Time : 50 minutes
Answer : End of the problem
Problem 1
(2) Calculation
(3) Find X
(4) There are four balls of red, blue, yellow, green and four boxes of red, blue, yellow. How many ways to put each ball in each box so that it may be different from the color of the ball and the box are there?
(5) Taro runs 100m in 12 seconds and Jiro runs in 15 seconds.
(6) Equilateral triangle ABC is rotated counterclockwise by 15 degrees around the vertex A as a rotation center to be ADE as shown in a figure below.
(7) Find the volume of the solid which is made by the rotation of the figure below around the line AB as a rotation axis.
Problem 2
Problem 3
Problem 4
(1) Calculation
3 + 4.6 × 5 - (11 - 1/5 / 0.1) =
(2) Calculation
13/39 + 5 / (12 + 72 / 4) =
(3) Find X
(0.6 × 7 - 6/5) × X - 7 = 5/(3 × 2) / 1/6
(4) There are four balls of red, blue, yellow, green and four boxes of red, blue, yellow. How many ways to put each ball in each box so that it may be different from the color of the ball and the box are there?
(5) Taro runs 100m in 12 seconds and Jiro runs in 15 seconds.
Taro and Jiro ran 100m race.
When Taro did a goal, how many meters behind Taro did Jiro run?
(6) Equilateral triangle ABC is rotated counterclockwise by 15 degrees around the vertex A as a rotation center to be ADE as shown in a figure below.
Find the angle of X.
(7) Find the volume of the solid which is made by the rotation of the figure below around the line AB as a rotation axis.
Pi is assumed to be 3.14.
Problem 2
There are sequences P and Q arranged according to a certain rule.
P 3, 9, 27, 81,-----
Q 2, 4, 6, 8, 10,-----
Two sequences P and Q are matched to make a sequence R in a small order.
R 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22,-------
Answer the following questions on the sequence R.
(1) Which number is 30?
(2) Which number is 300?
(3) Find the 81st number.
(4) Find the 2014th number.
Problem 3
There are cubes of ① and ② as shown in a figure below.
Cube ① is assembled with 8 pieces of small size cubes and cube ② is assembled with 27 pieces of small size cubes with the same small size. Answer to the following questions.
(1) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the name of the figure of a cut surface as correctly as possible.
(2) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.
(3) When you cut cube ② by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.
Problem 4
There are 12 points are put at equal intervals on the circumference of a circular truck as shown in Fig.1.
Boy runs clockwise and girl runs counterclockwise with a fixed speed, respectively.
Girl goes around a truck in 24 minutes and boy runs by double speed of the girl.
For example when one boy and two girls begin to run from the location of Fig.2 at the same time, boy and girls pass each other once at each of P point and Q point before the boy reaches spot R for the first time.
Thus boy and girls will pass each other twice all together.
Answer the following questions.
(1) One boy and two girls begin to run from the location of Fig.2 at the same time.
How many times do they pass each other all together before girls go around a truck?
(2) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many times do boys and girls pass each other at the point P all together before girls goes around a truck?
(3) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many hours and minutes later do they become to be the position of Fig.4 after started running?
Noted that boys and girls passed each other 9 times at the point P all together after they begin to run until they become to the position of Fig.4.
Answer
Problem 1
(2) Calculation
(3) Find X
(4) There are four balls of red, blue, yellow, green and four boxes of red, blue, yellow. How many ways to put each ball in each box so that it may be different from the color of the ball and the box are there?
(5) Taro runs 100m in 12 seconds and Jiro runs in 15 seconds.
(6) Equilateral triangle ABC is rotated counterclockwise by 15 degrees around the vertex A as a rotation center to be ADE as shown in a figure below.
(7) Find the volume of the solid which is made by the rotation of the figure below around the line AB as a rotation axis.
Problem 2
Solution
Problem 3
Problem 4
(1) Calculation
3 + 4.6 × 5 - (11 - 1/5 / 0.1) =
Answer
17
(2) Calculation
13/39 + 5 / (12 + 72 / 4) =
Answer
1/2
(3) Find X
(0.6 × 7 - 6/5) × X - 7 = 5/(3 × 2) / 1/6
Answer
4
(4) There are four balls of red, blue, yellow, green and four boxes of red, blue, yellow. How many ways to put each ball in each box so that it may be different from the color of the ball and the box are there?
Answer
9 ways
Solution
When putting a blue ball in a red box as shown in the table below, number of ways putting other balls in other boxes is three altogether.
Since there are three kind of balls of blue, yellow and green to put in red box, there are 3 × 3 = 9 in total.
(5) Taro runs 100m in 12 seconds and Jiro runs in 15 seconds.
Taro and Jiro ran 100m race.
When Taro did a goal, how many meters behind Taro did Jiro run?
Answer
20 m
Solution
Because Jiro takes 15 seconds to run 100 m, he runs 100 / 15 × 12 = 80 m in 12 seconds.
Therefore it's 100 - 80 = 20 m until a goal.
(6) Equilateral triangle ABC is rotated counterclockwise by 15 degrees around the vertex A as a rotation center to be ADE as shown in a figure below.
Find the angle of X.
Answer
75 degrees
Solution
∠CAE = ∠DAB ＝ 15 degrees
X=∠CAE + ∠AED ＝ 15 ＋ 60 ＝ 75 degrees
(7) Find the volume of the solid which is made by the rotation of the figure below around the line AB as a rotation axis.
