Math Problem : HH.16 Tournament games of soccer in 7 teams

The competition of soccer was held in 7 teams.

The game was held by the round-robin tournament (game system which every team matches one game with every other 6 teams).

When winning in each game, the team is given two points, when losing, given zero point and when drawing, given one points.

These points are called winning points in the standings.

When all the games were finished, the sum total of winning points in the standings of each team other than A team became as it is shown in the table.

Answer each of following questions at this time.

(1) Find all the numbers of games.

(2) Find the sum total of the winning point in the standings of A team.

(3) Find the number of victory, defeat and draw of A team.

Math Problem : G.20 Bring all poles to one point

The pole stands every 5 m between 2 points A and B which are 50 m away.

I am at A point now thinking that all poles would be brought together to A point.
Whenever I collect one pole, I return to A point.

Find the distance where I will walk by the time I bring all poles together to A point.

Moreover, when I walk 280 m to collect pole in order from the place near A, find the number of the poles which can be collected.

Noted that the pole stands on two points of A and B as well.

Math Problem : DD.25 Pass traffic jam section

A motorcycle and a car departed from A point simultaneously toward 105 km away B point.
There is a the traffic jam section (section which it is crowded and cannot be run fast) on the way.
It took two hours for a car and one hour for a motorcycle to pass through the section.
The motorcycle arrived at B point 45 minutes earlier than the car.
Except the traffic jam section, a car advances at 75 km/h and a motorcycle advances at 60 km/h.
They progress with fixed speed respectively also in the traffic jam section.
Find the distance of the traffic jam section.

Math Problem : G.12 Integer divided by 4 and 5

When a integer is divided by 4, remainder is 2 and when it is divided by 5, remainder is 3.

Find the largest integer of tens digit applicable to this result.

Math Problem : G.10 Fraction denoted by the decimal

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.

Math Problem : LLL.3 Volleyball tournament of five teams

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. 

Math Problem : KKK.21 Vessel, water and bar

Water is contained in the vessel of a rectangular prism to a certain height as shown in Fig.1.
The bar of a rectangular prism whose length and width at the bottom is 10 cm and 6 cm respectively was put into this vessel to 24 cm from the bottom of the vessel for the both bottom might be parallel.
In this case, the water surface was up to the place of the height 9/10 of the vessel.

Furthermore, when the bar is put in until it reach to the bottom of the vessel, 600 cm³ of 

water was overflowed. 

Finally when the bar was taken out, the water surface came to 7/10 of the height of the vessel.



Answer the following questions.

(1) Find the ratio of the base area of the bar and the vessel.

(2) Find the depth of the vessel.

(3) Find the height of the water in the vessel first.

Math Problem : CC.9 Water moves in three tanks

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.

Math Problem : KKK.15 Cut rectangular prism with planes parallel to faces

There is a rectangular prism 3 cm in length, 4 cm in width and 5 cm in height. 

As for faces of this rectangular prism, the face of the rectangle with 3cm and 4cm side is set to face A, the face with 4cm and 5cm is set to face B and the face with 5cm and 3cm is set to 
face C.




Answer the following questions.

(1) Make small rectangular prisms by cutting with planes which are parallel to face A, face B and face C, once, once and twice respectively.

① Find the number of small rectangular prism.

② Find the sum total of the surface area of these small rectangular prisms.

Noted that the surface area of a rectangular prism is the sum total of the area of all the faces of the rectangular prism.

(2) This rectangular prism was cut with planes which are parallel to face A, face B and face C, X times, Y times and Z times respectively.

In this case, there are 90 small rectangular prisms made and the sum total of the surface area of these rectangular prisms was 462 cm². 

Find the number applicable to X, Y, and Z.

Math Problem : III.1 Lottery of "Amidakuji"

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.

Math Problem : HHH.5 Arrangement of squares in order

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 ?

Math Problem : JJ.53 Partition of right hexagon

The side AB of the right hexagon ABCDEF is equally divided into two and the side CD is equally divided into four.

Find the ratio of the area of the quadrangle BCNM of a figure, and Hexagon AMNDEF in the figure.

The answer should be by the least integer.

Math Problem : GGG.1 Put 48 balls into five boxes

There are 48 balls.

