2021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2021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8页。

Questin 1:
There is a table in frnt f yu with ne hundred quarters n it. Yu have been blindflded and are wearing a thick pair f glves. Yu are nt able t see whether the quarters are heads r tails because yu are blindflded. And yu are nt able t feel whether the quarters are heads r tails because f the thick glves. Yur friend tells yu that twenty f these quarters are tails and remaining eighty are heads, but yu d nt knw which are which. He tells yu that if yu are able t split the quarters int tw piles where the number f tails quarters is the same in each pile, yu will win all f the quarters. Yu are free t mve the quarters, flip them ver, and arrange them int tw piles f any number. Fr yu t win all f the quarters, hw many quarters are in each f the tw piles? (Please enter yur answers in ascending rder.)
Nte:
heads: the frnt side f a cin. tails: the back side f a cin.
Figure belw, heads and tails f a quarter.
Answer:
20
80
Slutin:
Yu can win this game with just ne elegant mve: Take any twenty quarters frm the hundred, put them int a separate pile, and flip them ver.
N matter hw many f the quarters are tails in the grup flip ver, yu will always wind up with tw grups f quarters with the same number f tails. If yu happened t select all twenty f the quarters that were already tails, yu wuld separate them and flip them, resulting in tw grups f quarters with zer tails in them. If yu happened t select twenty quarters with nly six tails amng them, yu wuld leave furteen tails in the riginal pile, and after flipping the twenty quarters yu selected, yu wuld have furteen tails and six heads in the new pile. This will always wrk as lng as yu nly take twenty quarters, put them int their wn pile, and flip them all ver.
Questin 2:
The pages f a bk are cnsecutively numbered frm 1 thrugh 384. Hw many times des the digit ‘8’ appear in this numbering?
Answer: 73 Slutin:
One way: We can lk at the nes and tens digits separately:
The nes digit f 8 frm 1 thrugh 384: since it ccurs ne time in every set f 10 cnsecutive numbers, there are 38 cmplete sets f 10 cnsecutive numbers. S, the digit ‘8’ appears 38 times as a nes digit.
The tens digits f 8 frm 1 thrugh 384: since it ccurs 10 times in every set f 100 cnsecutive numbers, there are 3 cmplete sets f 100 (1-100, 101-199, 200-299). The digit 8 appears 30 times as a tens digit. In additin, the numbers 380-384 cntain 5 mre tens digit f 8. In all, the digit ‘8 appears a ttal f 38 + 30 + 5 = 73 times.
Or if yu are lking at the first hundred numbers there will be 20 digits 8 in them: 8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 98.
Since we have 384 pages it means 20 × 3 + 13 = 73 times digit 8 will be happening in the page numeratin 1-384.
Questin 3:
Annabella is visiting her grandma wh resides 295 miles away frm her. She started her driving trip at a speed f 65 mph (miles per hur) fr the first 3 hurs. Fr the rest f the trip, she drve at a speed f 50 mph. Hw many hurs did she take t drive t her grandma?
Answer: 5 Slutin:
Fr the first 3 hurs, she cmpletes 65 mph × 3 hurs = 195 miles. She then has 295 – 195 = 100 miles t g. Since her speed is 50 mph, it will take her 100 ÷ 50 = 2 hurs t cmplete the rest f her jurney. Ttal hurs: 2 + 3 = 5 hurs.
Questin 4:
Sarah is using ppsicle sticks t build sme grids fr her city planning prject. She needs 4 ppsicle sticks t make a 1 by 1 grid, and 12 sticks t make a 2 by 2 grid as shwn belw. Hw many sticks des she need t make a 10 by 20 grid?
Answer: 430 Slutin:
Analyze the 2 by 2 grid. T create it Sarah used 3 rws f 2 sticks each and 3 clumns f 2 sticks each. Similar t it, there wuld be 11 rws and there shuld be 20 sticks in each rw. S, there are 20 × 11 = 220 sticks (placed hrizntally).
There wuld als be 21 clumns and there shuld be 10 sticks in each. Therefre, there are 10 × 21 = 210 sticks placed vertically.
Ttal = 220 + 210 = 430 ppsicle sticks
Questin 5:
The sides f the large rectangle are 20 m and 16 m, figure belw, nt drawn t scale.
Answer:
192
All six shaded rectangles are identical. What is the ttal area f all the shaded regins, in square meters?
