Time : 45 minutes
Passing mark : 65%
Answer : End of problem
Problem 1
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Problem 2
Problem 3
Problem 4
Problem 5
(1) ①15.7 20 + 62.8 5 + 314 =
② Find X.
{20 - 8/5 × (X - 1)} × 13/3＝52
(2)
I set the list price with 15 % profit on the cost price of a certain article.
But it did not be sold.
Then I decided to sell at 2070 yen which is 10 % discount of the list price.
Find the cost price.
(3)
I walked from A point to B point at 4 km/h and returned running at 8 km/h.
The time of a round trip was 45 minutes.
Find the distance between AB.
(4)
There are two articles A and B in a certain store and if you buy four pieces of A and three pieces of B, it is 1000 yen.
Moreover, the price of three A is 70 yen higher than the price of two B.
Find a price of one piece of A.
(5)
It takes 12 minutes for Taro and 8 minutes for Jiro to run the surroundings of a pond 1 round.
When they started from the same point to the same direction, every how many minutes is Taro passed by Jiro ?
(6)
There is a square ABCD whose length of one sideof is 1.
First, the point P is in the position of A.
When I throw one coin and head comes out, the point P moves 1 and tail comes out, P moves 2 in the direction of an arrow on the side.
When I throw coin 5 times, the point P came to the position of A.
In this case, how many kinds of way of head and tail coming out are there?
<Example> When throwing twice and coming to the position of D, there are two kinds, (Head, Tail),(Tail, Head).
(7)
Three fractions 28/3, 35/6, and 77/8 are multiplied by the same fraction and each product becomes to be integer.
Find the fraction more than 10 and the nearest to 10 among such fractions.
(8)
Paper of a right triangle is rolled around a pillar whose bottom is square as shown in a lower figure.
Find the area of the portion which only one-fold paper has rolled around the side of the square pillar.
Problem 2
Some sheets of origami (paper folding) is divided into three persons, Taro, Jiro, and Hanako.
Taro gets seven sheets less than three times of sheets Jiro gets.
Hanako gets five sheets more than twice of sheets Jiro gets.
Answer the next questions.
(1) When you distribute 100 sheets of origami to all three persons, find the number of sheets Jiro gets.
(2) There are some sheets of origami. First, 17 sheets of origami are distributed to Jiro.
When the remaining origami was divided according to the conditions described above, all of three persons get the same number of sheets.
In this case, find the number of the distributed origami.
Problem 3
As shown in a lower figure, cubes whose one side is 1cm are accumulated.
Answer the next questions.
(1) When you accumulate five layers, find the number of cubes used.
(2) When the number of cubes used is 140 pieces, find the number of cubes in the bottom layer.
(3) Remove cubes from a top to the 4th layer after accumulating cubes to ten layers.
Find the surface area of the remaining solid.
Problem 4
As for triangle ABC of a lower figure, D and E are the points of dividing the side AB into three equally and F and G are the points of dividing the side AC into three equally.
Answer the next questions.
(1) Find the ratio of BQ : QP : PF by the least integer.
(2) Find the ratio of the area of a quadrangle PQRS to the area of triangle ABC by the least integer.
Problem 5
There is a rectangular prism as shown in a figure below.
This rectangular prism is cut according to following ①, ② and ③.
① It is cut with a plane parallel to the plane ABCD.
② It is cut with a plane parallel to the plane DCGH.
③ It is cut with a plane parallel to the plane ADHE.
Noted that it is cut so that the form of original rectangular prism may not be broken down and the rectangular prism is separated off after cutting.
Answer the next questions.
(1) When it is cut according to ①, the sum of the surface area of solids increased 3 times of the surface area of the original rectangular prism.
In this case, find the number of times of cutting.
(2) When it is cut according to ①, ② and ③ in total several times and divide into 30 solids.
Find the total number of times with the least times of cutting.
(3) When it is cut according to ①, ② and ③ in total several times and divide into 40 solids.
When the sum of the surface area of all solids becomes the least, find the sum of the surface area.
<Answer>
Problem 1
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Problem 2
Problem 3
Problem 4
Problem 5
(1) ①15.7 20 + 62.8 5 + 314 =
② Find X.
{20 - 8/5 × (X - 1)} × 13/3＝52
Answer
① 942
② 6
(2)
I set the list price with 15 % profit on the cost price of a certain article.
But it did not be sold.
Then I decided to sell at 2070 yen which is 10 % discount of the list price.
Find the cost price.
