NADA-2003-Day2

Time : 60 minutes Passing marks : 70%

Answer

Problem 1

I divide integers more than 6 by 6 and add a remainder to a quotient.

I express this calculation in → and continue it until it becomes 5 or less.

<Example 1> 12 / 6 = 2 remainder 0, then 2 + 0 = 2. When begin with 12, it is expressed as 12 → 2.

<Example 2) 50 / 6 = 8 remainder 2, then 8+2=10, 10 / 6 = 1 remainder 4, then 1+4 =5. When begin with 50, it is expressed as 50 → 10 → 5.

Answer the following questions.

(1) When I begin this calculation with 2003, express the result by using →.

(2) Find the number of integers more than 6 to be suitable for A when it is expressed in one arrow as A → B.

(3) Find the smallest integer in integers more than 6 to be suitable for A when it is expressed in two arrows as A → B → C.

In addition, find the biggest integer.

(4) Find the smallest integer in integers more than 6 to be suitable for A when it is expressed in four arrows as A → B → C → D → E.

In addition, find the biggest integer.

Problem 2 (JJJ1)

A rectangle departs from the position in a figure and is moving in the direction of an arrow at 2 cm/s.

(1) Find the time when the rectangle is completely inside of the triangle.

(2) Find the area of the portion of the rectangle out of a triangle 12 seconds after leaving.

(3) Find the time when the area of the portion of the rectangle which is contained in the triangle is 30 cm² or more.

Problem 3

The figure shows a cube whose length of one side is 6cm.

Two points of P,Q moves on the side of the cube with a speed of 2cm per second, 3cm per second respectively.

P,Q leave A at the same time.

P moves in order of A → E → H → G → C → B → A and repeats this cycle.

Q moves in order of A → D → C → G → F → E → A and repeats this cycle.

Answer the following questions.

(1) Find the time when P and Q meet for the first time after leaving A.

(2) Find the time when P and Q meet for the fifth time after leaving A.

Problem 4 (DD18)

There is a flower clock in a certain town and the long hand advances one round in one hour with a fixed speed.

The hour hand stands still first 59 seconds in every minute and advances 1 of 720 round with a fixed speed in last one second.

The long hand and the hour hand have overlapped exactly at 0:00 a.m.

Answer to the following questions.

(1) Find the time when the long hand and the hour hand become right-angled during 0:00 a.m. and 1:00 a.m.

(2) Find the time when the long hand and the hour hand become right-angled during 8:00 a.m. and 9:00 a.m. except exactly at 9:00 a.m.

Problem 5 (JJJ2)

There are some parallelograms whose length of two sides is 2 cm and 1 cm, and one inside angle is 60 degrees.

These are connected at the vertex so that all of the 2 cm sides are parallel as shown in Fig.1 and Fig.2.

(1) A and B were connected as shown in Fig.1.

Find the area ratio of the sum of the area of two triangles painted black and the area of one parallelogram.

(2) C and D were connected as shown in Fig.2.

Find the area ratio of the sum of the area of four triangles painted black and the area of one parallelogram.