2022 Math League International Summer Challenge, Grade 5 (Unofficial version, for reference only)Note: There are nine questions in total. Seven questions are worth 10 points each. Two questions are worth 15 points each. The total points are 100.Question 1 (10 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 = 6.12You 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 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 only ladybugs that are numbered with powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, ) 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 limit.Answer: 0(b) If only ladybugs that are numbered with square numbers (1, 4, 9, 16, 25, 36, 49, 64, ) 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 limit.Answer: 0(c) If only ladybugs that are numbered with cube numbers (1, 8, 27, 64, 125, 216, ) 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 limit.Answer: 0(d) If only ladybugs that are numbered with non-multiples of 5 (1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 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 limit.Answer: 0Question 2 (15 Points)Objective:Find the gold bar in as few weighings as possible.Rules:1. You can use a balance scale to compare the weights of two groups of bars.2. Only one bar is gold, and all other bars are counterfeits (fakes).3. The gold bar is a little heavier than each of the counterfeits.4. All counterfeits weigh the same.5. The appearances of all bars are identical. The only way to find out the gold bar is using a balance scale.Example 1:There are two bars. One is gold. The other is a counterfeit. The gold bar is a little heavier than the counterfeit. You need one weighing to find out the gold bar. In the figure below, bar 2 is the gold bar.Example 2:There are four bars. One is gold. The other three are counterfeits. The gold bar is a little heavier than each of the counterfeits. All counterfeits weigh the same. The minimum number of weighings to guarantee to find out the gold bar is 2, figure below. In the figure below, the gold bar is bar 2.Another way to do this is that you first compare two bars. If one is heavier, you are done. Otherwise, you compare the other two bars. Again, the minimum number of weighings to guarantee to find out the gold bar is 2.(a) There are 8 bars. One is gold. The other 7 are counterfeits. The gold bar is a little heavier than each of the counterfeits. All counterfeits weigh the same. What is the minimum number of weighings to guarantee to find out the gold bar?Answer: 2(b) There are 9 bars. One is gold. The other 8 are counterfeits. The gold bar is a little heavier than each of the counterfeits. All counterfeits weigh the same. What is the minimum number of weighings to guarantee to find out the gold bar?Answer: 2(c) There are 10 bars. One is gold. The other 9 are counterfeits. The gold bar is a little heavier than e