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
16、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 fill a 3 x 3 garden with plots
17、of carrots, figure below?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 0(b) You have a 3 x 3 garden that has a big boulder blocking one of the corner squares, figure below. Is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if you
18、r answer is Yes, and 0 if your answer is No.Answer: 1(c) You have a 3 x 3 garden that has a big boulder blocking the center square, figure below. Is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 1(d) You
19、 have a 3 x 3 garden that has a big boulder blocking one of the side squares, figure below. Is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No. Answer: 0(e) You have a 19 x 19 garden that has a big boulder (repres
20、ented by a red X) blocking the top left square, figure below. Is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 1(f) You have a 19 x 19 garden that has a big boulder (represented by a red X) blocking one
21、of the squares, figure below. Is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 0(g) You have a 19 x 19 garden that has a big boulder (represented by a red X) blocking one of the squares, figure below. Is
22、 it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 1(h) You have a 19 x 19 garden that has a big boulder (represented by a red X) blocking one of the squares, figure below. Is it possible to fill your remain
23、ing garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 0(i) If a 20 x 20 garden has one of its corners blocked by a big boulder (represented by a red X), figure below, is it possible to fill your remaining garden with plots of carrots?Note: Pl
24、ease enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 0(j) If a 20 x 20 garden has two boulders (represented by two red Xs), figure below, is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer
25、: 1(k) If a 20 x 20 garden has two boulders (represented by two red Xs), figure below, is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No.Answer: 0(l) If a 20 x 20 garden has two boulders (represented by two red X
26、s), figure below, is it possible to fill your remaining garden with plots of carrots?Note: Please enter 1 if your answer is Yes, and 0 if your answer is No. Answer: 0Question 3 (10 Points)How many diagonals does a regular decagon have?Note: A regular decagon is a polygon. It has 10 sides (edges) and
27、 10 angles. All sides have the same length, and all angles are equal in measure.Answer: 35Question 4 (10 Points)What is the ones digit of the sum 1+ 9 + 92 + 93 + + 92022 + 92023 ?Answer:0Question 5 (10 Points)How many ways are there for a horse race with three horses to finish if ties are possible?
28、 Note: Two or three horses may tie.Suppose there are three horses, A, B, and C. One possible way to finish the horse race is that A finished the first, B finished the second, and C finished the last. Another possible way to finish the horse race is A, B, and C all finished at the same time, i.e. all
29、 three horses tie. The question is asking the total number of all possible ways for a horse race with three horses to finish if ties are possible.Answer: 13Question 6 (10 Points)Each of the 4 cards shown at the below has a letter on one side and a digit on the other side.Now read the test sentence b
30、elow:Whenever there is a vowel on one side of a card, there is an even number on the other side of that card.What is the minimum number of cards that you must turn over to determine if the test sentenceabove is true or false for this set of 4 cards?Note:There are five vowels in English alphabet: A,
31、E, I, O, U. Answer: 2Question 7 (10 Points)There are five men in front of you. One of them is a truth-teller, a man who always tells the truth, and four of them are what are called “switchers” that behave in the following way: The first time you ask them a question they will answer with either the t
32、ruth or a lie, randomly. The second time you ask them a question, they will answer in the opposite way of how they answered for the first question. So if they answer the first question truthfully, they will answer the second question with a lie, the third question truthfully, the fourth question wit
33、h a lie, and so on and so forth.What is the minimum number of questions you need to ask so that you can determine which of the five men in front of you is the truth-teller?Note:(1) Each question can only be directed to one person at a time. And only the person whom you ask the question can answer th
34、e question.(2) When you ask a person a question, none of the other four men can hear your question or the answer.(3) No communication of any kind is allowed among the five men.(4) No man can say anything except when answering your questions.(5) It is not necessary that your question has to be a yes/
35、no question. You can ask something like “Who is the truth-teller?”, “Who are the switchers?”, etc.Answer: 2Question 8 (20 Points)(a) True or False: Among any 225 consecutive positive integers (each of them greater than 225), there is exactly one positive integer divisible by 225.Note: Please enter 1
36、 if your answer is True, and 0 if your answer is False.(b) True or False: Among any 225 consecutive positive integers (each of them greater than 225), there may be more than one positive integer each divisible by 225.Note: Please enter 1 if your answer is True, and 0 if your answer is False.(c) True
37、 or False: Among any 225 consecutive positive integers (each of them greater than 225), there may be no one divisible by 225.Note: Please enter 1 if your answer is True, and 0 if your answer is False.(d) True or False: If 10 integers are selected from the first 18 positive integers, there must be a
38、pair of these integers with a sum equal 19.Note: The first 18 positive integers are 1, 2, 3, 4, , 17, 18.Note: Please enter 1 if your answer is True, and 0 if your answer is False.(e) True or False: If 11 integers are selected from the first 18 positive integers, there must be a pair of these intege
39、rs with a sum equal 19.Note: Please enter 1 if your answer is True, and 0 if your answer is False.(f) True or False: If 12 integers are selected from the first 18 positive integers, there must be a pair of these integers with a sum equal 19.Note: Please enter 1 if your answer is True, and 0 if your
40、answer is False.(g) True or False: The minimum number of integers that must be selected from the following fourteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, to guarantee that at least one pair of these integers add up to 28 is 7.Note: Please enter 1 if your answer is True, and 0
41、if your answer is False.(h) True or False: The minimum number of integers that must be selected from the following fourteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, to guarantee that at least one pair of these integers add up to 28 is 8.Note: Please enter 1 if your answer is True
42、, and 0 if your answer is False.(i) True or False: The minimum number of integers that must be selected from the following fourteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, to guarantee that at least one pair of these integers add up to 28 is 9.Note: Please enter 1 if your answer
43、 is True, and 0 if your answer is False.(j) True or False: Assuming that no one has more than 1,000,000 hairs on the head of any person and that the population of New York City was 8,008,278 in 2010, there had to be at least eight people in New York City in 2010 with the same number of hairs on thei
44、r heads. (People could be bald, without any hair at all.)Note: Please enter 1 if your answer is True, and 0 if your answer is False.(k) True or False: Assuming that no one has more than 1,000,000 hairs on the head of any person and that the population of New York City was 8,008,278 in 2010, there ha
45、d to be at least nine people in New York City in 2010 with the same number of hairs on their heads. (People could be bald, without any hair at all.)Note: Please enter 1 if your answer is True, and 0 if your answer is False.(l) True or False: Assuming that no one has more than 1,000,000 hairs on the
46、head of any person and that the population of New York City was 8,008,278 in 2010, there had to be at least ten people in New York City in 2010 with the same number of hairs on their heads. (People could be bald, without any hair at all.)Note: Please enter 1 if your answer is True, and 0 if your answer is False.Note:“divisible”: for example, 6 is divisible by 3.Answer:(a) 1(b) 0(c) 0(d) 1(e) 1(f) 1(g) 0(h) 1(i) 0(j) 1(k) 1(l) 0