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Daily Quiz #6: March 27, 2014

I hope that after all of you finish Mathcounts, you keep up with what you learned here. You’re all really excited about how cool math is, and that’s something I really enjoy seeing. Because believe it or not, it is the frantic studying, the voracious scrambling for problems, it is that kind of math that really makes you good at it. It’s getting engaged with every problem that meets your eyes. That’s why I’m giving you these problems and going over them with you. And if you have any questions, I would love to discuss them with you in the comments ad nauseam.

This will be the last daily quiz before the competition. Try to get as high a score as you can on it (both the quiz and the competition) – the problems might be hard, but remember that you can always think of a way to make sure your answer is right! Good luck, and have fun at state!

The scores as of this morning are as follows:

Solutions to yesterday’s quiz.

21. A useful tidbit to know before solving this problem is that only perfect squares have an odd number of factors. This is quite simply because factors come in pairs, but in a perfect square one factor is repeated. So by listing the factors of the first several perfect squares, we find that the answer is 144.

22. As you know, the area of a parallelogram is the product of the base and height. We’re given the base, but the height has to be calculated from the other side length of the parallelogram. If you draw the picture, you can see that the 2-inch side is a hypotenuse of a 30-60-90 triangle. Therefore, the height is \sqrt{3} inches. Now we can find the area, 6\sqrt{3} square inches. (Everyone missed this problem, but I think for most of you it was the special right triangle triangle that screwed things up. Remember, 1-2-square root of 3).

23. Like we solved #9 earlier, we need to write an expression that represents the combined rates of both cores. The rate of the first core is 1.6 million computations per second, while the second’s is 1.5 million computations per second. They add up to 3.1 million computations per second. Writing an equation using the desired 8 million computations, 3.1 = \frac{8}{x}, we find that the answer is 2.6 seconds.

24. Let x be the width of the pool, and x + 4 the length of the pool. We need to write an equation representing the area of the pool and deck area: (x + 10)(x + 4 + 10) = 1440. Honestly, at this point you can choose whether you want to solve a quadratic or not. If you don’t, you could list factors of 1440 that have a difference of 4. You’ll come up with 36 and 40, or x = 26. If you do write a quadratic, you’ll get x2 + 24x – 1300 = 0. Then you’ll look for factors of 1300 that have a difference of 24, and come up with 50 and 26. Either way, the side lengths of the pool are 26 and 30, and the perimeter is 112.

25. I sincerely apologize for not doing this before with you guys, I just assumed you knew how to do it. You need to fill out a table with rows for the solution before and after diluting. The first column will represent the percent of hydrochloric acid (HCl), the next the size of the solution, and the next the amount of HCl. Remember that the unknown is the amount of solution that we need of the concentrated HCl.

Substance Percent HCl Solution (mL) Amt. HCl
Input Solution 0.80 x 0.80x
Output Solution 0.20 250 50

Now, we know that the amount of HCl has to be the same before and after, since Ms. Clements just added water. So setting the two expressions equal to each other, 0.80x = 50 and x = 62.5.

Today’s quiz.

26. Blaise wrote each factor in the prime factorization of 10! on an index card (separately, so two factors of 3 would be written on two index cards). If Caroline selects a card from this deck, what is the probability that she will choose a 3? (10 pts.)

27. Srikhar wants to get from building A to building B in New York City, and the way to get there is through roads that are arranged in a grid pattern. Building B is five blocks east and two blocks north of building A. Provided that the shortest distance to walk is 7 blocks, how many routes can Srikhar take that will be the shortest distance possible? (10 pts.)

28. You just lost all your financial records! Imagine that you have $15,000 in a CD that has been earning 6.1% interest, compounded annually since 2001. (Your parents deposited the money for you at your birth, presumably.) Use your math skills to figure out much money they deposited back then without your financial records. Don’t worry about inflation, by the way.

29. Bonus. This is a rerun of problem 25, since I didn’t teach you guys this before. So Ms. Clements is back at the lab, and she’s using silver nitrate this time. She has a bottle of concentrated silver nitrate (80%) and dilute silver nitrate (10%). How much of the 80% solution does she have to put into the volumetric flask (along with the 10% solution) to get 500 mL of a 40% solution? Express your answer to the nearest tenth of a milliliter. (Hint: fill in the below table using the information you know, and follow the general way we did #25.) (20 pts.)

Substance Percent AgNO3 Solution (mL) Amt. AgNO3
80% Solution x
10% 500 – x
Output Solution

30. Bonus. The product of one less than four times a number and three more than the number is 17. What is the sum of the two possible values for the number? Express your answer as a common fraction. (20 pts.)

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Daily Quiz #5: March 26, 2014

The scores as of this morning are as follows: Srikhar, 130 points; Caroline, 180 points. Johnum is starting off with us at 20 points. Again, keep up the good work and have fun with today’s quiz!

