During the month of February, the 2020 Chapter Competitions took place across the country. Let’s look back at a few of the problems our competitors encountered.
2020 Chapter Sprint Round, #9
Gladys draws two polygons. Her second polygon has two fewer than twice as many sides as her first polygon. If Gladys’ first polygon is a triangle, how many sides does her second polygon have?
The first polygon, a triangle, has 3 sides. The other polygon has 2 × 3 – 2 = 6 – 2 = 4 sides.
2020 Chapter Target Round, #5
Addi, Subbi, Multi, and Divi are given two distinct nonzero numbers. Addi adds the two numbers. Subbi subtracts the lesser number from the greater. Multi multiplies the two numbers. Divi divides the greater number by the lesser. If the results obtained by Addi, Multi and Divi are the same, what is Subbi’s result? Express your answer as a common fraction.
Let x and y be two distinct nonzero numbers. We have x + y = xy = x/y. xy = x/y implies xy2 = x, so y2 = 1, meaning y = +1 or y = -1. Now, y = 1 requires x + 1 = x, which is nonsense, so we must have y = -1. Then, x – 1 = -x, so x = 1/2 and -y = 1/2 + 1 = 3/2.
2020 Chapter Team Round, #3
Marko’s heart beats 72 times per minute during normal activity. It beats 65 times per minute during sleep, and 118 times per minute when Marko is exercising. During a 24-hour period, Marko slept from 10:00 p.m. until 5:00 a.m. and then exercised from 5:30 a.m. to 7:30 a.m. For the remaining time, he had normal activity. During that 24-hour period, how many times did Marko’s heart beat?
Marko slept from 10:00 p.m. to 5:00 a.m., a total of 7 hours. At a rate of 65 times per minute, which equals 3900 times per hour, his heart beat 7 × 3900 = 27,300 times. He exercised from 5:30 a.m. to 7:30 a.m., a total of 2 hours. At a rate of 118 times per minute, which equals 7080 times per hour, his heart beat 2 × 7080 = 14,160 times. During the remaining 24 – 9 = 15 hours, Marko had normal activity. At a rate of 72 times per minute, which equals 4320 times per hour, his heart beat 15 × 4320 = 64,800 times. So, his heart beat a total of 27,300 + 14,160 + 64,800 = 106,260 times.
2020 Chapter Countdown Round, #64
Last night, 5/8 of the students in the dorm looked at Facebook, 3/8 looked at Twitter, and 1/4 looked at both. What fraction of the students looked at neither? Express your answer as a common fraction.
If 1/4 of students looked at both Facebook and Twitter, we know that in the 5/8 + 3/8 = 8/8 of students who looked at Facebook or Twitter, 1/4 of them (those who looked at both) have been counted twice. So, as 1/4 = 2/8, 8/8 - 2/8 = 6/8. Therefore, 6/8 of students looked at Facebook, Twitter, or both, and 2/8 = 1/4 of the students looked at neither Facebook nor Twitter.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