Answer: 192 Slutin:
Find the dimensin f the small rectangle. The length: 16 ÷ 2 = 8 m. The width: 20 – 8 – 8 = 4 m. The ttal area f all the shaded regins: 6 × (8 × 4) = 192 m2.
Questin 6:
Alice, Bibi, and Clary are three intelligent girls. One day, their teacher decides t play a simple game and award sme prize t the first girl wh slves it:
A black r white hat is placed n each girl’s head, and each girl can see the ther girls’ hats, but nt her wn. They are instructed t raise their hands if they see at least ne black hat. And the first girl t tell the teacher what clr hat she is wearing and hw she figured this ut will get the prize. After the hats are placed n Alice, Bibi, and Clary’s heads, each girl raises her hand. After a few secnds, Alice says: “I have figured it ut!”
What clr hat is Alice wearing? Three chices:
Black
White
Nn-deterministic Answer: (a)
Slutin:
Alice knws that she is wearing a black hat and this is hw she is able t deduce it: It is clear, because all three persns raised their hands, that there are at least tw black hats. Hwever, if there were tw black hats and ne white hat, either f the peple with the black hats wuld see the white hat, and see that everyne had their hand raised which wuld allw her t instantly deduce the clr f their hat is black.
After a few secnds had passed withut anyne speaking abut the clr f their hat, it became clear t Alice, the brightest f the bunch, that they must all be wearing black hats, therwise either Bibi r Clary, wh are intelligent, wuld have said smething by nw.
Questin 7:
Yu have a balance scale and six weights. There are tw red weights, tw range weights, and tw blue weights. In each pair f clred weights, ne weight is slightly heavier than the ther, but is therwise identical. The three heavier weights all weigh the same, and the three lighter weights all weigh the same.
What is the fewest number f times yu need t use the balance scale in rder t psitively identify the heavier weight in each pair?
Answer: 2 Slutin:
T find the heavier weight f each clr, yu nly need t use the balance scale twice. First, yu must weigh a red and an range weight against a blue and an range weight.
If the pans balance: Yu can be sure that there is a heavy and a light weight n each pan. Take bth the red and blue weights ff the scale and leave the range weights n
each side. This will shw yu which f the range weights is the heavier ne, which will then shw yu whether the red r blue weight that was just n the pan was the heavier ne, which will then allw yu t, by prcess f eliminatin, determine the weight f the red and blue weight that were never n the pan.
If the pans d nt balance: Yu can be sure that the range weight n the side f the scale that went dwn is the heavier f the tw range weights. T find the heavier weights f the red and blue pair, yu must take the red weight that yu just weighed and weigh it against the blue weight that has nt yet been n the scale. Seeing what happens here, plus remembering what happened when yu made the first weighing, will allw yu t crrectly label each weight.
Questin 8:
In a class f 5th grade students, 15 have pet cats, 12 have pet dgs, 5 have bth pet cats and pet dgs, and 8 have neither pet cats nr pet dgs. Hw many ttal students are in the class?
Answer: 30 Slutin:
5 have bth cats and dgs. 15 – 5 = 10 have cats nly.
12 – 5 = 7 have dgs nly. 8 have neither cats r dgs. 5 + 10 + 7 + 8 = 30.
Questin 9:
What is the 5-digit mystery number that fits all the cnditins belw?
The sum f the first tw digits (cunting frm left t right, same belw) is ne smaller than the third digit.
The third digit is duble the furth digit.
The furth digit is duble the last digit.
The third digit is the prduct f the furth and fifth digits.
The secnd digit is five mre than the first digit.
The first digit is ne-eighth f the third digit and als ne-furth f the furth digit.
Answer: 16842 Slutin:
“The first digit is ne-eighth f the third digit.”: the first digit has t be 1, and the third digit has t be 8.
“The first digit is ne-furth f the furth digit.”: the furth digit is 4.
“The sum f the first tw digits is ne smaller than the third digit.”: the secnd digit is 6.
“The furth digit is duble the last digit.”: the last (fifth) digit is 2. S the answer is 16842.
Please nte sme cnditins are redundant.
Questin 10:
If tw sides f a square field were increased by five feet, as seen in the diagram belw, nt drawn t scale, the area f the field wuld increase by 245 square feet. Find the area f the riginal square.
Answer: 484 Slutin:
The area f the shaded square is 5 × 5 = 25, figure belw. The area f each f the tw “new” rectangles is (245 – 25) ÷ 2 = 110. Thus the length f the “new” rectangle, r the side length f the riginal square, is 110 ÷ 5 = 22. S the area f the riginal square is 22 × 22 = 484.