2018 Sprint Solutions

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1 08 Sprint s. $ 30 If computers are worth 00 dollars and 3 monitors are worth 90 dollars, what is the cost of computer and monitors in dollars? One computer is worth 00/ = 0 dollars and monitors are worth 90 /3 = 60 dollars, so the total cost is $0 + $60 = $ What is the value of? Epress your answer as a common fraction 0.0 in simplest form The sum of two prime numbers is 0. What is the value of the greater prime number? The sum is an odd number, so one of the primes must be odd, while the other is even. Since the only even prime number is, we know that odd prime must be 0 =99. So, the answer is 99 (which is prime) cups A pound sack of sugar was donated to a birdwatcher s club for their members to mi hummingbird nectar. The nectar recipe calls for a hot water to sugar volume ratio of 4 to. One cup of sugar weighs 0.4 pounds. How many cups of hot water are needed to mi the entire sack of sugar? 4 cups water cup sugar pounds sugar = 0 cups water cup sugar 0.4 pounds sugar Notice how the sugar units cancel from numerator to denominator.. 3 years Si years ago, Bill s dad was times as old as Bill. If his dad is 4 now, how old is Bill? Bill's dad was 4 6 = 3 years old 6 years ago. This means that Bill was 3/ = 7 years old 6 years ago, so Bill is = 3 years old now. Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

2 6. 84 Devin has twelve homework papers for every tetbook he owns. If he has a sum total of 9 homework papers and tetbooks, how many homework papers does Devin have? Let the number of tetbooks he has be. Then he has homework papers. The papers plus books add up to 9, so 3 = 9, meaning Devin has 7 tetbooks. Thus, he must have 7 = 84 homework papers in total How many positive integers are factors of 36 or 63, but not both? The factors of 36 are,, 3, 4, 6, 9,, 8, and 36. There are 9 factors. Also, the factors of 63 are, 3, 7, 9,, and 63 a total of 6 factors. Note that 3 factors are shared by 36 and 63 (, 3, and 9). So 36 has 9 3 = 6 factors that 63 does not have, and 63 has 6 3 = 3 factors that 36 does not have. So, there are a total of 9 integers that are factors of either 36 or 63, but not the other Alvin has taken four tests so far in his math class, getting a 86, 88, 88, 89 on his tests. What is the lowest score that Alvin can get on his net test in order to raise his average test score to at least a 90? Since Alvin wants at least a 90 score average across tests, the sum of his tests must be at least 90 = 40. The sum of the scores of his first four tests is = 3, so in order to reach the sum of 40, he needs to get a 40 3 = 99 on his net test. 9. Sunday The shortest possible route from Earth to Mars is 34 million miles. If a spacecraft, traveling this route, leaves Earth at :00 AM, Monday, maintaining an average speed of 34,000 miles per hour, on what day of the week would the craft reach Mars? The spacecraft takes 000 hours to reach Mars, which is around 4.67 days ( weeks days). So, the craft would reach Mars on Sunday A standard 6-sided die is rolled. If the number on the die is odd, it is rolled again. If the number is even, nothing happens. What is the probability that the first or second roll is a 6? Epress your answer as a common fraction in simplest form nd roll 6 st roll Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

3 . 0 What is the measure of angle A, in degrees, given that lines and y are parallel? 30 A y 00 Add another parallel line through the verte of angle A. Parallel angles are the same, and supplementary angles add to 80. So angle A = = y 00. 3:36 PM Candle A is 4 inches long and candle B is inches long. Candle A melts down 8 inches per hour when lit, while candle B melts down inches per hour when lit. Candle A is lit on 3:00 PM. At what time must candle B be lit so that Candle A and B completely melt simultaneously? Candle A takes 4/8 = 3 hours to completely melt down, while candle B takes hours to completely melt down. By the time Candle A completely melts down, it will be 6:00 PM, so Candle B must be lit hours, or hours and 4 minutes, before 6:00 PM. So, the time at which Candle B must be lit is 3:36 PM., if, if If f( ), what is the value of 9, if 0 8, if 0 f( f( f( f( f( f( f(9)))))))? f(9) =, f() = 3, f(3) =, f() =, f() = -4, f(-4) = 6, f(6) = 7 Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

4 4. 6 Circles A and B overlap each other and share an area K. K is of circle A and of circle B. What is the ratio of the area of circle A to the area of 8 circle B? Epress your answer to the simplest fraction. Let equal to area of circle A and y equal to area of circle B. The overlap K is of circle A y and of circle B. So, we can epress this by writing:. The ratio of the area of 8 8 circle A to the area of circle B is, so by moving around variables,. y y Consider the following two sequences: an arithmetic sequence, a, in which a = and a = 4, and a geometric sequence, b, in which b = and b = 4,. Compute a 0 + b 0. The formula for an arithmetic sequence can be found from just two values. In this case, the formula is given by a n = 6(n ) = 8 6n. So, a 0 =. Likewise, the formula for a geometric series can be found from just two values. In this case, b n = ( ) n-, and b 0 = 04. Hence, a 0 + b 0 = 04 = Si people are evenly divided into three groups. How many different ways can the three groups be formed? For group, there are 6!/(!4!) = ways to choose the out of 6. Then, for group, there are 4!/(!!) = 6 ways to choose out of 4. The remaining are automatically in group 3, so only way. Overall, 6 = 90 ways. Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