Pi is assumed to be 3.14.
Answer
1554.3 cm3
Solution
This figure is divided into two part of M and N as shown in the figure below.
The volume of M + N is 7 × 7 × 3.14 × 5 + 5 × 5 × 3.15 × 10 = 495 × 3.14 = 1554.3 cm3.
Problem 2
There are sequences P and Q arranged according to a certain rule.
P 3, 9, 27, 81,-----
Q 2, 4, 6, 8, 10,-----
Two sequences P and Q are matched to make a sequence R in a small order.
R 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22,-------
Answer the following questions on the sequence R.
(1) Which number is 30?
(2) Which number is 300?
(3) Find the 81st number.
(4) Find the 2014th number.
Answer
(1) 18th
(2) 155th
(3) 154
(4) 4014
(1) 30 is the number of the sequence Q and is 30 / 2 = the 15th number.
Because there are three numbers of 3, 9, 27 of the sequence P until 30, 30 is 15 + 3 = the 18th number.
(2) The sequence of P is 3, 9, 27, 81, 243, 729, 2187, 4374----.
300 is the number of the sequence Q and is 300 / 2 = the 150th number.
Because there are five numbers of 3, 9, 27, 81, 243 of the sequence P until 150, 150 is 150 + 5 = the 155th number.
(3) The 81st number of Q is 81 × 2 = 162.
There are 4 numbers of P until 162. 81 - 4 = the 77th number is 77 × 2 = 154.
(4) The 2014th number of Q is 2014 × 2 = 4028.
There are 7 numbers of P until 4028.
2014 - 7 = the 2007th number is 2007 × 2 = 4014.
Problem 3
There are cubes of ① and ② as shown in a figure below.
Cube ① is assembled with 8 pieces of small size cubes and cube ② is assembled with 27 pieces of small size cubes with the same small size. Answer to the following questions.
(1) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the name of the figure of a cut surface as correctly as possible.
(2) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.
(3) When you cut cube ② by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.
Answer
(1) Rhombus
(2) 6
(3) 15
Solution
(1) The cut surface is as shown in the figure below, this figure is a rhombus.
(2) The number of cubes being cut can be expressed as shown below figure.
There are three pieces both in an upper and a lower step, so there are six pieces altogether.
(3) The number of cubes being cut can be expressed as shown below figure.
There are four pieces in an upper step, seven in a middle step and four in a lower step, so there are 15 pieces altogether.
Problem 4
There are 12 points are put at equal intervals on the circumference of a circular truck as shown in Fig.1.
Boy runs clockwise and girl runs counterclockwise with a fixed speed, respectively.
Girl goes around a truck in 24 minutes and boy runs by double speed of the girl.
For example when one boy and two girls begin to run from the location of Fig.2 at the same time, boy and girls pass each other once at each of P point and Q point before the boy reaches spot R for the first time.
Thus boy and girls will pass each other twice all together.
Answer the following questions.
(1) One boy and two girls begin to run from the location of Fig.2 at the same time.
How many times do they pass each other all together before girls go around a truck?
(2) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many times do boys and girls pass each other at the point P all together before girls goes around a truck?
(3) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many hours and minutes later do they become to be the position of Fig.4 after started running?
Noted that boys and girls passed each other 9 times at the point P all together after they begin to run until they become to the position of Fig.4.
Answer
(1) 6 times
(2) 2 times
(3) 1 hour 48 minutes
Solution
(1) Boy moves one piece per one minute according to 12/12 = 1.
Girl moves 0.5 piece per one minute according to 12 / 24 = 0.5.
According to 3/ (1 + 0.5)= 2, 6/(1 + 0.5) = 4 and 12 / (1 + 0.5) = 8, there are six times of 2 minutes later, 4 minutes later, 10 minutes later, 12 minutes later, 18 minutes later and 20 minutes later when boy and girls meet in 24 minutes.
(2) In the figure below the time when girls come to P in 24 minutes are 18 minutes later for G1, 24 minutes later for G2 and 3 minutes later for G3.
The time when boys come to P in 24 minutes are 10 minutes 22 minutes later for B1 and 6 minutes and 18 minutes later for B2.
Therefore it's twice when a man and a girl meet at P.
(3) Explore time when boys and girls come to P.
B1 comes 10, 22, 34, 46, 58, 70, 82, 94 ---- minutes.
B2 comes 6, 18, 30, 42, 54, 66, 78, 90, 102 ----- minutes.
G1 comes 18, 42, 66, 90, 114 ---- minutes, and G2 comes 24, 48, 72, 96, 130 ------ minutes, and G3 comes 6, 30, 54, 78, 102 ---- minutes. Therefore the position when a boy and a girl passed each other 9th times is as shown in the Fig. 8 and this will be 102 minutes later.
When checking the time boys come to S,U and time girls come to T,U,V thereafter, it is as following.
B1 comes to S in 108 minutes and comes to U in 112 minutes.
B2 come to S in 104 minutes and comes to U in 108 minutes.
G1 come to T in 108 minutes, U in 102 minutes and V in 120 minutes.
G2 comes to T in 114 minutes, U in 108 minutes and V in 102 minutes.
G3 comes to T in 124 minutes, U in 114 minutes and V in 108 minutes.
Therefore it'll be 108 minutes later = 1 hour and 48 minutes later when the position of boys and girls become as Fig.4 after 102 minutes.
