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Time : 60 minutes
Passing marks : 70%
Problem 1
Water is contained in the three tanks A, B, and C.
First, after moving 1/3 of the water of A to B, 1/3 of the water remaining in A was moved to C.
Next, after moving 1/3 of the water of B to C, 1/3 of the water remaining in B was moved to A.
As a result, the volume ratio of water remaining in A, B, and C was 2 : 3 : 4.
Find the volume ratio of the water which was contained in the tank of A, B, and C at first by the least integer ratio.
Problem 2
The gas rate of every month of a certain gas company is the sum total of the fixed amount of basic charge and the charge corresponding to the amount of the gas used.
The charge corresponding to the amount used is calculated from the unit price per m3 provided in stages corresponding to the amount used.
The unit price is as follows.
The amount used Up to 10 m3 160 yen per 1 m3
Over 10 m3 up to 30 m3 x yen per 1m3
Over 30 m3 240 yen per 1 m3
In addition, the gas rate is calculated by 1 m3 increments.
For example, when it is used 40 m3, the charge is calculated as (basic charge) + 10 x 160 yen + 20 x x yen + 10 x 240 yen.
At Taro’s house, 28 m3 gas was used in October and the charge was 6,310 yen.
Moreover, 40 m3 gas was used in December and the charge was 9,150 yen.
Answer the following questions.
(1) Find the basic charge of gas and the unit price of x yen of a secondary stage.
(2) As for Taro’s house, the least amount of the gas used in a year is in August and the most is in February.
In February the amount of gas used was exactly 3 times in August.
Moreover, the gas rate in February also became exactly 3 times in August.
Find the amount of gas used and the gas rate in February.
Problem 3
In a volleyball tournament the league match (round-robin matches) of 5 teams, A, B, C, D, and E is held.
There is one court and the schedule is for two days.
The game of this tournament is organized in accordance with the following administration rules.
<Rule 1>
In addition to two teams under game, there are one team that referee a game and two teams that are waiting to play a next game.
<Rule 2>
There are five games held a day.
<Rule 3>
One team does not hold two games continuously among one day. One team may hold the 5th game on Day1 and the first game on Day2.
<Rule 4>
Each team will referee a game one time with the 1st day on the 2nd, respectively.
A part of schedule organized according to this rule is presented as it is shown in the next table.
Fill in all blanks in the table for answer.
Problem 4
Quadrangle PQRSA was made by extending each side of quadrangle ABCD as follows.
① The length of the side AP is 3 times of the length of the side AB.
② The length of the side BQ is twice of the length of the side BC.
③ The length of the side CR is 3 times of the length of side CD.
④ The length of the side DS is twice of the length of the side DA.
Answer the following questions.
(1) Find the area ratio of the area of triangle APS and the area of the triangle ABD.
(2) Find the area ratio of the area of quadrangle PQRS and the area of quadrangle ABCD.
(3) When the multiplying factor is 50 times of ① and ③, and 100 times of ② and ④, find the area ratio of the area of quadrangle PQRS and the area of quadrangle ABCD.
Problem 5
By putting the equilateral triangles with 1cm one side in order without any gap nor overlap, a big equilateral triangle is made.
(1) How many equilateral triangles with 1cm one side are used for the equilateral triangle with 4cm one side?
(2) The equilateral triangle of 1cm, 2cm, 3cm, 4cm ---- one side is named as the 1st, the 2nd, the 3rd, the 4th, ----- equilateral triangle, respectively.
Answer the following questions.
① How many equilateral triangles with 1cm one side are used for the 26th equilateral triangle?
② I made two equilateral triangles, one is a certain numerative number and another is the following.
There are in total 1013 pieces of equilateral triangles with 1cm one side are used for the two equilateral triangles.
Find the numerative number of each equilateral triangle.
Problem 6
The rows of letters can be put in order using a Lottery “Amidakuji”. For example, the row ABCDE of letters is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.
The thick line in the figure is a route showing the way of A.
Answer the following questions.
(1) Connect two Lotteries of Fig. 1 lengthwise and make a Lottery as shown in Fig. 2.
How is the row of letters ABCDE located in a line is replaced with this Lottery?
Answer the new row of letters.
(2) With the Lottery which connected some Lotteries of Fig. 1 lengthwise, when the row of letters ABCDE are rearranged, it became the same row of letters as original ABCDE.
How many Lotteries of Fig. 1 are connected lengthwise in this case?
Answer the least number.
(3) There is a Lottery which replaces the row of letters ABCDEFGHIJ along with the row of letters BCAJFGHIED.
With the Lottery which connected this Lottery lengthwise, when the row of letters ABCDEFGHIJ are rearranged, it became the same row of letters as original ABCDEFGHIJ.
How many Lotteries of this Lottery are connected lengthwise in this case?
Answer the least number.
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