KAISEI-2007

Time : 60 minutes

Passing marks : 70%

Answer

Problem 1

There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.

Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.

The point A is at point P and point B is on the side PS at first.

① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.

② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.

③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.

In this case, find the surrounding length of the portion by which triangle ABC moved.

The answer should round off the 2nd decimal place and should be found to the 1st decimal place.

Problem 2

The number of integer from 0 is written down clockwise in whorl to small order in a grid as shown in a figure as 0, 1, 2, -----, 26, 27, --------.

It is considered eight directions which added slanting direction to the direction of vertical and horizontal.

For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.

Answer the following questions.

(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further.

(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.

(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.

Problem 3

There are three points A, B, and C which move around on the one circumference with a fixed speed, respectively.

A and B move to the same direction and C moves to the opposite direction of A and B.

These three points A, B, and C departed from the same point exactly at 1:00.

A and C met at 1:02 and B and C met for the first time after the start at 1:07.

Moreover, A returned to the original point for the first time at 1:02 30 seconds.

(1) Find the first time for B to return to the original point.

(2) Find the first time for A to catch up with B.

(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.

Problem 4

There is a rectangular prism as shown in a figure.

I is a point on the side FG.

AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.

The intersection of the plane which passes along the three points A, H, and I, and the side BF is set to J.

The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.

Answer the following questions.

(1) Find the length of FJ and FK.

(2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.

< Cautions > Triangle AEH is already drawn.

Draw as not to overflow the frame in which the answer sheet is defined.

(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.