Math Problem : KKK.12 Square pyramid and homothetic ratio

There is a square pyramid with a square In the bottom and an isosceles triangle in the side as shown in Fig. 1. 

The intersection of AC and BD is set to O. 

When O is connected to the vertex P, PO will become vertical to AC and BD, respectively. 

The point in the middle of PO is set to M. 

The intersection of the extension of CM and the side PA is set to E. 

Moreover, the point which divides the side PB into 3 : 1 is set to F. 

The intersection of the plane EPC and side PD is set to G. 

Three point G, M, and F are located on a straight line. 

Answer the following questions at this time.

The ratio should be by the least integer.

(1) With reference to Fig. 2 and find PE : EA.



(2) With reference to Fig. 3 and find PG : GD.



(3) Find the ratio of the volume of triangular pyramid P-ECG to the volume of square pyramid P-ABCD.

(4) Cut the square pyramid P-ABCD with Plane EFCG.

The volume of the upper solid is set to U and the lower solid is set to V.

Find the ratio of U to V at this time.

Math Problem : III.2 Two machines displaying numbers on the rule

There is the machine A which displays the number of double-digit figures, and when a switch is turned on, a number will be displayed as the following rule for every second.

[Rule]
As for tens digit, it changes every second in order of 1, 2, 3, 4, 5, 6, 7, 8 and after 8 it return to 1 again. It changes in a same cycle.

As for ones digit, it changes every second in order of 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and after 0 it return to 1 again. It changes in a same cycle.

For example, when a switch is turned on, one second after 11 will be displayed, two seconds after 22 will be displayed, three seconds after 33 will be displayed, nine seconds after 20 will be displayed and eleven seconds after 31 will be displayed.

Answer the following questions.

(1) After turning on a switch, in how many seconds is it that 73 will be displayed for the first time ?

(2) If each number of double-digit figures which can be displayed by this machine A is added altogether, how much is it ?

(3) There is the machine B which displays the number of triple-digit figures according to the rule as machine A.

It displays both tens digit and ones digit as same as the machine A.

As for hundreds digit, it changes every second in order of 7, 6, 5, 4, 3, 2, 1 and after 1 it return to 7 again.

It changes in a same cycle.

For example, when a switch is turned on, one second after 711 will be displayed, two seconds after 622 will be displayed and eight seconds after 788 will be displayed.

(a) After turning on a switch, in how many seconds is it that 711 is displayed on the 2nd times?

(b) After turning on a switch, in how many seconds is it that 773 is displayed for the first time?

Math Problem : JJ.52 Triangle in rectangle

As shown in the figure, the triangle AEF is in rectangle ABCD. 

The area of the triangle AEF is 30cm². 

The points E and F are on the side BC and CD, respectively and are BE = 8cm and DF = 4cm.

Find the area of rectangle ABCD at this time.

JJ.44 Moving point on the side of rectangle

There is a rectangle ABCD as shown in the figure.

AB = 6 cm and AD = 12 cm. AM : MB = 2 : 1 and DN : NC = 1 : 2.

The point P moves on the side AD.

The intersection of the extension of PM and the extension of BC is set to Q.

The intersection of the extension of PN and the extension of BC is set to R.

Answer the following questions.

(1) When the triangle PQR turns into an isosceles triangle of PQ = PR, find the length of AP.

(2) Find the length of AP at the time of BQ = CR.

(3) When the area of the triangle PQR is 90 cm², find the area of pentagon PMBCN.

CC.11 Homework of summer vacation

Taro and Jiro respectively decided to make a plan to do homework of the mathematics of the summer vacation.

Taro decided to solve three problems a day from the first day of the summer vacation.

Jiro decided not do its homework for six days of the beginning of the summer vacation and from seventh day to solve [ ] problems a day until he finish the homework.

It was found that several days after Jiro started to solve, the number of problems solved by 
Taro and Jiro are the same. 

Since the day above Taro solved five problems a day until the day he would finish, but it was two days after the day Jiro finished that Taro finished.