Answer
2000 yen
Solution
Assuming the cost price is 1, list price = 1 × (1 + 0.15) = 1.15.
1.15 × ( 1 - 0.1) = 1.035 = 2070 yen
1 = 2070 / 1.035 = 2000 yen.
(3)
I walked from A point to B point at 4 km/h and returned running at 8 km/h.
The time of a round trip was 45 minutes.
Find the distance between AB.
Answer
2 km
Solution
Ratio of the speed = 1 : 2.
Ratio of the time = 2 : 1.
Time of going is 45 minutes × 2/3 = 30 minutes.
Therefore, 4 km/h × 30/60 = 2 km.
(4)
There are two articles A and B in a certain store and if you buy four pieces of A and three pieces of B, it is 1000 yen.
Moreover, the price of three A is 70 yen higher than the price of two B.
Find a price of one piece of A.
Answer
130 yen
Solution
4A + 3B = 1000,
8A + 6B = 2000
3A = 2B + 70,
2B = 3A - 70,
6B = 9A - 210
8A + 9A -210 = 2000
17A - 210 = 2000
17A = 2210
A = 130 yen
(5)
It takes 12 minutes for Taro and 8 minutes for Jiro to run the surroundings of a pond 1 round.
When they started from the same point to the same direction, every how many minutes is Taro passed by Jiro ?
Answer
24 minutes
Solution
The surrounding of a pond is set to 24.
The speed of Taro is 24 / 12 = 2 and Jiro is 24 / 8 = 3.
24 / (3 - 2) = 24 minutes
(6)
There is a square ABCD whose length of one sideof is 1.
First, the point P is in the position of A.
When I throw one coin and head comes out, the point P moves 1 and tail comes out, P moves 2 in the direction of an arrow on the side.
When I throw coin 5 times, the point P came to the position of A.
In this case, how many kinds of way of head and tail coming out are there?
<Example> When throwing twice and coming to the position of D, there are two kinds, (Head, Tail),(Tail, Head).
Answer
10 kinds
Solution
When coin is thrown 5 times, the shortest is 1 × 5 = 5 and the longest is 2 × 5 = 10.
As P comes to A when it moves 4, 8, 12 -----, it is found total number is 8.
(8 - 1 × 5) / (2 - 1) = 3 which is the number head comes out.
Then (5 × 4 × 3) / (3 × 2 × 1) = 10 kinds.
(7)
Three fractions 28/3, 35/6, and 77/8 are multiplied by the same fraction and each product becomes to be integer.
Find the fraction more than 10 and the nearest to 10 among such fractions.
Answer
72/7
Solution
GCM of 28, 35 and 77 = 7
LCM of 3, 6 and 8 = 24
The fraction is 24/7, 48/7, 72/7 which is over 10 and nearest to 10.
(8)
Paper of a right triangle is rolled around a pillar whose bottom is square as shown in a lower figure.
Find the area of the portion which only one-fold paper has rolled around the side of the square pillar.
Answer
42 cm2
Solution
AA' : GE = 12 cm : 6 cm = 2 : 1
A'P = 6cm 2/(2+1) = 4cm and P'E = 6 - 4 = 2cm.
Area of rectangle AEE'A' = 6 12 = 72 cm2.
△APA' = 4 12 / 2 = 24 cm2
△P'EG = 2 6 / 2 = 6 cm2
Then 72 - ( 24 + 6) = 42 cm2
Problem 2
Some sheets of origami (paper folding) is divided into three persons, Taro, Jiro, and Hanako.
Taro gets seven sheets less than three times of sheets Jiro gets.
Hanako gets five sheets more than twice of sheets Jiro gets.
Answer the next questions.
(1) When you distribute 100 sheets of origami to all three persons, find the number of sheets Jiro gets.
(2) There are some sheets of origami. First, 17 sheets of origami are distributed to Jiro.
When the remaining origami was divided according to the conditions described above, all of three persons get the same number of sheets.
In this case, find the number of the distributed origami.
Answer
(1) 17 sheets
(2) 87 sheets
Solution
(1)
T = 3J - 7
H = 2J + 5
3J - 7 + J + 2J + 5 = 100
6J = 102
J = 102/6 = 17 sheets
(2)
J + 17 = 3J -7
J = 24 / 2 = 12 sheets
J + 17 = 29
29 × 3 = 87 sheets
Problem 3
As shown in a lower figure, cubes whose one side is 1cm are accumulated.
Answer the next questions.
(1) When you accumulate five layers, find the number of cubes used.
