NRP Math Challenge Club

Transcription

1 Week 7 : Manic Math Medley 1. You have exactly $4.40 (440 ) in quarters (25 coins), dimes (10 coins), and nickels (5 coins). You have the same number of each type of coin. How many dimes do you have? 2. In a triangle, the measure of one of the angles is 45. The measures of the other two angles in the triangle are in the ratio 4 : 5. What is the measure of the largest angle in the triangle? 3. Lara ate 1 of a pie and Ryan ate 3 of the same pie. The next day Cassie ate 2 of the pie that was left What fraction of the original pie was not eaten? 4. The graph shows points scored by Riley-Ann in her first five basketball games. What is the difference between the mean and the median of the number of points that she scored? 5. Toothpicks are used to make rectangular grids, as shown. Note that a total of 31 identical toothpicks are used in the 1 10 grid. How many toothpicks are used in a grid? 6. Three pumpkins are weighed two at a time in all possible ways. The weights of the pairs of pumpkins are 12 kg, 13 kg and 15 kg. How much does the lightest pumpkin weigh? 7. The product of three different positive integers is 72. What is the smallest possible sum of these integers?

2 8. The sum of four numbers is T. Suppose that each of the four numbers is now increased by 1. These four new numbers are added together and then the sum is tripled. What is the value of this final result (in terms of T)? 9. Distinct points are placed on a circle. Each pair of points is joined with a line segment. An example with 4 points and 6 line segments is shown. If 6 distinct points are placed on a circle, how many line segments would there be? 10. In the addition of the three-digit numbers shown, the letters A, B, C, D, and E each represent a single digit. What is the value of A + B + C + D + E? 11. A pool has a volume of 4000 L. Sheila starts filling the empty pool with water at a rate of 20 L/min. The pool springs a leak after 20 minutes and water leaks out at 2 L/min. Beginning from the time when Sheila starts filling the empty pool, how long does it take until the pool is completely full? 12. The positive integers are arranged in rows and columns as shown. More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000? 13. Steve begins at 7 and counts forward by 3, obtaining the list 7, 10, 13, and so on. Dave begins at 2011 and counts backwards by 5, obtaining the list 2011, 2006, 2001, and so on. Which of the following numbers appear in each of their lists? (A) 1009 (B) 1006 (C) 1003 (D) 1001 (E) 1011

3 14. An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2, 4, 6, 8 and 1, 4, 7, 10 are arithmetic sequences. In the grid shown, the numbers in each row must form an arithmetic sequence and the numbers in each column must form an arithmetic sequence. What is the value of x? 15. Molly assigns every letter of the alphabet a different whole number value. She finds the value of a word by multiplying the values of its letters together. For example, if D has a value of 10, and I has a value of 8, then the word DID has a value of = 800. The table shows the value of some words. What is the value of the word MATH? 16. The numbers 1 through 25 are arranged into 5 rows and 5 columns in the table below. What is the largest possible sum that can be made using five of these numbers such that no two numbers come from the same row and no two numbers come from the same column? 17. A bicycle travels at a constant speed of 15 km/h. A bus starts 195 km behind the bicycle and catches up to the bicycle in 3 hours. What is the average speed of the bus in km/h? 18. The average of four different positive whole numbers is 4. If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers? 19. A palindrome is a positive integer that is the same when read forwards or backwards. The numbers 101 and 4554 are examples of palindromes. What is the ratio of the number of 4-digit palindromes to the number of 5-digit palindromes?

4 20. Five students wrote a quiz with a maximum score of 50. The scores of four of the students were 42, 43, 46, and 49. The score of the fifth student was N. The average (mean) of the five students scores was the same as the median of the five students scores. How many values of N are possible? 21. Two different 2-digit positive integers are called a reversal pair if the position of the digits in the first integer is switched in the second integer. For example, 52 and 25 are a reversal pair. The integer 2015 has the property that it is equal to the product of three different prime numbers, two of which are a reversal pair. Including 2015, how many positive integers less than have this same property? 22. A bicycle at Store P costs $200. The regular price of the same bicycle at Store Q is 15% more than it is at Store P. The bicycle is on sale at Store Q for 10% off of the regular price. What is the sale price of the bicycle at Store Q? 23. Each face of a cube is painted with exactly one colour. What is the smallest number of colours needed to paint a cube so that no two faces that share an edge are the same colour? 24. How many different pairs (m, n) can be formed using numbers from the list of integers {1, 2, 3,, 20} such that m < n and m + n is even? 25. On a coordinate grid, Paul draws a line segment of length 1 from the origin to the right, stopping at (1, 0). He then draws a line segment of length 2 up from this point, stopping at (1, 2). He continues to draw line segments to the right and up, increasing the length of the line segment he draws by 1 each time. One of his line segments stops at the point (529, 506). What is the endpoint of the next line segment that he draws? 26. Tanner wants to fill his swimming pool using two hoses, each of which sprays water at a constant rate. Hose A fills the pool in a hours when used by itself, where a is a positive integer. Hose B fills the pool in b hours when used by itself, where b is a positive integer. When used together, Hose A and Hose B fill the pool in 6 hours. How many different possible values are there for a? 27. The number N is the product of all positive odd integers from 1 to 99 that do not end in the digit 5. That is, N = What is the units digit of N?

5 HARDER QUESTIONS 28. One face of a cube contains a circle, as shown. This cube rolls without sliding on a four by four checkerboard. The cube always begins a path on the bottom left square in the position shown and completes the path on the top right square. During each move, an edge of the cube remains in contact with the board. Each move of the cube is either to the right or up. For each path, a face of the cube contacts seven different squares on the checkerboard, including the bottom left and top right squares. How many different squares will not be contacted by the face with the circle on any path? 29. A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected? 30. Ten circles are all the same size. Each pair of these circles overlap but no circle is exactly on top of another circle. What is the greatest possible total number of intersection points of these ten circles? 31. Kira can draw a connected path from M to N by drawing arrows along only the diagonals of the nine squares shown. One such possible path is shown. A path cannot pass through the interior of the same square twice. In total, how many different paths can she draw from M to N? 32. The sum of all the digits of the integers from 98 to 101 is = 38. What is the sum of all of the digits of the integers from 1 to 2008? 33. There are 100 lockers in a line, numbered Locker 1, Locker 2, to Locker 100. All the lockers start of initially open. Person 1 goes down the line and for Lockers 2,4,6,8, he closes the locker. Person 2 goes down the line and for Lockers 3,6,9,12, he opens the locker if it s closed and closes it if it s open. Person 3 goes to lockers 4,8,12, and opens it if it s closed and closes it if it s open. This process repeats for 99 people, ending with Person 99 opening or closing Locker 100. After all 99 people go down the line, how many lockers are open?