2022年大联盟(Math League)国际夏季四年级数学挑战活动一练习（含答案）

这是一份2022年大联盟(Math League)国际夏季四年级数学挑战活动一练习（含答案），共18页。

Questin 1 (15 Pints)
Objective:
Help as many ladybugs as pssible land n the leaves.
Rules:
Ladybugs arrive in numerical rder: Ladybug 1, Ladybug 2, Ladybug 3, etc.
Yu help each ladybug chse whether t land n the left leaf r the right leaf.
If n ne leaf, the number f dts n tw ladybugs adds up t the number f dts n a third ladybug, all f the ladybugs fly away.
In the figure belw, yu put Ladybug 1 n the right leaf. Then yu put Ladybug 2, Ladybug 3, and Ladybug 4 n the left leaf. Then yu put Ladybug 5 n the right leaf.
Nw there is nwhere t put Ladybug 6.
If Ladybug 6 lands n the left leaf, all f the ladybugs will fly away, because 2 + 4 = 6. If Ladybug 6 lands n the right leaf, all f the ladybugs will fly away, because 1 + 5 = 6.
Yu saw hw t help 5 ladybugs. What is the largest number f ladybugs that yu can help in this case? The answer is 8, figure belw.
Nte: Yu can’t skip any ladybugs. Fr example, the fllwing is nt allwed. Yu place Ladybug 1 and Ladybug 2 n the left leaf. Then yu skip Ladybug 3, and place Ladybug 4 n the left r right leaf. This skipping Ladybug 3 is nt allwed. Ladybugs arrive in numerical rder, and yu must place each f them n either leaf in numerical rder. This is true fr all the fllwing questins.
If yu started with Ladybug 2 instead f Ladybug 1, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder: Ladybug 2, Ladybug 3, Ladybug 4, etc.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 11
If nly the dd-numbered ladybugs are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder: Ladybug 1, Ladybug 3, Ladybug 5, etc.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 0
If nly the even-numbered ladybugs are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder: Ladybug 2, Ladybug 4, Ladybug 6, etc.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 8
If nly ladybugs that are numbered with multiples f 3 (3, 6, 9, 12, 15, …) are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 8
If nly ladybugs that are numbered with prime numbers (2, 3, 5, 7, 11, 13, 17, 19, …) are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 0
If nly ladybugs that are numbered with nn-multiples f 3 (1, 2, 4, 5, 7, 8, 10, 11, 13,
14, …) are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 0
If nly ladybugs that are numbered with nn-multiples f 4 (1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …) are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 0
If nly ladybugs that are numbered with Fibnacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …) are ut flying, what is the largest number f ladybugs that yu can help? Ladybugs still arrive in increasing rder.
Fibnacci numbers: starting frm the third number, each number is the sum f the tw preceding numbers.
Nte: Please enter 0 if yur answer is infinitely many, which means yu can place as many ladybugs n the leaves as yu want. There is n limit.
Answer: 0
Questin 2 (15 Pints)
Objective:
Fill yur entire garden with plts f carrts.
Rule:
A plt f carrts cnsists f tw squares f yur garden that are either vertically r hrizntally next t each ther.
Example:
Is it pssible t fill a 3 x 3 garden with plts f carrts, figure belw?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 0
Yu have a 3 x 3 garden that has a big bulder blcking ne f the crner squares, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 1
Yu have a 3 x 3 garden that has a big bulder blcking the center square, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 1
Yu have a 3 x 3 garden that has a big bulder blcking ne f the side squares, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N. Answer: 0
Yu have a 19 x 19 garden that has a big bulder (represented by a red X) blcking the tp left square, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 1
Yu have a 19 x 19 garden that has a big bulder (represented by a red X) blcking ne f the squares, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 0
Yu have a 19 x 19 garden that has a big bulder (represented by a red X) blcking ne f the squares, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 1
Yu have a 19 x 19 garden that has a big bulder (represented by a red X) blcking ne f the squares, figure belw. Is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 0
If a 20 x 20 garden has ne f its crners blcked by a big bulder (represented by a red X), figure belw, is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 0
If a 20 x 20 garden has tw bulders (represented by tw red X’s), figure belw, is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 1
If a 20 x 20 garden has tw bulders (represented by tw red X’s), figure belw, is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N.
Answer: 0
If a 20 x 20 garden has tw bulders (represented by tw red X’s), figure belw, is it pssible t fill yur remaining garden with plts f carrts?
Nte: Please enter 1 if yur answer is Yes, and 0 if yur answer is N. Answer: 0
Questin 3 (10 Pints)
Hw many diagnals des a regular decagn have?