(2) When the number of cubes used is 140 pieces, find the number of cubes in the bottom layer.
(3) Remove cubes from a top to the 4th layer after accumulating cubes to ten layers.
Find the surface area of the remaining solid.
Answer
(1) 55 pieces
(2) 49 pieces
(3) 380 cm2
Solution
(1)
One layer = 1
Two layer = 2 × 2 = 4
Three layer = 3 × 3 = 9
Then 1 + 4 + 9 + 16 + 25 = 55 pieces
(2)
Until six layer = 55 + 36 = 91
Seven layer = 91 + 49 = 140 pieces
Bottom layer = 49 pieces
(3)
10 cm × 10cm × 2 = 200 cm2
1 cm 1cm (5+6+7+8+9+10) = 180 cm2
Total = 200 + 180 = 380 cm2
Problem 4
As for triangle ABC of a lower figure, D and E are the points of dividing the side AB into three equally and F and G are the points of dividing the side AC into three equally.
Answer the next questions.
(1) Find the ratio of BQ : QP : PF by the least integer.
(2) Find the ratio of the area of a quadrangle PQRS to the area of triangle ABC by the least integer.
Answer
(1) 12 : 9 : 7
(2) 9 : 70
Solution
(1)
△DPF and △PBC are homothetic and FP : BP = 1 : 3.
△BDF and △BEX are homothetic and BX : XF = BE : ED = 1 : 1.
Moreover, EX : DF = 1 : 2 and DF : BC = 1 : 3 = 2 : 6, then EX : BC = 1 : 6.
△QEX and △QBC are homothetic and XQ : BQ = 1 : 6.
Relation of ratio of Fig.2 can be adjusted as Fig.3.
Therefore BQ : QP : PF = 12 : 9 : 7.
(2)
CS : SP : PD = BQ : QP : PF = 12 : 9 : 7.
CR : RQ : QE is also calculated to be 21 : 9 : 5.
△BCF = △ABC × 2/3
△PCQ = △BCF × 9/(12+9+7)
= △BCF × 9/28
= △ABC × 2/3 × 9/28
= △ABC × 3/14
CS = CP × 12/21, CR = CQ × 21/30
△SCR = △PCQ × 12/21 × 21/30
= △ABC × 3/14 × 12/21 × 21/30
= △ABC × 3/35
PQRS = △ABC × (3/14 - 3/35)
= △ABC × 9/70
PQRS : △ABC = 9 : 70
Problem 5
There is a rectangular prism as shown in a figure below.
This rectangular prism is cut according to following ①, ② and ③.
① It is cut with a plane parallel to the plane ABCD.
② It is cut with a plane parallel to the plane DCGH.
③ It is cut with a plane parallel to the plane ADHE.
Noted that it is cut so that the form of original rectangular prism may not be broken down and the rectangular prism is separated off after cutting.
Answer the next questions.
(1) When it is cut according to ①, the sum of the surface area of solids increased 3 times of the surface area of the original rectangular prism.
In this case, find the number of times of cutting.
(2) When it is cut according to ①, ② and ③ in total several times and divide into 30 solids.
Find the total number of times with the least times of cutting.
(3) When it is cut according to ①, ② and ③ in total several times and divide into 40 solids.
When the sum of the surface area of all solids becomes the least, find the sum of the surface area.
Answer
(1) 11 times
(2) 7 times
(3) 6800 cm2
Solution
(1)
Total surface area of original rectangular prism is (10 × 20 + 20 × 30 + 10 × 30) × 2 = 1100 × 2 = 2200 cm2.
The area increased by one cut is 10 × 20 × 2 = 400 cm2.
2200 × 3 = 6600
(6600 - 2200) / 400 = 11 times.
(2)
30 = 1 × 1 × 30 = 1 × 2 × 15 = 1 × 3 × 10 = 1 × 5 × 6 = 2 × 3 × 5
The number of cutting of 2 × 3 × 5 is (2 - 1) + (3 -1) +(5 - 1) = 7 which is the least.
(3)
40 = 1 × 1 × 40 = 1 × 2 × 20 = 1 × 4 × 10 = 1 × 5 × 8 = 2 × 2 × 10 = 2 × 4 × 5
Cutting of ③ should be as less as possible and ① should be as many as possible in order to make the total surface area be least.
In case of 2 × 4 × 5, ① is 4 times , ② is 3 times and ③ is 1 time, in total 7 times.
Total surface area is 2200 + 400 × 4 + 600 × 3 + 1200 × 1 = 2200 + 4600 = 6800 cm2.
