Math Problem : G.14 Five rectangles of same area

It is considered such rectangles (including squares) as the length of each side is integers (measure is cm).
There are five kinds of rectangles of which the area is same.
Find the smallest area among such areas.
Noted that two rectangles which can overlap exactly should be considered as the same kind.

Math problem : KKK.19 Cut piling up eight cubes

As shown in a figure, eight cubes of 1 cm³ in volume are accumulated. 

When this solid is cut by the plane which passes along the side AB and CD, find the volume of the solid of the smaller one. 

Math problem : CC.17 Donation by three persons

The ratio of the money which three persons, Taro, Jiro, and Hanako, have was 5 : 3 : 1.

Since Taro passed 800 yen to Hanako and Jiro also passed some amount of money to Hanako, the ratio of three persons' money was 7 : 4 : 4.

Since three persons donated the same amount of money after that, the ratio of the money they have turned to 5 : 2 : 2.

Find the total amount of money three persons donated.

Math Problem : III.4 Triangle of integers written counterclockwise

As shown in a figure, an integer is arranged sequentially from 1.

The case where 2007 steps are put in order is considered.

    


(1) Find the largest number.

(2) In which steps from the top and which number from the left is the largest number ?

Math Problem : LL.16 Rabbits in the hut

In a certain zoo, rabbits are kept in the hut which looks like the form as shown in Fig. 1 when it is seen from right above. 

In the inside of the hut, there are transparent boards at the position of the dotted line in the figure and it is divided into eight rooms. 

The shadow portion in the middle is not a room. 

There are four windows in the hut and we can see rabbits by looking through the windows. 

For example, when rabbits are kept as shown in Fig. 2, four rabbits are visible from any window and there are nine in total.

(1) There may be a room in which no rabbit is. 

Rabbits should be kept to be seen nine from any window. 

In this case, how many rabbits are there at least and how many rabbits at most?

(2) There is at least one rabbit kept in every room and 17 rabbits in total are kept in the hut. 

In order for 7 rabbits to be seen from any window, how many rabbits should be kept in each room? 

Answer one pattern in the figure.
For example, when putting in a rabbit as shown in Fig. 2, it should be answered as shown in Fig. 3. 

GG.32 Number of sheets of origami

We are going to make some origami cranes by some sheets of origami (paper folding).
In order to fold all cranes, the time does not change even if it is done by five persons group or six persons group.

However if it is done by seven persons party, the time will be shorter.
How many sheets of origami is there in all?

Answer all the possible number of sheets considered.
Noted that the time for one crane to be folded does not change by people.

GG.30 Skip every 2 piece on the stones around a pond

Stones are arranged around the pond.
I walk around the pond clockwise by skipping every two pieces on the stones.

I start from a certain stone and when I stop on the stone at 2 round exactly, the number of stones stepped on while I was walking will become 18 in the stones around the pond.

Find the number of the stones arranged around the pond.

Answer all the probable numbers considered.

When exceeding the starting stone by the last one step, I shall stop with the starting stone without carrying out skipping every two pieces.

I.18 What day of the week six years later

February 2, 2009 is Monday.
Find the day of the week of February 28, 2015.
Noted that there is 29th in February in the year when the number of the year is divided by 4.

FFF.4 Put water in four vessels

There are the vessels A, B, C, D, and E.

The capacity of A is 200 mL and B is 40 mL.

Answer the following questions.

When you put water into a vessel, suppose that water is full in vessel.

(1) When it was repeated that water in C moving into empty A, A filled in the middle of the 4th time moving.

All water in A is pulled out and the water which remains in C is moved to A.

And when it was repeated that water in B moving into A, A filled exactly at the 4th time of moving.

Find the capacity of C.

(2) When it was repeated that water in D moving into empty A, A filled in the middle of the 5th time.

All water of A is pulled out and the water which remains in D is moved to A.

And when it was repeated that water in E moving into A, A filled in the middle of the 3rd time.

Furthermore, all water of A was pulled out and after moving the water remaining in E to A, when it was repeated water in D moving into A, A filled exactly at the 3rd time.

Moreover, after moving water in D into empty A only once, it was repeated that water in E moving into A, A filled exactly at the 2nd time.

Find the capacity of D and E, respectively.

A.1 Number added to be over 1000 of the sum

A number is added in an order from 1 as 1 + 2 + 3 + ------. 

When a certain number was added, the sum exceeded 1000 for the first time. 

Find both of the number and the sum. 

KK.16 Shortest line on solid

There is a solid OABC surrounded with the four sheets of the equilateral triangle whose length of one side is 10 cm.

The point of the middle of the side AC is set to M and the points P and Q are on the sides OB and OC, respectively.

As shown in a figure, the straight line AP, PQ, and QM are drawn.

Find the ratio of AP : PQ : QM in case that the sum of the length of the straight line AP, PQ, and QM becomes the minimum.

KK.8 Triangular pyramid from rectangular paper

Some rectangular pasteboard is cut off along a diagonal line, as shown in Fig. 1. 

By using this I would like to make the glass which has the form of the triangular pyramid as shown in Fig. 2.

However, it may be unable to make or able to make depending on the length of the side AB.

Answer the conditions of the length of the side AB in the case of being able to make.