Solution to yesterday’s quiz.

16. We write an equation representing the new class average after the 7 students make up the test: \frac{1638+7x}{25} = 93. Solving this equation, we find that x = 98.1.

17. To find the lowest common multiple of the three expressions, all we need to do is list the factors and include the greatest number of each that exists in any one factor. For example, a factor of 2 appears once, x appears up to twice, 3 appears once, and y appears up to twice. Multiplying these together, we get 6x2y2.

18. The number of possible magnets that could have fallen on the floor is 10 x 9 = 90. I know we didn’t account for reversed arrangements, but it’s okay because we’ll also not account for them in the numerator of the probability. The numerator, then, is 4 x 3 = 12 vowel arrangements. So the answer is 12/90 or 2/15.

19. To solve this problem, we can write an equation that represents all the students in the class. Let x represent the number of students with freckles: x + 2x + x + 4 – 7 – 3 = 30. We subtracted 7 and 3 because, like in #12, those were repeats in the hat of names. The solution is 9.

20. Using Pascal’s triangle, we can immediately tell that the coefficients of the expansion of (a + b)6 are 1, 6, 15, 20, 15, 6, 1. The probability of selecting an odd number from this list is 4/7.

Today’s quiz.

21. What is the smallest integer with exactly 15 distinct factors? (10 pts.)

22. The new iPad has a screen shaped like a parallelogram with side lengths 6 in and 2 in and angle measures 60° and 120°. What is the area of this screen? Express your answer in simplest radical form. (10 pts.)

23. The new MacBook Pro uses multiple cores to achieve the most performance; however, due to strikes in their factory in China, the cores don’t all work at the same speed. One core processes 1.6 million floating point operations in a second, and the other processes 3 million computations in two seconds. If Call of Duty requires 8 million floating point operations to load on your computer, how long will it take the computer when both cores work together? Express your answer as a decimal to the nearest tenth. (10 pts.)

24. Bonus. Each pool at the Hyperbola Hotel is planned to have a 5-foot wide sunbathing deck surrounding it. The pool will be 4 feet longer than it is wide. If 1,440 square feet of floor space have been set aside for both the pool and deck, what is the perimeter of the pool? (20 pts.)

25. Bonus. Ms. Clements, the chemistry teacher at Magnet, needs 250 mL of 20% hydrochloric acid for a Chem I lab. Her chemical cabinet has a bottle of 80% hydrochloric acid. How many mL of the stronger hydrochloric acid should she put in the volumetric flask, which she will then fill with water to the 250 mL mark? Express your answer as a decimal to the nearest tenth. (20 pts.)

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Daily Quiz #4: March 25, 2014

Here are the scores as of this morning: Srikhar, 100 points; Caroline, 140 points. Keep up the good work, and don’t forget to check your answers!

Solutions to yesterday’s quiz.

11. We simply write an equation, \sqrt{3\cdot x^\frac{3}{5}} = 9 and solve. Remember to raise both sides to the reciprocal power when you have a fractional exponent. The answer is 243.

12. Drawing a Venn diagram to help you visualize the problem is a good idea. Let’s also think of the ninth graders as a hat with names in it. We know there are 408 different names in the hat, but there are actually 164 + 208 + 99 = 471 slips of paper in it. That means there are 471 – 408 = 63 names that are repeated at least once. Also remember that 11 names are in the hat three times, so we can subtract that from the number of repeated names to get 52 names that are in the hat at least twice. The answer, then, is 52/408 or 13/102.

13. To solve this problem, we need to know the area of the circle that Polly can reach and the area of the hexagonal yard. The area of the circle is \pi(30)^2, or 900\pi square feet. We know the yard has an apothem of 30 feet, so imagine that as the height of the six equilateral triangles that make up a hexagon. Since equilateral triangles can be divided into two 30-60-90 triangles, we can use that special right triangle rule to determine that the side length of the hexagon is 20\sqrt{3} feet. That means the area of each triangle is \frac{1}{2}20\sqrt{3}(30) = 300\sqrt{3}, and the area of the entire hexagon is 6 times that. Dividing the two areas, we get an answer of 90.7%. Be sure to round properly, it might be counted wrong otherwise.

14. Like Srikhar observed the other day with Robert’s college math problem, sometimes the best way to solve these boggling problems is to start with smaller examples. By listing the remainders of various powers of 2 when divided by 7, you’ll find a pattern of 2, 4, 1, 2, 4, 1,… This way, it’s easy to see that the 2014th term of this sequence is 2.

15. The intersection of the four red circles with the blue square together comprise one red circle. Since the radius of the red circles is 1/4 the large circle, we know that the area will be 1/16 of the total area. Then, we must consider the area of the one green circle, whose radius is 1/8 the large circle. Therefore, its area is 1/64 of the large circle. Adding 1/16 and 1/64, we get 5/64.

