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1、2022 Math League International Summer Challenge, Grade 4 (Unofficial version, for reference only)Note: There are eight questions in total. Five questions are worth 10 points each. Two questions are worth 15 points each. One question is worth 20 points. The total points are 100.Question 1 (15 Points)
2、Objective:Help as many ladybugs as possible land on the leaves.Rules:1. Ladybugs arrive in numerical order: Ladybug 1, Ladybug 2, Ladybug 3, etc.2. You help each ladybug choose whether to land on the left leaf or the right leaf.3. If on one leaf, the number of dots on two ladybugs adds up to the num
3、ber of dots on a third ladybug, all of the ladybugs fly away.In the figure below, you put Ladybug 1 on the right leaf. Then you put Ladybug 2, Ladybug 3, and Ladybug 4 on the left leaf. Then you put Ladybug 5 on the right leaf.Now there is nowhere to put Ladybug 6.If Ladybug 6 lands on the left leaf
4、, all of the ladybugs will fly away, because 2 + 4 = 6. If Ladybug 6 lands on the right leaf, all of the ladybugs will fly away, because 1 + 5 = 6.18You saw how to help 5 ladybugs. What is the largest number of ladybugs that you can help in this case? The answer is 8, figure below.Note: You cant ski
5、p any ladybugs. For example, the following is not allowed. You place Ladybug 1 and Ladybug 2 on the left leaf. Then you skip Ladybug 3, and place Ladybug 4 on the left or right leaf. This skipping Ladybug 3 is not allowed. Ladybugs arrive in numerical order, and you must place each of them on either
6、 leaf in numerical order. This is true for all the following questions.(a) If you started with Ladybug 2 instead of Ladybug 1, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing order: Ladybug 2, Ladybug 3, Ladybug 4, etc.Note: Please enter 0 if your answer
7、 is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 11(b) If only the odd-numbered ladybugs are out flying, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing order: Ladybug 1, Ladybug 3, Lady
8、bug 5, etc.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 0(c) If only the even-numbered ladybugs are out flying, what is the largest number of ladybugs that you can help? Ladybugs still arrive i
9、n increasing order: Ladybug 2, Ladybug 4, Ladybug 6, etc.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 8(d) If only ladybugs that are numbered with multiples of 3 (3, 6, 9, 12, 15, ) are out fly
10、ing, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing order.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 8(e) If only ladybugs that are numbered wit
11、h prime numbers (2, 3, 5, 7, 11, 13, 17, 19, ) are out flying, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing order.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no
12、 limit.Answer: 0(f) If only ladybugs that are numbered with non-multiples of 3 (1, 2, 4, 5, 7, 8, 10, 11, 13,14, ) are out flying, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing order.Note: Please enter 0 if your answer is infinitely many, which means y
13、ou can place as many ladybugs on the leaves as you want. There is no limit.Answer: 0(g) If only ladybugs that are numbered with non-multiples of 4 (1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, ) are out flying, what is the largest number of ladybugs that you can help? Ladybugs still arrive in increasing
14、 order.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 0(h) If only ladybugs that are numbered with Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ) are out flying, what is the largest
15、 number of ladybugs that you can help? Ladybugs still arrive in increasing order.Fibonacci numbers: starting from the third number, each number is the sum of the two preceding numbers.Note: Please enter 0 if your answer is infinitely many, which means you can place as many ladybugs on the leaves as you want. There is no limit.Answer: 0Question 2 (15 Points)Objective:Fill your entire garden with plots of carrots.Rule:A plot of carrots consists of two squares of your garden that are either vertically or horizontally next to each other.Example:(a) Is it possible to f