Nte: A regular decagn is a plygn. It has 10 sides (edges) and 10 angles. All sides have the same length, and all angles are equal in measure.
Answer: 35
Questin 4 (10 Pints)
What is the nes digit f the sum 1 9  92  93    92022  92023 ?
Answer:0
Questin 5 (10 Pints)
Hw many ways are there fr a hrse race with three hrses t finish if ties are pssible? Nte: Tw r three hrses may tie.
Suppse there are three hrses, A, B, and C. One pssible way t finish the hrse race is that A finished the first, B finished the secnd, and C finished the last. Anther pssible way t finish the hrse race is A, B, and C all finished at the same time, i.e. all three hrses tie. The questin is asking the ttal number f all pssible ways fr a hrse race with three hrses t finish if ties are pssible.
Answer: 13
Questin 6 (10 Pints)
Each f the 4 cards shwn at the belw has a letter n ne side and a digit n the ther side.
Nw read the test sentence belw:
Whenever there is a vwel n ne side f a card, there is an even number n the ther side f that card.
What is the minimum number f cards that yu must turn ver t determine if the test sentence
abve is true r false fr this set f 4 cards?
Nte:
There are five vwels in English alphabet: A, E, I, O, U. Answer: 2
Questin 7 (10 Pints)
There are five men in frnt f yu. One f them is a truth-teller, a man wh always tells the truth, and fur f them are what are called “switchers” that behave in the fllwing way: The first time yu ask them a questin they will answer with either the truth r a lie, randmly. The secnd time yu ask them a questin, they will answer in the ppsite way f hw they answered fr the first questin. S if they answer the first questin truthfully, they will answer the secnd questin with a lie, the third questin truthfully, the furth questin with a lie, and s n and s frth.
What is the minimum number f questins yu need t ask s that yu can determine which f the five men in frnt f yu is the truth-teller?
Nte:
Each questin can nly be directed t ne persn at a time. And nly the persn whm yu ask the questin can answer the questin.
When yu ask a persn a questin, nne f the ther fur men can hear yur questin r the answer.
N cmmunicatin f any kind is allwed amng the five men.
N man can say anything except when answering yur questins.
It is nt necessary that yur questin has t be a yes/n questin. Yu can ask smething like “Wh is the truth-teller?”, “Wh are the switchers?”, etc.
Answer: 2
Questin 8 (20 Pints)
True r False: Amng any 225 cnsecutive psitive integers (each f them greater than 225), there is exactly ne psitive integer divisible by 225.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: Amng any 225 cnsecutive psitive integers (each f them greater than 225), there may be mre than ne psitive integer each divisible by 225.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: Amng any 225 cnsecutive psitive integers (each f them greater than 225), there may be n ne divisible by 225.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: If 10 integers are selected frm the first 18 psitive integers, there must be a pair f these integers with a sum equal 19.
Nte: The first 18 psitive integers are 1, 2, 3, 4, …, 17, 18.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: If 11 integers are selected frm the first 18 psitive integers, there must be a pair f these integers with a sum equal 19.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: If 12 integers are selected frm the first 18 psitive integers, there must be a pair f these integers with a sum equal 19.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: The minimum number f integers that must be selected frm the fllwing furteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, t guarantee that at least ne pair f these integers add up t 28 is 7.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: The minimum number f integers that must be selected frm the fllwing furteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, t guarantee that at least ne pair f these integers add up t 28 is 8.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: The minimum number f integers that must be selected frm the fllwing furteen integers, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, t guarantee that at least ne pair f these integers add up t 28 is 9.
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: Assuming that n ne has mre than 1,000,000 hairs n the head f any persn and that the ppulatin f New Yrk City was 8,008,278 in 2010, there had t be at least eight peple in New Yrk City in 2010 with the same number f hairs n their heads. (Peple culd be bald, withut any hair at all.)
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: Assuming that n ne has mre than 1,000,000 hairs n the head f any persn and that the ppulatin f New Yrk City was 8,008,278 in 2010, there had t be at least nine peple in New Yrk City in 2010 with the same number f hairs n their heads. (Peple culd be bald, withut any hair at all.)
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
True r False: Assuming that n ne has mre than 1,000,000 hairs n the head f any persn and that the ppulatin f New Yrk City was 8,008,278 in 2010, there had t be at least ten peple in New Yrk City in 2010 with the same number f hairs n their heads. (Peple culd be bald, withut any hair at all.)
Nte: Please enter 1 if yur answer is True, and 0 if yur answer is False.
Nte:
“divisible”: fr example, 6 is divisible by 3.
Answer:
1
0
0
1
1
1
0
1
0
1
1
0