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Time : 60 minutes
Passing mark : 70 %
Problem 1
As shown in a figure, there are three water tank of different sizes. Every tank is a rectangular prism.
In addition, A and B, B and C are connected by a thin pipe with a cock X, Y respectively.
When the cock is opened, water in connected two water tanks moves through the pipe until height of the water surface in each tank is same.
Answer the following questions.
Pay attention that the amount of the water in the pipe shall not be considered.
(1) At first, water is contained up to a height of 100cm from bottom in A and no water in B and C.
When the cock X is opened while cock Y being closed, the height of the water surface of A and B comes to be 40 cm.
Indicate the ratio of the amount of water in the tank of A and B as the integral ratio at this moment.
(2) Next, when the cock X is closed and the cock Y is opened, the height of the water surface of B and C comes to be 25cm.
Indicate the ratio of the amount of water in the tank of B and C as the integral ratio at this moment.
(3) Next, with cock Y being opened, cock X is opened again, the height of the water surface of the water in the three tanks comes to be same.
Find the height of the water surface at this moment.
Problem 2
As shown in a figure, there is a point A on the straight line ① and the straight line ② is drawn from point A.
Line ② is drawn as the angle with line ① to be 10 degrees.
Next, the point B is set on the straight line ② and the straight line ③ is drawn from point B.
Line ③ is drawn as the angle with line ② to be 20 degrees.
In the same way, the straight line ④ is drawn increasing 10 degrees of angles at a time.
Moreover, the point on a straight line and its straight line is set like the point D on line ④.
When this operation was continued several times, the straight line drawn newly overlapped with the straight line ①.
There was no line that is parallel to the straight line ① in the continuous operation.
Answer the following questions.
(1) Find the number of the straight line drawn.
The last straight line which has overlapped with the straight line ① is not counted.
(2) There are some straight lines which are in a vertical position each other.
Answer all numbers of pairs of lines.
(3) There are some straight lines which are in a parallel position each other.
Answer all numbers of pairs of lines.
Problem 3
According to the order as shown in a figure, a figure is made by adding a square whose one side is 1cm one by one.
For example, the figure which added the 23rd square becomes as follows.
Answer the following questions.
(1) Find the number of all the squares in the figure which added the 9th square.
For example, in the case of the figure which added the 6th square, a square will be eight pieces in all.
(2) Find the number of all the squares in the figure which added the 75th square.
(3) What number of square is added when there are 76 pieces of square in all ?
Problem 4
Tournament game was held in four persons, A, B, C, and D.
This game is held in every two persons' groups.
There are three matches of AB group versus CD group, AC group versus BD group, and AD group versus BC group.
Two winning points in case of win, one point of draw and zero point of lose is goven to every person of the group by one match.
When three matches are completed, answer the following questions.
(1) Find the sum total of four persons' winning point.
(2) When the number of a certain person's winning points is six, what are four persons' winning point? Answer all the cases.
Answer should be in the large order of a winning point, for example as [6, 4, 2, 2].
(3) When the number of a certain person's winning points is three, what are four persons' winning point?
Answer all the cases according to (2).
(4) What kinds of cases are there in four persons' winning point?
Answer all the cases other than (2) and (3) according to (2).
Problem 5
When a certain fraction was denoted by the decimal and the 3rd decimal place was rounded off, it was to be 0.32.
Find the smallest denominator among such fractions.
Problem 6
There is a circular clock.
There is a scale showing the “minutes” which divided the circumference into 60 equally in the dial plate.
(1) Choose all the cases where both the hour hand and the long hand have indicated the scale among the following time.
①0:15 a.m. ②2:36 a.m. ③4:44 a.m. ④8:03 a.m. ⑤10:12 a.m.
Furthermore, the 3rd hand (it is not the second hand) that rotates with fixed speed is supposed.
The center of rotation and the direction of this hand are the same as the hand of the clock.
Answer the following questions.
(2) At a certain time in a.m., the 3rd hand pointed a certain scale.
At this time, the hour hand pointed the 12th scale counterclockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale clockwise from the 3rd needle.
Find this time.
(3) Afterward at a certain time within 12 hours of the time of (2), the 3rd hand pointed a certain scale.
At this time the hour hand pointed the 12th scale clockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale counterclockwise from the 3rd hand.
Find this time.
(4) Between the time of (2) and (3), the 3rd hand passed the hour hand 10 times.
Find the time for one time rotation of the 3rd hand.
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