Download HighFour Mathematics Round 5 Category A: Grades 4 – 5 Monday

HighFour Mathematics Category A: Grades 4 – 5 Round 5 Monday, January 16, 2017 The use of calculator is not allowed. Answer #1 Explanation: Answer #2 Explanation: Answer #3 Explanation: Answer #4 Explanation: 72900 seconds 20 hours is 20×60 = 1 200 minutes. Adding the 15 minutes to this gives the length of the chess game 1 215 minutes. This time, when converted to seconds, is 1 215×60 = 72 900 seconds. 100 The given expression may be grouped as follows: 200 − 198 + 196 − 194 + ⋯ + 8 − 6 + 4 − 2 The value of each of these pairs is 2 and as there are 100 numbers in this expression (there are 100 even numbers between 2 and 200, inclusive) there are 50 such pairs made in the grouped expression. Therefore, 200 − 198 + 196 − 194 + ⋯ + 8 − 6 + 4 − 2 = 2×50 = 100 999000 From 0 to 9, there are 10 different digits, giving 1 000 000 possible codes. However, as the code cannot begin with 9, 1, 1, some of these codes are impossible to have. The number of these restricted codes is 1000 (as there are 10 digits to choose from for the 4th, 5th, and 6th digits in the sequence). Therefore, the total number of possible six-­‐digit codes, after taking the restriction into account, is 1 000 000 − 1 000 = 999 000. 6 When completed, the magic square is as shown below: 9 2 25 18 11 3 21 19 12 10 22 20 13 6 4 16 14 7 5 23 15 1 24 17 8 Therefore, the number in the place of X is 6. HighFour Mathematics Category A: Grades 4 – 5 Round 5 Monday, January 16, 2017 The use of calculator is not allowed. Answer #5 Explanation: 119 Let 𝑛 represent the 8th of these 15 odd integers. Thus, the sum of these 15 consecutive integers may be represented by Answer #6 Explanation: Answer #7 Explanation: Answer #8 Explanation: 105 = 𝑛 − 14 + ⋯ + 𝑛 − 2 + 𝑛 + 𝑛 + 2 + ⋯ + 𝑛 + 14 This sum simplifies to 𝑛 = 105 , giving the largest integer 𝑛 + 14 =
105 + 14 = 119. 18 Let 𝑥 represent the total number of students, boys and girls combined. As 2
4
4
of them are boys, are girls. As such, 𝑥 = 30, which gives 𝑥 = 48. 3
3
3
Therefore, the number of boys participating in the 4th round of the competition was 48 − 30 = 18. 8000 meters Each lap Angela runs, Beth runs another quarter lap, which means Angela have run 4 laps when she is overtaken by Beth. By then Beth has run 5 laps. Therefore, the pass Angela the 2nd time, Beth has to run 10 laps in total, which is 8000 meters. 1600 m2 The original rectangle has 𝐴𝑟𝑒𝑎 = 100 𝑚 ×20 𝑚 = 2000 𝑚; and 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2 100𝑚 + 20𝑚 = 240𝑚 . Given that the perimeter ;?@A
remained the same, the sides of the square are each = 60𝑚. Thus, ?
