Time : 50 minutes
Passing mark : 70%
Answer : End of the problem
Problem 1
(4) As shown in a lower figure, there is a square exactly inside a big circle.
(5) As shown in Fig.1, there are three rectangular prisms whose lengths of three sides are 1 cm, 3 cm, and 9 cm.
(6) When three integers 1277, 1550, and 2278 are divided by a certain integer, all remainders will become the same.
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Find the number in X and Y.
(1) 20 / ｛7.2 - ( 16/5 - 5/4) /0.5｝ × 9/4 - 13 = X
(2) ｛(24/23 - Y) / 0.125 - 8/5 ｝ × 2.53 = 11
Problem 2
(1) Students stay at a hotel.
If the capacity of one room would set to be five persons, it becomes insufficient for 4 persons even if all rooms are used.
If the capacity of one room would set to be six persons, there is one room of five persons and one room will remain.
Find the number of students.
(2) Find the angle of A in the figure below.
(3) In the integers from 1 to 60, find the number of integers which cannot be divided by 2 or 3.
(4) As shown in a lower figure, there is a square exactly inside a big circle.
Moreover, there are four small circles with same size exactly inside the square.
Find the ratio of the area of a big circle to the area of one small circle.
(5) As shown in Fig.1, there are three rectangular prisms whose lengths of three sides are 1 cm, 3 cm, and 9 cm.
These are accumulated from the ground.
For example, if it puts as shown in Fig. 2, the height from the ground will be 13 cm.
How many kinds of height from the ground when these three rectangular prisms are accumulated is there in all?
(6) When three integers 1277, 1550, and 2278 are divided by a certain integer, all remainders will become the same.
Find the largest integer among such integers.
Problem 3
Three persons, Taro, Jiro, and Hanako walk the surrounding way of a pond 1 round.
All three persons start from the same place simultaneously.
Taro walks clockwise at 80 m/m and Jiro walks clockwise at 60 m/m. Hanako walks at fixed speed counterclockwise.
Hanako passed by Taro first 20 minutes after leaving and she passed by Jiro 4 minutes afterward.
Answer the following questions.
(1) Find the speed per minute of Hanako.
(2) Find the distance of the surrounding way of a pond.
Problem 4
When all the number of the dates of Wednesday in August of a certain year are added, the sum was 85.
Answer the following questions.
(1) Find the date of the first Wednesday in August of this year.
(2) Find the sum which added all the number of the dates of Wednesday in January of the following year.
Problem 5
There are right triangle ABC as shown in a lower figure and the point D on the side AC.
Answer the following questions.
(1) Find the length of AD and the length of BD, respectively.
(2) Find the volume of the solid made when rotating triangle ABC one time centering on the side AC.
Pi is assumed to be 3.14.
Problem 6
Three kinds of characters A, B and C are arranged sequentially from the left.
You may use the same character many times.
However, the right-hand of A must be C and the right-hand of B must also be C.
As for the way of arranging which fulfills this rule,when putting only one character in order from the left, there are three kinds of A, B, and C. When putting two characters in order from the left, it becomes five kinds, AC, BC, CA, CB, and CC.
Answer the following questions.
(1) How many ways of arrangement in the case of putting three pieces in order from the left are there?
(2) How many ways of arrangement in the case of putting four pieces in order from the left are there?
(3) How many ways of arrangement in the case of putting seven pieces in order from the left are there?
Problem 7
White stones arranged in the form of square whose one side is five stones as shown in (A) a lower figure are rearranged so that the number of a vertical side may become the same as (A) as shown in (B).
Then as for (B) there are three vertical sides and one piece remain.
When white stones arranged in the form of square whose one side is five or more stones are rearranged, remainder will be called Fraction.
In the case of the figure, it is "Fraction is 1."
Answer the following questions.
(1) Find Fraction when white stones arranged in the form of square whose one side is six stones are rearranged.
(2) Find the sum total of white stone in case Fraction is set to 4.
(3) The sum total of white stone is calculated as (Fraction) × X + Y. Find the number applicable to X and Y, respectively.
<Answer>
Problem 1
Answer
(4) As shown in a lower figure, there is a square exactly inside a big circle.
(5) As shown in Fig.1, there are three rectangular prisms whose lengths of three sides are 1 cm, 3 cm, and 9 cm.
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Find the number in X and Y.
(1) 20 / ｛7.2 - ( 16/5 - 5/4) /0.5｝ × 9/4 - 13 = X
(2) ｛(24/23 - Y) / 0.125 - 8/5 ｝ × 2.53 = 11
Answer
(1) 7/11
(2) 3/10
Problem 2
(1) Students stay at a hotel.
If the capacity of one room would set to be five persons, it becomes insufficient for 4 persons even if all rooms are used.
If the capacity of one room would set to be six persons, there is one room of five persons and one room will remain.
Find the number of students.
59 persons
Solution
6 -5 =1
(4 + 7) / 1 =11
5 × 11 + 4 = 59 persons
(2) Find the angle of A in the figure below.
Answer
100 degrees
Solution
180 × (6 - 2) = 720 degrees
720 - (90 + 90 + 120 + 80 + 80) = 260 degrees
360 - 260 = 100 degrees
(3) In the integers from 1 to 60, find the number of integers which cannot be divided by 2 or 3.
Answer
20 pieces
Solution
60 / 2 = 30
60 / 3 = 20
60 / 6 = 10
60 - (30 + 20 - 10) = 20 pieces
(4) As shown in a lower figure, there is a square exactly inside a big circle.
Moreover, there are four small circles with same size exactly inside the square.
Find the ratio of the area of a big circle to the area of one small circle.
