# Honor Pre-Calculus – Problems of the Week

 POW Answer Sheet  `Due September 7` Let A, B, and C represent different digits greater than zero.  Determine the minimum value of the expression ABC A+B+C (Note: for ABC, A is the hundreds digit, B is the tens digit, and C is the ones digit – – they are not multiplied) `Due September 13` When Kyle drives to work at an average speed of 48 miles per hour, he arrives 9 minutes late. When he drives to work at an average speed of 64 miles per hour, he arrives 9 minutes early. At what average speed should Kyle travel in order to arrive at work exactly on time? `Due September 20` The digits 1, 2, 3, 4, 5 can be arranged to form more than one hundred different five digit numbers.  If these five-digit numbers are arranged from least to greatest, in what position is 42513? Due September 28 Due October 4 Each spring, a 12 meter by 12 meter garden has its length increased by 2 meters but its width decreased by 50 centimeters. What will be the maximum attainable area of the garden? Due October 12 How Many Rectangles The above figure is composed of congruent squares. How many rectangles are in the figure?Due October 18 Newton High School has a hall with 500 lockers, all of which are closed. Five hundred students start down the hall. The first student opens every locker. The second student closes all the lockers that are multiples of 2. The third student changes (closes an open locker or opens a closed one) all multiples of 3. The fourth student changes all multiples of 4. And so on. After all students have finished with the lockers, how many lockers will be open? Due October 25 At noon, a train leaves New York for Toronto while another leaves Toronto for New York. It takes one train 8 hours and the other 22 hours to make the trip. If both maintain constant speeds and travel along parallel tracks, at what time do they pass each other? Due November 1 On his last test, Kyle scored a 95, raising his average by one point to 89.  What must he score on his next test to raise his average to 90? Due November 8 Find three different two-digit prime numbers such that the average of all three prime numbers is also prime and the average of any two of the prime numbers is also prime Due November 15 The little town of Radiusville has only one public library.  Many of the citizens feel that recent growth in the town’s population warrants the construction of an additional library.  The current population of the town is 15000 people.  According to the town’s bylaws, construction of a new library will begin when the population reaches 26000 people.  If the annual growth rate of the town has been consistently 4.5% and the year is 2005, in what year will the town need to build a new library? Due November 22 What is the smallest number that is divisible by 13 but when divided by the whole numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 yields a remainder of 1? Due November 30 In a far off galaxy on a distant planet, some extra-terrestrial children were busy in math class.  The students completed the following equations: 13 + 24 = 40 10 x 10 = 100 5 x 3 = 21 If the computations are correct on their planet, how many fingers do the children have on this planet? Due December 6 A and B are whole numbers and neither of which are divisible by 10. If the product of A and B equals 20000, find the absolute value of the difference of A and B Due December 13 Determine the two missing values that represent the blanks in following series of numbers. F, 4, 27, 4, 25, 216, 7, 64, 729, 10, 121, ___, ___ Due December 20 The four circles in the figure above each have a radius of 3 units.  The centers of each circle form a square.  Find the area of the green shaded region to the nearest tenth of a unit. Due January 4 Imagine that a new mathematical operation has been implemented.  Its symbol is &. Given that: 1 & 1 = 0 4 & 3 = 55 5 & 9 = 44 4 & 7 = 15 8 & 6 = 476 10 & 14 = 804 Find the value of  11 & 29. Due January 10 Compute the value of the above expression. Due January 18 If three hens lay four eggs in five days, how many days will it take a dozen hens to lay eight dozen eggs?  Please round your answers to the nearest day (or egg if needed)! Due January 25 Determine the ones digits for the following expression: 0! + 1! + 2! + 3! + 4! + . . . + 9998! + 9999! Due January 31 You are given a right triangle with integral side lengths (the lengths of the sides are integer values) that has an area of 756 square meters. What is the exact perimeter of the right triangle? Due February 7 A jar contains 50 red gumballs and 30 blue gumballs. Seven balls are randomly selected from the box without replacement. What is the probability of selecting a red gumball followed by a blue gumball, then a red gumball followed by a blue gumball, then a red gumball followed by a blue gumball, then followed by another red gumball? Please round your answer to the nearest thousandths. Due February 14 A man walks into a hardware store. He buys : 1 for a dollar, 20 for two dollars, and 300 for three dollars. What is the man buying? Due February 28 Suppose that the surface of the earth is smooth and spherical and that the distance around the equator is 25,500 miles. A steel band is made to fit tightly around the earth at the equator, then the band is cut and a piece of band 21 feet long is inserted.  To the nearest inch, what will be the gap, all the way around, between the band and the earth’s surface? (Use 3.14 as an approximate value of ∏ ) Due March 7 Fresh grapes contain 80 percent water by weight, whereas dried grapes contain 15 percent water by weight. How many pounds of dried grapes can be obtained from 34 pounds of fresh grapes? Due March 14 The number 21328 has a unique feature: The sum of the first four digits equals the units digit.  How many EVEN five-digit numbers have this unique property? Due March 21 One vertex angle of a regular octagon that has side length 4 units is joined with two other vertex angles to form a kite as shown in the above figure. Determine the area of the kite?  Round your answer to the nearest tenth of a unit. Due March 28 Three friends have a tradition of buying each other a New Year’s gift each year. Each person is bought a gift by the other two friends who split the cost equally. Therefore, friend A and friend B each pay half the cost of the gift for friend C, friend B and friend C each pay half the cost of the gift for friend A, and friend C and friend A pay half the cost of the gift for friend B. The final rule for this gift giving process is that the gifts are purchased together on the same day where each person spends all the money they bring to the event. If friend A brings \$50, friend B brings \$58, and friend C brings \$62, what are the prices for each of the three gifts? Due April 4 Find the sum of the next three numbers of 1, 2, 5, 12, 27, 58, ___, ___, ___. Due April 11 When the expression (358)(447) (559) is evaluated, the product ends with a string of zeros.  How many zeros will be at the end of this product? Due April 26 On a television show, a player receives 5 points for answering an easy question and 11 points for a hard one. What is the largest integer that cannot be a contestant’s total score of the game? Due May 2 A novel has 527 pages (pages 1 – 527).  How many digits will it take to number all 527 pages from 1 to 527? Due May 9 Circle A has a radius of 5 and its center is at (0, 5). Circle B has a radius of 12 and its center at (12, 0). What is the length of the radius of a third circle (Circle C) that passes through the center of the second circle and both points of intersection of the first two circles? Due May 16 An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle remains. What is a length of a diagonal of the rectangle if the sum of the areas of the removed pieces is 200? Due May 23 What is the only pair of integers (x, y) for which twice the square of the first equals three times the square of the second? Due May 31 What is the integer n for which 5n + 5n + 5n+ 5n + 5n = 525? Due June 6 Winkin said, “We’re all liars.” Blinkin replied, “Only you are a liar.” Finkin said, “You’re both liars.” If each of them always tells the truth or always lies, how many of the three are liars? Due June 13 An artist has been commissioned to create 540 scale replicas, each one-foot tall,  of her most famous sculpture, which stands six feet tall.  If the original statue weighs 750 pounds, what is the total weight  of the 540 scale replicas?  Assume that the replicas are made from the same material and have the same density  as the original. Most problems copyright © 2001-7 National Council of Teachers of Mathematics.