Math Problem : HH.13 Hexagon is divided by two diagonal lines

There is a hexagon ABCDEF as shown in a figure below.

How many ways of method of the division which divides this hexagon into three parts by two diagonal lines is there in all?

Math Problem : HH.12 Sum of each digit is 10

As for the integer of three digit that the sum becomes 10 when add the number of hundreds digit, the number of tens digit and the number of ones digit like 235 and 307, answer the following questions.

(1) In such integers, find the number of integer that hundreds digit is 1.

(2) In such integers, find the number of integer that hundreds digit is 5.

(3) Find the all the number of such an integer.

(4) In such integers, find the number of integer that tens digit is 1.

(5) Find the total sum of such an integer.

Math Problem : JJ.19 Length in a triangle

In a figure, AG is vertical to BC.
The points D and E are the middle points of AG and AC, respectively.
The point F is an intersection of BE and CD.

The length of DE is 2 cm.

The area of triangle ABC is 36 cm². 

The area of the triangle CEF is 2 cm². 

Find the length of AG.

Math Problem : JJ.18 Overlapped right angle triangles

There is a right-angled isosceles triangle ABC.
There is another right triangle DBE width is shorter than ABC by 3 cm and length is longer than ABC by 6 cm.
The figure shows two triangles overlap.
The point F is an intersection of AC and DE.
The ratio of the area of triangle CEF to ADF is 3 : 8.
Find the area of triangle ABC.

Math Problem : JJ.17 Partition of square

The quadrangle ABCD in a figure is a square with 5 cm one-side.
All the length of AE, BF, CG, and DH is 2 cm.
Find the area of a shadow area.

Math Problem : G.13 Change one yen coins to five and ten coins

There are some one-yen coins.
When I change these into five-yen coins as many as possible, the total number of coins is decreased by 60 pieces.
Furthermore, when I changes those coins into ten-coins as many as possible, the total number of coins becomes ten pieces.
Find the number of one-yen coins first.

Math Problem : JJ.16 Combination of five equilateral triangles

As shown in the figure, five pieces equilateral triangle of the same size are put in order without a gap.
Among quarters point of BC, point D is the point closest to the B.
The length of AE is 9 cm.
Find the length of AD.

Math Problem : JJ.15 Triangle in hexagon

3 sets of sides of the hexagon in the figure faced each other are parallel.
As for each of 3 sets, the ratio of the length of a short side and a long side is 1 : 3.
Find the area ratio of the area of a shadow area, and the area of a hexagon.

Math Problem : G.11 Sum and product of three integers

There are three different integers and the product of the three integers is larger than the sum of the three integers by four. 

There are two sets of group of such three integers.

Largest integer is 4 in one group. 

Find the each product of two groups. 

Math Problem : JJ.14 Area ratio of triangles

Both triangle ABC and ADE in a figure are a right-angled isosceles triangle.
(length of BD) : (length of DC) = 1 : 3.
Find the area ratio of triangle DCE and triangle ABC.

Math Problem : JJ.13 Area ratio of parallelogram

There is a parallelogram ABCD as shown in a figure.
The points E and F are points of dividing the side BC into three equally.
The point G is a point of the middle of the side CD.
Find the area ratio of the area of a shadow area and the area of parallelogram ABCD.

Math Problem : JJ.12 Fold a rectangular paper

There is a paper of rectangle as shown in a figure.
The length of vertical side, horizontal side and diagonal line are 15 cm, 20 cm and 25 cm respectively.
This paper is folded so that A and C may overlap.
Find the sum total of a triangular area with which paper has not overlapped in this case.

Math Problem : JJ.11 Disk rolls around another disk

The radius of the disk A is 4 cm and that of the disk B is 3 cm.

As shown in a figure, the arrow is written on the disk B.

The disk A is not moved.

The disk B rolls clockwise without sliding along with the circumference of the disk A until the arrow and the disk B come back to the original position same as in the figure.

Find the number of times that the disk B rotates around the center B.

Math Problem : G.9 Product of two integers is 44448888

A is an integer of 4 figures and each digit is the same number.
As for B, each digit consists of two kinds of numbers for the integer of 4 figures.
The product of A and B is 44448888.
Find A and B.

Math Problem : KKK.22 Assembling development views of two solids

Fig.1 and Fig.2 show development views of solids.

The development view of Fig.1 consists of one rectangle, two equilateral triangles, and two trapezoids.

The development view of Fig.2 consists of four equilateral triangles.

Find the volume ratio of the solid made by assembling Fig.1 and the solid made by assembling Fig.2.

Math Problem : KKK.23 Image solid from development view

Find the volume of the solid which is made by assembling the development view of the figure.

Each triangular face is an equilateral triangle.

Each face of a hexagon is a made figure where one side cuts out two right-angled isosceles triangles whose equal lengths of two sides are 1 cm from the square which is 2 cm one side.

Math problem : CC.8 Admission ticket of a theater

As for the admission ticket of a certain theater, the advance tickets and the day tickets are available at a rate of 7 : 3.
The advance tickets are on sale three weeks before to the previous day and the day tickets are on sale 2 hours before opening of the performance.
The day ticket audience of one day began to stand in a line 3 hours before opening and joined the sequence four persons per one minute.
The day tickets were sold at a sales window of the theater to six persons per one minute.
Answer the following questions.

(1) How many day tickets audience had stood in a line 10 minutes before opening of the performance?

(2) How many minutes did the person take to enter the theater who is the day tickets visitor and was in the 120th from the beginning of the line?
(Find the time this person was in the line.)

(3) Before opening of the performance, ten percent of scheduled number of the advance tickets remained unsold and 80 percent of schedule number of of the day tickets were sold.
Find the total number of audience this day.

Math Problem : AA.2 Six digits integer

When integer of 6 digits 5ABC15 becomes a multiple of 999, find the integer ABC of 3 digits.

Math Problem : J.10 Folding back of square

The quadrangle ABCD in figure is a square.

When the triangle BCE is folded at CE as crease, B overlaps with F.

When you connect E and F, A and C with a line, find the angle of the smaller one among the angles in the corners where EF and AC cross.

Math Problem : KKK.9 Number of dice to be cut or not

Make a large cube of 5cm one side by piling up 125 dices of 1cm one side without a gap.

This large cube is cut by the plane passing through the three vertex of the cube as shown in a figure.

As for the portion of the triangular pyramid below the cutting plane, find the following each number.

(1)The number of the dices which is not cut.

(2)The number of a solid with the volume larger than 0.5 cm3 in the solid made by cutting a dice.

(3)The number of a solid with the volume smaller than 0.5 cm3 in the solid made by cutting a dice.