AZABU-2009

Time : 60 minutes

Passing mark : 70%

Answer

Problem 1

As for two integers, calculation by < , > is defined as the following [Example].

[Example]

< 3, 4 > = 1 / 3×(3+4) =1 / 3×7 = 1/21

< 5, 3 > =1 / 5×(5+3) =1 / 5×8= 1/40

(1) Execute the following calculation in accordance with the rule of a [Example].

① < 8,3 > + < 3,8 >

② < 3,6 > + < 7,2 > + < 6,3 > + < 5,2 >

(2) Find all Integer A which is applicable to the formula,

< A,B > = 1 / 72.

Problem 2

Answer the following questions.

(1) Taro continues walking first 3 km at 4 km/h and afterwards at 3 km/h.

Find the distance he walks in 2 hours.

(2) 2 hours after Taro leaves, Jiro pursues Taro from the same point.

Jiro moves first 4 km at 15 km/h and afterwards at 12 km/h by bicycle.

Find the time when Jiro catches up with Taro after Jiro’s leaving.

Problem 3

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.

Problem 4

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.

Problem 5

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 and the hexagon AMNDEF in the figure.

The answer should be by the least integer.

Problem 6

Answer the following questions.

Express an answer by the ratio of the least integer.

(1) There is a waterway through which it flows downward from upside on as shown in the figure below.

The water through the entrance A is divided into the same volume at the fork.

There are five exits located in four stories of waterways but the volume of the water which comes out of exits is not the same.

Find the ratio of the volume of the water of the least and the most.

(2) Next, as shown in Fig. P, the water through A is divided into three and there is a waterway of the solid shape which the same volume of water comes out of B, C, and D.

Waterway as shown in Fig.Q made by combining four of waterways is called waterway of two stories.

Like this, one waterway after another is connected.

The exits of one story, two stories, three stories and four stories of waterways becomes as it is shown in the Fig.1 ~ Fig.4, respectively.

① As for the volume of the water which comes out of exits of the waterway of three stories, find the ratio of the volume of the water of the least and the most.

② As for the volume of the water which comes out of exits of the waterway of four stories, find the ratio of the volume of the water of the least and the most.