Math Problem : HH.17 Move of the side of regular tetrahedron

There is a solid surrounded with four equilateral triangles as shown in a figure.

The point P is at the vertex A at first and moves to one of other three vertex for every second.

For example, as for ways of movement where the point P is at the vertex A in 2 seconds, there is A→B→A, A→C→A and A→D→A.

(1) In accordance with the example, write ways of movement altogether where the point P is at the vertex A in 3 seconds.

Moreover, how many ways of movement where the point P is at the vertex B, C, and D in 3 seconds, respectively?

(2) How many ways of movement where the point P is at the vertex A in 4 seconds?

(3) How many ways of movement where the point P is at the vertex A in 5 seconds?

Math Problem : HH.14 Four figures containing 5

Find the number of numbers which contain two or more 5 such as 0545 among 4 figures number which are used for the telephone numbers from 0000 to 9999.

Math Problem : KKK.11 Development view of solid with six faces of lozenge

Draw the development view of the solid which are surrounded by six faces of lozenge in which degree of the angle of one pair of angles which face each other is 60 degrees.

Draw within the area in the figure using the dotted line.

Math Problem : C.26 Weight of the salt solution

8% of salt solution was mixed with 5% of salt solution 250 g to become 6.8% of salt solution. 

Find the weight of the salt solution made.　　　

Math Problem : GGG.4 Multiple of every number of each digit

Answer the following questions.

(1) As for two digit integers of 10, 11, 12.13, -----, 99, I make make 90 pieces of numbers by multiplying the number of tens digit and ones digit together, respectively.

Find the sum total of 90 numbers.

(2) As for four digit integers of 1000, 1001, 1002.1003, -----, 2013, I make make 1014 pieces of numbers by multiplying the number of thousands digit, hundreds digit, tens digit and ones digit all together, respectively.

Find the sum total of 1014 numbers.

(3) Among 1014 pieces of numbers made at (2), find the number of numbers of which ones digit is 7.

Math Problem : GGG.3 Multiple of continued integers

Answer the following questions.

(1) When all the integers from 1 to 2007 are multiplied, how many is 0 lined continuously from ones digit?

(2) When all continuous integers from A to B are multiplied, it is expressed as (A,B) that the number other than 0 appears first in the multiplied number checked sequentially from the ones digit.

For example, since 1 × 2 × 3 × 4 × 5 = 120, (1, 5) = 2.

(2)-1 Find (1, 10).

(2)-2 Find (121,130).

Math Problem : JJ.49 Six squares overlap sectors

As shown in a figure, there are six squares whose one side is 4 cm and some circles there.

Find the area of the shadow area.

Pi is assumed to be 3.14.

Math Problem : JJ.48 Partition of parallelogram to find area

As for parallelogram ABCD of 252 cm² in the area as shown in a figure, it is referred as AE :
ED = 1 : 1, BF : FC = 2 : 5 and BG : GE = 11 : 7.

Answer the following questions.

(1) Find the area of a quadrangle ABFE.

(2) Find BF : FI.

(3) Find the area of the triangle EGH.

GG.29 1989 expressed by continuous integers

1989 can be expressed as the sum of two or more continuous integers such as 1989 = 994 + 995 = 662 + 663 + 664.
Answer the following questions about such 1989.

(1) When 1989 is expressed by the sum of continuous integers among four or more pieces and eight or less, which is the number of continuous integer?
Moreover, what is the largest integer in the continuous integer?

(2) When arranging the continuous integer in small order, the integer in the middle was 51.
What is the largest integer at this time?

(3) When the number of the continuous integer is 13 pieces and 26 pieces, what is the largest integer, respectively?

GG.26 Quotient and remainder are same

Even when the integer X which is larger than 49 is divided by 29 or divided by 49, each quotient and remainder became the same integer.

Find the sum of such all the integers X.

JJ.33 Three sectors in trapezoid

The figure shows trapezoid whose side AD and BC are parallel and AB = CD = 5cm, BC = 8cm and angle C = angle B = 60 degrees.

Three sectors were drawn in this trapezoid.

Find the area of the shadow area.

Pi is assumed to be 3.14.

C.22 Consumption tax inclusive price

The tax-inclusive price which is added 5% of the consumption tax to the product cost is considered.

The product cost is considered as an integer and tax-inclusive price is made of omitting figures below a decimal point.

For example, when the product cost is 10 yen, the tax-inclusive price is also 10 yen and when the product cost is 100 yen, the tax-inclusive price is changes to 105 yen.

Find the product cost when the tax-inclusive price is 10000 yen.

II.12 Sequences on a certain condition

Regarding the integers from 212 to 2009, such numbers as when it is divided by 3, remainder is 1 and when it is divided by 4, remainder is 3 are arranged in small order.

Find the number in the middle exactly.

J.26 Partition of rectangle

Figure shows a rectangle 6 cm in vertical length and 10 cm in width. 

Find the area of a shadow area. 

EE.8 Special digital clock

It expresses as A < B that B is larger than A.
There is a relation among the six integers A, B, C, D, E, and F called 0 < A < B < C < D < E < F and A + B + C + D = 15.

(1) How many kinds of combination of (A, B, C, D) is considered in this situation ?

(2) Furthermore, there is additional relation as A + C + E = 21, B + D + F = 42 and D + E + F = 56 to be considered further.

I made a digital clock of 24 hour display of which upper part expresses o’clock with the sum total of ● mark and lower part expresses minutes with the sum total of ● mark.

For example, the table below expressed 15:21.

Then, what time does the next table express?



(3) What does display of the digital clock express 2009 minutes after the time when it was found by (2)?
Write in the following digital clock.

KK.20 Cut line on the development view

As for the cube in a figure, the points P and Q are middle points of the side AB and AD, respectively.

This cube was cut by the plane which passes along the three points P, Q, and G, and the side PQ was drawn as a part of the cut surface.

Draw other sides of the cut surface in a development view.