Today’s quiz.

16. Today, Madame Mathematique told her French students that if their class average was 93 or above on their last test, they would get a creme glacee party. The 18 students who have already taken the test had an average of 91, but 7 students still have to make up the test. What is the minimum average score those 7 students must make to bring the class average to a 93? Express your answer as a decimal to the nearest tenth. (10 pts.)

17. What is the lowest common multiple of 2x, x2y, and 3xy2? Write your answer as an algebraic expression in terms of x and y. (10 pts.)

18. The word CALCULATOR is spelled out on a refrigerator using magnets. If two letters fall off, what is the probability that they are both vowels? Express your answer as a common fraction. (10 pts.)
A
19. Bonus. Look at #12’s solution for help with this one. The 30 students in Mrs. Matrice’s class all have either dark hair, green eyes, or freckles. 7 of them have at least two of those traits, and 3 have all three traits. If there are twice as many students with green eyes as students with freckles, and there are four more brown-haired students than freckled students, how many students have freckles? (20 pts.)

20. Bonus. If a term is selected at random from the expansion of (a + b)6, what is the probability that its coefficient is odd? (20 pts.)

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Daily Quiz #3: March 24, 2014

Keep up the great work, everyone! Don’t forget to check your answers on the last post. Here’s the math du jour:

11. If a number is raised to the three-fifths power and multiplied by 3, and then the square root is taken, the result is 9. What is the number? (10 pts.)

12. At Half-and-Half High School, there are 408 ninth-graders who are involved in at least one of these activities: playing a musical instrument, playing a sport, or acting in the school drama productions. 164 students play a musical instrument, 208 play a sport, and 99 are in drama. 11 students are in all three activities. If a ninth-grader is selected at random from this group, what is the probability that he or she is involved in more than one activity? (10 pts.)

13. A dog named Polly is tethered by a 30-foot rope to a pole in the middle of a yard shaped like a regular hexagon. If she can just reach the midpoint of one the yard’s sides when she stretches the rope to its full length, what percentage of the yard can Polly reach? Express your answer as a decimal to the nearest tenth. (10 pts.)

14. Bonus. What is the remainder when 22014 is divided by 7? (20 pts.)

15. Bonus. Dartboards on the planet Zygote look slightly different than the ones on planet Earth, as shown here. The centers of the red circles are positioned halfway from the center to the edges of the board, and their radii are one-fourth of of the board’s radius and twice that of the green circles. What is the probability that when a Zygotean throws a dart at the board, it will hit on a region that is within the blue square but also within a red or green circle?

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Daily Quiz Solutions #1 and #2

Here is how to work the quizzes from this weekend. Unfortunately, this means if you haven’t posted your answers for these questions yet, you don’t get points for them. But even so, work the problems anyway – they’re great practice!

Got questions? Post them in the comments!

1. We want to find the number of visits at which the amount paid by a subscriber equals the amount paid by the non-subscriber. If the subscriber’s cost is given by 99 + 5x and the non-subscriber’s by 18x, solving the equation 99 + 5x = 18x for x yields the answer of 8 visits. Remember to round up, because rounding down (7 visits) would still cost me more.

2. We’ll work this problem using the blanks method. There are two blanks, because two books are being chosen. The first blank contains 70/150, and the second blank contains 80/149. But wait – we also have to multiply by 2 because we could have chosen a nonfiction book first instead of a fiction book. Evaluating (70/150) x (80/149) x 2 yields the answer, 224/447. Yuk.

3. Here we’re solving a system of equations in three variables. The equations are (1) bac – 8, (2) abbc – 14, and (3) b = 3a. We can subtract a from both sides in (1) and subtract b from both sides in (2). Then we substitute for b in (1) from (3) to get (4) 3ac – 8. Then, we subtract the modified (2) and (4), eliminating the c term, to get 2a = 6 and a = 3. From (3), b = 9, and from (1) c = 17. The sum abc29.

4. To convert 17268 to base 10, we determine what place value each digit represents. Since the place values are powers of 8, the value in base 10 is 1 x 83 + 7 x 82 + 2 x 81 + 6 x 80 = 982. Performing the same steps on 324 using base 6, we get 124. 982 + 124 = 1 106.

5. The only way to imagine this trapezoid (call it ABCD) is like this: AB and BC are 300 meters, angles B and C are right angles, and the diagonal DB is 500 meters. If it doesn’t make sense, I recommend making a drawing. We are looking for diagonal AC, and we quickly realize that since both sides of the triangle (AB and BC) measure 300 meters, the hypotenuse must be 300\sqrt{2}, or 424.3 meters.