Answer
8 : 1
Solution
Length of radius of small circle is set to 1.
The area of small circle is 1 × 1 × 3.14 = 3.14.
The area of square is 4 × 4 = 16.
Length of radius of big circle is set to X, the are of big circle is expressed as X × X × 3.14.
The area of square is expressed as X × X × 2 by using X.
X × X × 2 = 16
X × X = 8
Then the area of big circle is 8 × 3.14.
Therefor ratio of the area is (8 × 3.14) : 3.14 = 8 : 1.
(5) As shown in Fig.1, there are three rectangular prisms whose lengths of three sides are 1 cm, 3 cm, and 9 cm.
These are accumulated from the ground.
For example, if it puts as shown in Fig. 2, the height from the ground will be 13 cm.
How many kinds of height from the ground when these three rectangular prisms are accumulated is there in all?
Answer
10 kinds
Solution
| 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 9 |
| 1 | 1 | 1 | 3 | 3 | 9 | 3 | 3 | 9 | 9 |
| 1 | 3 | 9 | 3 | 9 | 9 | 3 | 9 | 9 | 9 |
(6) When three integers 1277, 1550, and 2278 are divided by a certain integer, all remainders will become the same.
Find the largest integer among such integers.
Answer
91
Solution
1550 - 1277 = 273
2278 - 1550 = 728
G.C.M. of 273 and 728 is 91.
Problem 3
Three persons, Taro, Jiro, and Hanako walk the surrounding way of a pond 1 round.
All three persons start from the same place simultaneously.
Taro walks clockwise at 80 m/m and Jiro walks clockwise at 60 m/m. Hanako walks at fixed speed counterclockwise.
Hanako passed by Taro first 20 minutes after leaving and she passed by Jiro 4 minutes afterward.
Answer the following questions.
(1) Find the speed per minute of Hanako.
(2) Find the distance of the surrounding way of a pond.
Answer
(1) 40 m/m
(2) 2400 m
Solution
(1)
(80 - 60) × 20 =400 m
400 / 4 = 100 m/m
100－60 = 40 m/m
(2)
1600 + 400 × 20 = 2400 m
Problem 4
When all the number of the dates of Wednesday in August of a certain year are added, the sum was 85.
Answer the following questions.
(1) Find the date of the first Wednesday in August of this year.
(2) Find the sum which added all the number of the dates of Wednesday in January of the following year.
Answer
(1) (August) 3rd
(2) 58
Solution
(1)
The first day of Wednesday is set to X.
When there are four times of Wednesday, X + X + 7 + X + 14 + X + 21 = 85.
X × 4 = 43 which is not suitable.
When there are five times of Wednesday, X + X + 7 + X + 14 + X + 21 X + 28 = 85.
X × 5 = 15
Then X = 3.
(2)
29 + 30 + 31 + 30 + 31 + 1 = 152 days
152 / 7 = 21 remainder 5
Then Jan. 1st is Sunday.
First Wednesday is Jan. 4th.
Then 4 + 11 + 18 + 25 = 58.
Problem 5
There are right triangle ABC as shown in a lower figure and the point D on the side AC.
Answer the following questions.
(1) Find the length of AD and the length of BD, respectively.
(2) Find the volume of the solid made when rotating triangle ABC one time centering on the side AC.
Pi is assumed to be 3.14.
Answer
(1) AD = 1.8 cm (9/5 cm),
BD = 2.4 cm (12/5 cm)
(2) 30.144 cm2
Solution
(1)
AD = 3 cm × 3/5 = 9/5 cm = 1.8 cm.
BD = 3 cm × 4/5 = 12/5 cm = 2.4 cm.
(2)
2.4 × 2.4 × 3.14 × AD × 1/3 + 2.4 × 2.4 × 3.14 × CD × 1/3
= 2.4 × 2.4 × 3.14 × 5 × 1/3 = 30.144 cm3Problem 6
Three kinds of characters A, B and C are arranged sequentially from the left.
You may use the same character many times.
However, the right-hand of A must be C and the right-hand of B must also be C.
As for the way of arranging which fulfills this rule,when putting only one character in order from the left, there are three kinds of A, B, and C. When putting two characters in order from the left, it becomes five kinds, AC, BC, CA, CB, and CC.
Answer the following questions.
(1) How many ways of arrangement in the case of putting three pieces in order from the left are there?
(2) How many ways of arrangement in the case of putting four pieces in order from the left are there?
(3) How many ways of arrangement in the case of putting seven pieces in order from the left are there?
Answer
(1) 11 ways
(2) 21 ways
(3) 171 ways
Solution
① = 43
② = 43
③ = 21
④ = 21
⑤ = 43
43 + 43 + 21 + 21 + 43 = 171 ways
Problem 7
White stones arranged in the form of square whose one side is five stones as shown in (A) a lower figure are rearranged so that the number of a vertical side may become the same as (A) as shown in (B).
Then as for (B) there are three vertical sides and one piece remain.
When white stones arranged in the form of square whose one side is five or more stones are rearranged, remainder will be called Fraction.
In the case of the figure, it is "Fraction is 1."
Answer the following questions.
(1) Find Fraction when white stones arranged in the form of square whose one side is six stones are rearranged.
(2) Find the sum total of white stone in case Fraction is set to 4.
(3) The sum total of white stone is calculated as (Fraction) × X + Y. Find the number applicable to X and Y, respectively.
Answer
(1) 2
(2) 28 pieces
(3) X = 4, Y = 12
Solution
(1)
(6 - 1) × 4 = 20 pieces
20 / 6 = 3 remainder 2
(2)
(3)
4 × X + Y = 28
3 × X + Y = 24
X = 4
Y = 12