the new square has 𝐴𝑟𝑒𝑎 = 60 𝑚×60 𝑚 = 3600 𝑚; . Therefore, the increase in area is 3600 𝑚; − 2000 𝑚; = 1600 𝑚; . HighFour Mathematics Category A: Grades 4 – 5 Round 5 Monday, January 16, 2017 The use of calculator is not allowed. Answer #9 Explanation: 48 The largest sum occurs when the largest of the 5 numbers is placed in the center. This is 26. As such, the five numbers are arranged as follows: Answer #10 Explanation: Answer #11 Explanation: Answer #12 Explanation: 2 14 26 8 20 Therefore, the largest possible sum in the vertical column is 48. 7 The only way to get a negative product (which would be needed to get the smallest possible) is to multiply one negative number with two positive numbers or multiply three negative numbers. Only two reasonable choices exist: −12 × −10 × −8 = −960 or −12 ×7×12 = −1008 Of these two, −1008 is the smallest, which means the sum of the three numbers is −12 + 7 + 12 = 7. 30° The third angle in the triangle containing the 100° and the 40° is also 40° (given that the sum of the three interior angles in a triangle must be 180°). Angle A is the third angle in the triangle containing the newly found 40° and the given 110°. Therefore, the measurement of angle A is 30°. 35 carrots At the rate of 3 carrots per minute, Peggy peeled 6×3 = 18 carrots in the 6 minutes before James joined her. If they peeled 74 carrots together then they peeled 74 − 18 = 56 carrots working together. As together they 4B
peeled at a rate of 3 + 5 = 8 carrots per minute, they took = 7 3
minutes working together. In this time, James peeled 7×5 = 35 carrots. HighFour Mathematics Category A: Grades 4 – 5 Round 5 Monday, January 16, 2017 The use of calculator is not allowed. Answer #13 Explanation: Answer#14 Explanation: Answer #15 Explanation: 17 The three lines intersect one another a maximum of 3 times. The two circles intersect each other a maximum of 2 times. Each of the three lines intersect the two circles a maximum of 4 times, giving 3×4 = 12 more points of intersection. Therefore, the largest possible number of intersections is 17. 8 2016 is divisible by 9 because the sum of its digits (2 + 0 + 1 + 6 = 9) is divisible by 9. The coming years with digits 201𝑋 will not be divisible by 9. The next time it happens will be in 202𝑋, when 𝑋 = 5, which is 2025. This will be the case 2025 − 2017 = 8 years from now. 72 The following table helps summarize the pattern: Order in sequence Number of square tiles Answer #16 Explanation: 1 2 3 4 … 𝟏𝟐
= 𝟏 𝟑𝟐
= 𝟗 𝟓𝟐
= 𝟐𝟓 𝟕𝟐
= 𝟒𝟗 … 9 10 𝟏𝟕𝟐
𝟏𝟗𝟐
= 𝟐𝟖𝟗 = 𝟑𝟔𝟏 Thus, the 10th square require 361 − 289 = 72 tiles more than the 9th square. 63 M
As points E and F are midpoints, the area of triangle EBC is of the area of M
?
the rectangle. Similarly, the area of triangle FDC is also of the rectangle’s ?
M
area. Triangle AEF then is of the rectangle. Thus, the unshaded area of the rectangle is M
?
M
3
M
?
3
+ +
4
×168 = ×168 = 105. Therefore, the area 3
of the shaded triangle is 168 − 105 = 63. HighFour Mathematics Category A: Grades 4 – 5 Round 5 Monday, January 16, 2017 The use of calculator is not allowed. Answer #17 Explanation: 0 In order to get the largest possible average, we want the 5 negative odd integers as large as possible. They are −1, −3, −5, −7, −9 and their OM P O2 P O4 P OQ P OR
O;4
average is =
= −5 . As Peter’s younger 4
4
brother gave an answer that was 5 more than the correct answer, it must have been −5 + 5 = 0. Answer #18 Explanation: Answer #19 Explanation: Answer #20 Explanation: 4 Let 𝑐 represent the number of correct answers Mark gave. Thus, her total points may be written as 3 𝑐 + −1 × 20 − 𝑐 = 44 , which gives 4𝑐 − 20 = 44 and then 𝑐 = 16. Therefore, of the 20 problems, Mark gave 4 incorrect answers. 7 The only way the four numbers can be arranged to give the equation is 2
?
− = 1. Thus, 𝐴 = 3, 𝐵 = 1, 𝐶 = 4, and 𝐷 = 2. Therefore, 𝐴 + 𝐶 = 7. M
;
6 Given that Harry has 3 brothers and 3 sisters, there are altogether 4 brothers and 3 sisters. Thus, Eva has 4 brothers and 2 sisters (Eva being the 3rd girl). Therefore, 𝐵 = 4 and 𝑆 = 2, giving 𝐵 + 𝑆 = 4 + 2 = 6.