The number of days that Taro solved three problems a day is same as the number of days Jiro solved five days a day.

Answer the following questions.

(1) How many days did Taro take to finish the homework?

(2) How many days did Jiro take to finish the homework?

(3) Find the number applied in [ ].

II.17 Machine displays numbers of two digits

There is a machine A which displays the number of 2 digits and when a switch is turned on, numbers will be displayed according to the following rule for every second.
<Rule 1>

Tens digit changes for every second in order of 1, 2, 3, 4, 5, 6, 7, and 8, and after 8 returns to 1 again and changes in the same rule.
<Rule 2>

Ones digit changes for every second in order of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, and after 0 returns to 1 again and changes in the same rule.

For example, when a switch is turned on, 11 will be displayed one second after, 22 will be two seconds after, 33 will be three seconds after and 19 will be nine seconds after, 20 will be ten seconds after and 31 will be 11 seconds after. 

Answer the following questions.

(1) Find the number of seconds when 73 is displayed first after the switch is turning on.

(2) Find the sum total which added all of the number of two digits which can be displayed by this machine A.

(3) There is a machine B which displays the number of three digits like machine A. 

The rule displaying tens digit and ones digit is the same as the machine A. 

As for hundreds digit, it changes for every second in order of 7, 6, 5, 4, 3, 2, and 1 and after 1 returns to 7 again and changes in the same rule. 

For example, when a switch is turned on, 711 will be displayed one second after, 622 will be two seconds after and 788 will be eight seconds after.

(3)-1 Find the time when 711 is displayed 2nd time after the switch was turned on.

(3)-2 Find the time when 773 is displayed first time after the switch was turned on. 

I.17 Square with black and white stones

I made the square form by arranging black stones and white stones in order as shown in the figure of which numbers of both stones were same. 

The square was made when I finished arranging all black stones. 

At this time, 11 white stones remained. 

Find the number of white stones are there at first.

DD.37 Another triathlon race

Taro and Jiro participated in the competition of the triathlon.

A triathlon is a game which carry out three athletic games in order of swimming, a bicycle (49 km), and marathon (20 km), and all athletes compete for speed there.

In a swimming race, the ratio of the speed of Taro and Jiro is 15 : 13 and there is distance difference of 12 m in one minute.
It was 6 minutes after since Taro begins a bicycle race that Jiro made a goal by swimming.
In the bicycle race, the ratio of the speed of Taro and Jiro was 5 : 7.
Since the tire was flat on the way, Jiro took 12 minutes and 15 seconds for the repair.
It was 9 minutes and 45 seconds after Jiro started marathon game when Taro made a goal by the bicycle.
Answer the following questions.

(1) Find the speed per minute at which Taro swims.

(2) Find the distance of the swimming race.

(3) Find the speed per hour of Jiro in a bicycle race.

(4) In the last marathon, 7.2 km from the start is the course of ascent and remaining 12 km is flat course.
Through the course of ascent, the ratio of the speed of Taro and Jiro is 6 : 5.
Taro was able to shorten the time difference with Jiro in the course of this ascent for 5 minutes.
Find the speed per minute when Jiro runs a flat course to make a goal ahead of Taro.
The ratio of the speed in the course of ascent and flat of Taro is 9 : 10.

JJ.29 Ratio of area and length of sides

As shown in the figure, the point P is in the inside of triangle ABC.

The points that the straight line which passes along the vertex A, B, C and the point P crosses the sides BC, CA, and AB is set to D, E, and F, respectively.

The intersection of a straight line passing through the point F and being parallel to BP and AD is set to G.

AP = 3 cm, BP = 5 cm, CP = 2 cm, PD = 1 cm, PF = 2 cm.

Answer the following questions.

Noted that the triangle of which the ratio of three sides is 3 : 4 : 5 is a right triangle.

(1) Find the area ratio of triangle ABC, the triangle ABP, and the triangle PBC by the least integer.

(2) Find AF : PB by the least integer.

(3) Find the length of FG.

(4) Find the area of triangle ABC.