5 7. An equilateral triangle is circumscribed by a circle. If the ratio of the triangle s perimeter to the circle s circumference is a, and the ratio of the triangle s area to the circle s area is b, then what is the value of a/b? Lines from the triangle vertices through the circle center bisect the angles and subdivide the equilateral triangle into triangles. Altitudes of the equilateral triangle are 3r/, where r is the circle radius. The triangle side lengths are 3 r. The perimeter ratio is a. The triangle area is, so the area ratio 4 r 3 3 is b. Therefore, a/b =. 4 r/ r 3 r r 3 r 8. What is the remainder when is divided by 7? = Notice that is already divisible by 7. Therefore, we need to know the remainder when + is divisible by 7. Simply divide + (which is 3 = 89) by 7, which gives A square with side length 6 is inscribed in a square with side length 8. A verte of the smaller square divides a side of the larger square into two segments of lengths a and b. What is ab? a is the area of one of the four triangles. ab ab + 6 = 8 ab = 8 ab = 4 b ab4ab Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

6 What is the value of 3 3? (9 8). Find the smallest positive integer such that (!)(6!) is divisible by The prime factorization of is 9 4, so (!)(6!) must contain 9 4. Also, 6! is divisible by 6 3, so! must contain another factor of. So, must equal to.. 6 What is the smallest positive integer n such that 6n is a perfect square and 8n is a perfect cube? Since 8 is already a perfect cube, n has to be a perfect cube too. The factorization of 6n results in 3 n. For 6n to be a perfect square, the minimum n has to be ( 3) P, with an odd power, P. For n to be a perfect cube, n must be = 6 3 = In triangle ABC, BC = 4, CA =, and AB = 6. Draw three circles of radius two, each centered on a different verte of the triangle. What is the sum of the area of the overlapping regions of the triangle and each of the three circles? Since the overlapping areas are all arcs of circles with radius, we know that the area of all three of those arcs combined are A B C ( A B C) Since A B C 80, the overlapping region would equal (80 ). 360 C B A Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

7 4. 7 What is the value of if 3? The and are a clue on how to solve the problem. Try squaring If 009 f ( ) f ( ), then what is the value of f (08)? Plug = 08 and then = 08 in the formula to get two equations and two unknowns. (08) ( 08) f f ( 08) (08) f f Then, multiply the second equation by and add it to the first equation to eliminate f(-08). (08) (08) 0 f f (08) f f (08) Let f (n) = (n 7) (n + 3) +7(n + 3) + n + n. f (09) is the product of two large prime numbers. Find the larger prime number. Note that (n 7)(n + 3) + 7(n + 3) = (n 7 + 7)(n + 3) = n(n + 3), and that n + n = n(n + ). Hence, f (n) = n(n + 3) + n(n + ) = n(n + ). So then, f (09) = 09 ( 09 + ) = The two prime must be 09 and 3, and the larger one is 3. Or, just epand f (n) and combine like powers of n to get f (n) = n + n = n(n + ) Jack plays a game where if he flips heads he moves foot to the west, while if he flips tails he moves foot to the east. If Jack flips si coins, what is the probability that Jack ends up in his starting position? The total possible number of different moves that Jack can make is 6 = 64. If Jack wants to move back to his original position, he must flip 3 heads and 3 tails. The number of ways he 6 6! 0 can obtain 3 heads and 3 tails is 4 0. The answer is. 3 3!3! 64 6 Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s

8 Evaluate the following epression as a common fraction in lowest terms First, factor each term as the difference of two squares. Second, add or subtract within each term with common denominators. Third, cancel to reduce multiplications A 9. 0 Quadrilateral ABCD is inscribed in a circle of D radius 6. If BDA = 40 and AD = 6, what is the measure of BAD in degrees? O Let O be the center of the circle that circumscribes ABCD. Note that the radii OA = OD = 6 and AD = 6, so OAD is equilateral. AD AOD It follows that AOD = 60. Furthermore, ABD = 30 C Thus, from ABD we have BAD = 80 BDA ABD = = 0 B 30. What is the value of? Epress your answer as a common fraction A series that can be put in a form where terms cancel like this is referred to as a telescoping series. Copyright South Bay MOCK MATHCOUNTS 07. All rights reserved. 08 Sprint Round s