Time : 60 minutes
Problem 5
Passing mark : 70%
Answer : The end of the problem
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Problem 1
Problem 2
Problem 3
Problem 4
As 0, 1, 2, 3, 5, 7, 10, 11, 12, 13, 15, 17, 20, ----, the number made only of 0, 1, 2, 3, 5, 7 is arranged in small order.
Answer the following questions.
(1) Which number in this sequence is 2007?
(2) Find the 100th number in the multiple of 5.
0 is assumed as the multiple of 5.
(3) Find the 100th number in the multiple of 3.
0 is assumed as the multiple of 3.
Problem 2
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.
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.
Problem 3
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.
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 [ ].
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 [ ].
Problem 4
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 uphill course and remaining 12 km is flat course.
Through the uphill course, the speed ratio of Taro and Jiro is 6 : 5.
Taro was able to shorten the time difference with Jiro in the this up hill course for 5 minutes.
Find the speed per minute when Jiro runs in the flat course to make a goal ahead of Taro.
The speed ratio in the uphill and flat course of Taro is 9 : 10.
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 uphill course and remaining 12 km is flat course.
Through the uphill course, the speed ratio of Taro and Jiro is 6 : 5.
Taro was able to shorten the time difference with Jiro in the this up hill course for 5 minutes.
Find the speed per minute when Jiro runs in the flat course to make a goal ahead of Taro.
The speed ratio in the uphill and flat course of Taro is 9 : 10.
Problem 5
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.
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) Find PE : EA with reference to Fig. 2.
(2) Find PG : GD with reference to Fig. 3.
(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.