These balls are put into five boxes so that it may be applied to the following conditions 1 and 2.

<Condition 1>
Five or more balls are put into every box.

<Condition 2>
As for every two boxes, the common divisor of the number of ball in each box is 1 only.

Find all groups of the each number of balls in five boxes.

Math Problem : GG.9 Product of two integers is 44448888

A is a number with same number of each place in a four-digit integer.

B is a four-digit integer which consists of two kinds of numbers.

In case of A × B = 44448888, find the number of B.

Math Problem : LL.15 Round-robin tournament in five teams

Five teams, A, B, C, D, and E held the round-robin tournament to which every team plays a game against other teams 1 time respectively.

The number of victories of A and B is same and that of C and D is same, too.

The sum total of the number of victories of these four teams is six.

Supposing A won B and C, answer the team which C won.

Suppose that there was no draw.

Math Problem : DD.21 Change speed to catch up

The way from the house of Taro and Jiro to the school joins at the point A, as shown in a figure.

As for the distance to A point, Taro’s house is more 750 m far than Jiro’s house.

Moreover, it is 1000 m from A point up to the school.

Usually, Taro leaves home at 7:50 in the morning, goes at 100 m/m to school and reaches 5 minutes before the starting time.

Usually, Jiro passes along A point 7 minutes later than Taro and reaches exactly at the starting time.

Since it was 5 minutes late to leave home today, Jiro ran at the twice the speed of usual.
Jiro caught up with Taro at the middle point between A point and school. 

Answer the following questions.

(1) Find the speed at which Jiro usually walks.

(2) Find the time delayed Jiro passed through A point than Taro today.

(3) Find the distance between Jiro’s house and A point.

(4) Find the starting time of the school.

Math Problem : DD.20 Replacement of long and hour hand

When Taro went out a house in the evening, he saw the clock and the long hand pointed between 5:35 and 5:40.

Taro came back within 2 hours and saw the clock.

The position of the long hand and the hour hand was reverse exactly comparing to the position when he went out.

Answer the following questions.

(1) Find the sum total of the angle which the long hand and the hour hand turned around during going out.

(2) Find the angle which the hour hand turned around during going out.

(3) Find the time when he returned home.

Math Problem : FF.6 Gather flower petals in a schoolyard

Since the beautiful petals were scattered in the schoolyard, Taro decided to gather them together with some friends to make pressed flowers.

Although it was decided that each person gather 15 petals at a time, there were three persons who were able to gather only 14 sheets and three persons who were able to gather only 12 sheets at a time, respectively.

Then, when collecting the petals gathered and redistributing to everyone so that it might become the same number of sheets, eight sheets of petal remained.

In this case, find the number of persons who gathered petals.

Answer all the probable numbers considered.

Math Problem : DD.17 Two persons meet and janken

Taro and Jiro are in A and B, respectively in the figure at first.

They begin to walk at the same time with the same speed and they play rock-paper-scissors at the place where they met.

The person who won at rock-paper-scissors walks as it is and the person who lost runs with speed 3 times the speed of walking and returns to a starting point and walks with the original speed to partners.

And they play rock-paper-scissors at the place where they meet again.

After repeating as above several times and the points they met are set to C, D, E, and F at order.

Taro won at C and E and Jiro won at D.

Answer the following questions.

(1) Answer the ratio of the distance between EB and between AB by the ratio of the least integer.

(2) The distance of between AF was 10 m.

Find the distance of between AB.

Round off the 2nd decimal place and find to the 1st decimal place.

DD.16 Two persons moving around a pond

Taro walks around a certain pond with fixed speed and Jiro runs with fixed speed to opposite direction.
Taro passed by Jiro again 10 minutes after passing by Jiro.
Just after first passing, Taro also started running increasing the speed by 120 m/m more than the speed at walking.
Then he passed by Jiro 6 minutes afterward again.
Answer the following questions.

(1) Find the surrounding length of the pond.

(2) After passing first, in order to pass again in 4 minutes and 48 seconds, find the speed per minute which Taro increases from the speed at walking first.

(3) If Taro runs with speed twice the speed of walking first after passing first, he will pass again in 7 minutes and 12 seconds.
Find the speed per minute of Jiro.