Individual Questions – Part 1 of 2
Each question is worth 10 points. Calculators are PROHIBITED.
#1.
(Time Limit: 7 minutes ) If six circles are tangent as shown, and each has an area of π, what is the area of the rectangle that just surrounds them, as shown?
国外旅游题目翻译: 如果6个面积为π的圆排列成图中的形状，那么图中的长方形的面积是多少？
Solution:
Since the area of each circle is π, each circle has a radius of 1 and a diameter of 2. The length across the rectangle consists of 3 diameters, and the width across the rectangle consists of 2 diameters. The dimensions of the rectangle are 6 by 4 and its area is 24.稠州游戏
#2.
(Time Limit: 7 minutes ) What is the least possible integer greater than 2019 that has remainder 11 when divided by 20 and 19?
题目翻译: 请问大于2019的整数中满足除以20 和 19 都余11 的最小的数是多少？
et是什么
Solution:
The least common multiple of 20 and 19 is 380, so we need an integer that is 11 more than a
multiple of 380 and that is greater than 2019. Since 5 < 2019¸380 < 6, the least acceptable multiple is = 2280 and the number 11 more than this is 2291.
#3. (Time Limit: 7 minutes ) I wrote each of the first 2019 positive integers in increasing order. Each
odd integer was written three concutive times and each even integer was written two concutive times. The first five integers I wrote were 1, 1, 1, 2, 2. What was the 2019th integer that I wrote?
题目翻译：我把前2019个正整数按照从小到大的顺序写了出来。每一个奇数连续写3次，每一个偶数连续写2次。 我写的前5个数是1, 1, 1, 2, 2。请问我写的第2019个数是什么数？
Solution:
Start at 1 and make groups of 5 integers. Each such group consists of two different integers, the first 3 odd and the next 2 even. The first 2020 integers written consists of 404 groupings of 2 different integers, so the 2020th integer written was 808. Since every even integer is written twice, the 2019th integer that I wrote was also 808.
#4. (Time Limit: 7 minutes ) What is the sum of the positive integers a , b , and c that satisfy
a + 11
b c + = 3121
?
题目翻译: 求满足以下等式的a, b, c 的和是多少 (a, b, c 为正整数)？
Solution:
节约用水手抄报
Since 31¸ 21 = 11021, a = 1. We can rewrite 1021as 121
10= 11210+, so b = 2 and c = 10. Therefore, a + b
+ c = 13.
#5.
(Time Limit: 7 minutes ) How many different five-digit positive integers contain each of the digits 1, 2, 3, 4, and 5 once, and have both 1 and 3 to the left of 2?
教诲的意思
题目翻译: 由5个数字 1, 2, 3, 4, 5组成的5位数中(每个数字用一次而且只能用一次)，有多少个数是1 和 3 都在2的左边？
Solution:
The digits 1, 2, 3, appear in either the order _1_3_2_ or _3_1_2_, with the first of the two remaining digits in one of the 4 unoccupied spaces indicated. Once one of the remaining digits is placed, there are 5 possible unoccupied spaces in which final digit may be placed. In all there are 2⨯4⨯5 = 40 ways to place all five digits.
#6.
不要以貌取人英语(Time Limit: 7 minutes ) If two fair dice are rolled, what is the probability that the sum of the two numbers rolled is a prime number?
题目翻译: 扔两个骰子，这两个骰子的和为质数的概率是多少?
Solution:
Since each die can show any one of 6 numbers, there are 36 possible outcomes when tossing two di
ce. When two dice are tosd, there are 11 possible sums: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Of the 11 possible sums, 5 of them are primes. There is 1 way to toss a 2, 2 ways to toss a 3, 4 ways to toss a 5, 6 ways to toss a 7, and 2 ways to toss an 11. That’s a total of 15 ways to toss a sum that is a prime number. Hence, the probability of tossing a sum that is a prime number is 15/36 = 5/12.
#7.
(Time Limit: 7 minutes ) A germ culture incread its population by x % in one hour. In the next hour it lost x %. If the net result after two hours was a loss of 36%, what is the value of x ?
黄色老师题目翻译: 一个细菌群体在一个小时后其人口会增长x%，下一个小时后其人口会减少x%。如果在两个小时后，该细菌群体的人口减少了36%，那么x 的值是多少？
Solution:
Let’s begin with a germ population of 100. After one hour, the population is 100 + x . After two hours,
the population is (100 + x ) – 100
x (100 + x ) = 100 – 2
100x . We are given that the net result after two hours is a loss of 36%. Thus, the amount of the germ culture left is 100 – 36 = 64. So 100 – 2100
x = 64 and 2100
x = 36. Solving, x = 60.
杨云凯
