2022年大联盟(Math League)国际夏季四年级数学挑战活动一

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) 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 number 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, 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. ( 10 ) 学科网（北京）股份有限公司 You 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 can’t skip 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 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 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