6. This problem can be solved by remembering that the volume of a cone is proportional to the square of the radius and to the height. Since the tall cone is 3 times as tall as the regular, it has 3 times the volume. Since the wide one has twice the diameter, it therefore has twice the radius and four times the volume. So the wide cone holds more ice cream than the tall, at a factor of 4/3.

7. This problem is similar to #2. First, let’s figure out how many ways we could flip the coin: 2 x 2 x 2 = 8. Then, let’s draw three blanks, ____ ____ ____. If heads A and B each have to go into one of those blanks, there are 3 blanks for heads A and 2 left for heads B. 3 x 2 = 6 ways, but we have to divide by 2 to eliminate the ways where heads A and B are switched. The answer is 3/8.

8. We have ten consecutive numbers, n, n + 1, n + 2, and so on up to n + 9. Their sum is 395, and when we write this as an equation and collect like terms, we get 10n + 45 = 395 and n = 35. This is the lowest of the consecutive numbers. To quickly add up the 5 odd numbers {35, 37, 39, 41, 43}, remember that their average is 39 and multiply by 5 numbers to get 195.

9. First, figure out how much lawn each person could mow in an hour: 1/2 lawn and 2/3 lawn. The amount of lawn both would mow together is 1/2 + 2/3 = 7/6. Setting a proportion 7/6 = 10.5/t, we find that it will take them 9 hours to mow the lawn.

10. The quadratic equation that includes the given values is -16t2 + 40t + 24 = 0. We can divide the equation by -8 to make the numbers easier: 2t2 – 5t – 3 = 0. Factoring by grouping, we get (2t + 1)(t – 3) = 0. The positive solution is 3 seconds.

Algebra Problems, Geometry Problems, Math Problems, Number Theory Problems, Probability Problems

Daily Quiz #2: March 23, 2014

As of writing this post, it looks like many of you have not seen the contest yet. I’m hoping those of you that do know about it will tell the others, and I’ll give you a day to get points for the questions you answer on Quiz #1. Remember, I’m preparing you for your competition this weekend!

6. Willy Wonka’s famous ice cream shop has three types of ice cream cones: the Rocky Road Regular, the Tiger Stripe Tall, and the White Chocolate Wide. Yum. They all have the same dimensions, except the tall’s height is three times the regular’s, and the wide’s diameter is twice the regular’s. Does the tall cone or the wide cone hold more ice cream, and how much more? Express your answer as a common fraction (ratio of the bigger cone to the smaller one). (10 pts.)

7. Given a standard, fair coin, what is the probability of flipping the coin three times and getting exactly two heads? (10 pts.)

8. If the sum of 10 consecutive numbers is 395, what is the sum of the odd numbers among them? (10 pts.)

9. Bonus. Joh can mow half an acre of lawn in 1 hour. His good friend Num can mow a full acre in one and a half hours. If they start mowing a 10.5-acre lawn at 10:00 AM together, at what time will they finish? (20 pts.)

10. Bonus. The height of a falling object in feet as time passes is given by h = -16t2vts, where v is the initial velocity of the object and s is the initial height. If a projectile is launched off a 24-foot building at an initial velocity of 40 feet per second, how many seconds later will the projectile hit the ground? (20 pts.)

Algebra Problems, Geometry Problems, Math Problems, Number Theory Problems, Probability Problems

Daily Quiz #1: March 22, 2014

The state competition is coming up! We’re already making great progress towards doing well at state – now we just need to keep up the skills until next weekend. So, here’s what I have in mind: every day from now until Thursday, I’ll post five problems on this blog. You, the Mathlete, should solve everything you can and post your answers (along with your name) in the comments. The next day, the answers will be posted so you can check your work.

Here’s the game, though: you will receive points for each question you answer correctly. Every day, I’ll post each person’s rankings along with the answers. Who will be the best Mathlete of all? We shall see….

1. I bought a one-year membership to the Abscissa Nature Park the other day for $99. The membership lets me visit the park for only $5, while non-members must pay $18. How many visits do I have to make this year to make buying the membership worth my money? (10 pts.)

2. Srikar is looking for books at the Broadmoor Library, and there are 70 fiction books and 80 nonfiction books on the nearest shelf. If no two books are the same, what is the probability that he will randomly choose one book of each kind without replacement? (10 pts.)

3. Sunjay is thinking of three positive integers. He tells his friend Robert that the sum of the first and second numbers is 8 less than the sum of the first and third, and 14 less than the sum of the second and third. If the second number is three times the first number, what is the sum of all three numbers? (10 pts.)

4. Bonus. What is the sum of 17268 and 3246? Express your answer in base 10. (20 pts.)

5. Bonus. The plot of land for Caroline’s new house is a trapezoid with two right angles, two sides of length 300 m, and a diagonal of length 500 m. What is the length of the other diagonal? Express your answer as a decimal to the nearest tenth of a meter. (20 pts.)?