KAISEI-2009

Time : 60 minutes

Passing mark : 70 %

Answer

Problem 1

The quadrangle ABCD of the figure is a parallelogram. E, P and G are points on the sides AB, CD and DA, respectively and AE = EB and AG=GD and DF : FC = 4 : 1.

Moreover, the point H is an intersection of EF and BG.

The point I is an intersection of EF and CG.

Answer the following questions.

(1) The intersection of the straight line which extended side DA and the straight line which extended FE is set to P.

Moreover, the intersection of the straight line which extended the side BC and the straight line which extended EF is set to Q.

In this case, Find PA : AD and BC : CQ.

(2) Find GH : HB and GI : IC.

(3) Find the ratio of the area of the triangle AHG, the triangle GHI and the triangle GID.

Problem 2

Answer the following questions about the hour hand (short hand), the minute hand (long hand) and the second hand of the clock in 60 minutes from 10:10 to 11:10.

(1) Find the time when an hour hand and the minute hand overlap.

The second only should answer using a fraction.

(2) Considering the case that any two of three hands do not have overlapped either among three hands, an hour hand, the minute hand and the second hand.

In this case, the center of the clock is divided into three angles as shown in a figure.

Find the number of times that two angles become equal among these from the time when it is found at (1) until 11:10 and also find the last time when two angles become equal.

As for the time, the second only should answer using a fraction.

Problem 3

There are three integers A, B and C which are A + B + C = 1000.

B/A is calculated to the first decimal place and it is 7 when rounding off the first decimal place of the quotient.

Moreover, when C is divided by B, the quotient is 2 and remainder is 16.

Answer the following questions.

(1) When B/A is calculated, it can be divided at the first decimal place exactly and the quotient is 6.5.

In this case, find A, B and C, respectively.

(2) In other than (1), find all the numbers considered as a combination of A, B and C.

Problem 4

N sheets of card in which 1, 2, 3, ...., N is written each are clockwise arranged in the small order of the number circularly.

When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.

<Rule 1>

The card in which 1 is written is removed first.

<Rule 2>

When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.

This operation is repeated until only one card remains.

For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains.

(× mark expresses the removed card.)

Answer the following questions.

(1) Answer the number written to the card which remains at the end in case of N = 8.

(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain.

The card in which 2 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 16.

Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64.

(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain.

The card in which 7 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 35.

(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain.

Answer the number written to the card which remains at the end in case of N = 100.

(5) Answer the number written to the card which remains at the end in case of N = 2009.