the equations y = 2x2 + 5x − 1 and x = 7y − 132 is (x, y) , where x and y are positive integers. Find the area of triangle created by the solutions of the first equation and the intersection point. Express your answer in simplest radical form. 29. A border prime is a prime number such that it is directly before or after another prime. For example, 3 would be the border primes of 2 and 5, and vice versa. 7 men are arranged in a circle and each given primes from 1­100. What is the probability that adjacent men do not have border primes? Express your answer as a common fraction in simplest form. 30. Square AEFG is constructed such that it is on the interior of square ABCD and that the ratio of its diagonal to the diagonal of square ABCD is √21+1 . Another square is drawn on vertex F such that its diagonal length is half of square AEFG and that its diagonal coincides with the diagonal of Square ABCD. This pattern is continued until the the last square fills up the opposite corner. What is the combined area of all of the smaller squares if the side length of square ABCD is 1? Express your answer as a common fraction in simplest radical form. MATHCOUNTS ∎
2015 Mock State 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. In a workshop, workers are classified by two determinants: gender and perseverance. If the data is as shown in the table below, what is the probability that a female worker is hard­working? Express your answer as a decimal rounded to the nearest hundredth. Hardworking Lazy Male 35 15 Female 44 6 2. ​
The Body Mass Index of someone is calculated using the equation 703b
, where b is h2
the body weight in pounds and h is the height of the person in inches. Given that ​
Bob weighs 120 lbs and is 5' 6" tall, what is his BMI? Express your answer as a decimal to the nearest tenth. 3. How many ways are there to eat seven dumplings if two are beef­filled, three are shrimp­filled, and the others are vegetarian? Note: two dumplings with the same insides are considered indistinguishable. 4. ​
If you begin counting 3 consecutive whole numbers each second, starting January 1, 2000, at 12:00 a.m., in what year will you reach 6 billions ? 5. Let an = √(n + 2)an − 1 for n > 0 . Find a2015 ​
. 6. How many lattice points (points with x and y coordinates that are both integers) are strictly inside the triangle bordered with y = 3x − 2 , y = −x3 − 5 , and the x­axis? 7. The probability of getting at least two heads in three flips of an unfair coin is 1/6 . The probability of getting at least one head and one tail in four flips of an unfair coin is 3/8 . What is the probability of landing at least 5 heads in 7 flips of the same coin? Express your answer as a common fraction in simplest form. 8. The Star of David is a figure created when two equilateral triangles overlap to form a regular hexagon in the center. Inside the hexagon lies circle O, which is tangent to four of the sides of the hexagon. Find radius of circle O, if the side of length of each triangle is 3. Express your answer as a decimal to the nearest tenth.