Math Problem : DD.33 Two points between school and park

There are A point and B point in order on the way from the school to the park.
Hanako left the school at 9:00 a.m. and walked to the park.
She took a rest for 7 minutes at A point and took a rest for 12 minutes at B point.

Taro left the school at 9:20 a.m. and went to the park by bicycle.
He went to the park without taking a rest on the way.
After resting for 7 minutes, he left the park towards the school.
In the meantime, Taro passed Hanako at P point which is located before A point at 9:28 a.m.

Moreover, Taro passed through B point 10 minutes after leaving the park and then Hanako arrived at B point at the same time.
Answer the following questions.

Noted that the speed of the bicycle of Taro and the speed Hanako walks are constant, respectively.

(1) Find the ratio of the speed of Taro to the speed of Hanako.

(2) Find the ratio of the distance between P point and B point to the distance between B point and the park.

(3) Find the time when Hanako arrived at the park.

Math problem : III.6 Time of display of crossing for trains

There is a crossing between A station and B station.

Trains bound for A station in every 5 minutes and trains bound for B station in every 6 minutes passes along this crossing.

The crossing is closed anytime a train passes.

While being closed, The display A and B is turned on in fixed time when trains for A station and B station pass respectively.

In 30 minutes from 8:00 to 8:30, the time when display either A or B is turned on was 10 minutes and 37 seconds in all.

Moreover, there were 1 minute and 34 seconds while both of displays were turned on simultaneously in all.

Answer the following questions.

(1) In case the time when A or B is displayed by one train is the same, find the time.

(2) In case the time when B is displayed by one train for B station is 22 seconds longer than the time when A is displayed by one train for A station,

(2)-1 Find the time when A is displayed by one train for A station.

(2)-2 B began to be displayed at 8:00 sharp and display A was turned off at 8:02:02 (2 seconds).

Find the time while the display of both A and B was turned on simultaneously in 100 minutes from 8:30 to 10:10.

Math Problem : JJ.46 Partition of rectangle and three points

There is a rectangle ABCD of which area is 140cm². 

As shown in a figure, the point E is inside the rectangle. 

The area of the triangle ABE is 42cm² and the area of the the triangle BCE is 21cm². 

Answer the following questions.

(1) Find the area of triangle CDE.

(2) Find the area of triangle BDE.

(3) The intersection of the diagonal line AC and the diagonal line BD is set to F and the intersection of the diagonal line AC and the line DE is set to G.

Find the difference of the area of the triangle DFG and the area of the triangle CEG.

GG.27 One number after another is multiplied from 1

Numbers are being multiplied as follows.
Multiplying 1 by 2 first and multiply 3 to the product and further multiply 4 to the product--------.
Thus, one number after another is multiplied sequentially from 1.

Answer the following questions.

(1) When multiplied to a certain number, the multiplied product number contain four 0 continuously located in a line from one digit.
Find the smallest number in what is considered as a certain number.

(2) When multiplied to a certain number, the multiplied product number contain seven 0 continuously located in a line from one digit.
Find the smallest number in what is considered as a certain number.

Moreover, when you multiply from 1 to the number, find the number of 8th digit counted from one digit.

II.14 Piled up two-face coins

There are some two-faced coins and piles up every coin of which heads face upward initially.

Answer the following questions.

(1) In case there are three coins, as the 1st operation, flip one piece of the top of coins.

As the 2nd operation, flip piled two pieces from the top of coins.

As the 3rd operation, flip piled three pieces all. As the 4th operation, flip one piece of the top again.

After this, continue the same cycle of operation of flipping two pieces from the top, three pieces all, and one piece of the top-------.

(1)-1 Find the number of times of operation in case heads of all coin face upward for the first time.

(1)-2 When you perform operation 100 times, find the number of pieces of the coin of which heads face upward.

(2) In case there are two coins, as the 1st operation, flip one upper piece.

As the 2nd operation, flip plied two all.

As the 3rd operation, flip one upper piece again.

After this, continue the same cycle of operation of flipping two all, one upper piece, two all,----------.

When you perform operation 100 times, which face of heads and/or tails of two coins face upwards respectively?

FFF.3 Election of class representative

In a certain class of 40 students, three class representatives will be decided by an election.

All 40 students voted someone of this class by one vote per person.

As a result, Taro was in the 1st place, Jiro was in the 2nd place and Hanako was in the 3rd place.

The number of votes of three persons was different and there is no student who got same number of votes as Hanako in other 37 persons. Answer the following questions.

(1) Find the most number of votes which Taro gained among possible number of votes.

Moreover, also find the smallest number of votes.

(2) How many ways of number of votes which Hanako gained among possible number of votes?

(3) The sum total of the number of votes which three persons, Taro, Jiro, and Hanako, gained was 37 votes.

Moreover, the difference of the number of votes Jiro and Hanako gained was twice the difference of the number of votes Taro and Jiro gained.

In this case, find the number of votes Taro gained.

Find all numbers of votes possibly gained.

EEE.1 Three points moving on a rectangle

As shown in Fig. 1, there is a rectangle ABCD whose length of side AB is 18 cm and length of side AD is 42 cm.

The point P leaves A and moves at the speed of 3 cm/s clockwise rotation on the side of rectangle ABCD as A -> D -> C -> B -> A.

It will stop when it arrives at A again.

The point Q leaves B at the same time as P leaves A.

It moves counterclockwise until P stops at the speed of 2 cm/s on the side of rectangle.

Answer the following questions.


(1) In how many seconds after started is it that P and Q meet?

(2) The point R leaves A at the same time with P and moves counterclockwise on the side of the rectangle at the speed of 2 cm/s.
It moves until P stops.
The figure which connected three point P, Q, and R is considered.

(2)-1 A triangle is not made several times while three points are moving.

Find the time altogether when the triangle is not be made.

The time should be found the time since they begin to move.

For example, When the triangle is not be made all the time from x-second to y-second, you may answer as x~y seconds.

(2)-2 As shown in Fig. 2, the intersection of the line AC and the line BD is set to E.

Find the time altogether when E is on the inside and circumference of the triangle PQR.

In accordance with (2)-1, answer the time altogether.