Time : 60 minutes
Passing mark : 70 %
Answer : End of the problem
Problem 1
(1) 33/14 / ｛2.25 × (17/7 - 11/9) - 62/35｝- (11/6 / 11/2 - 1/2 × 1/3) / 4/3 =
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(2)
Figure shows a rectangle 6 cm in vertical length and 10 cm in width.
Find the area of a shadow area.
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(3)
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.
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(4)
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.
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Problem 2
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?
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Problem 3
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.
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Problem 4
Square paper is folded in order of Fig. 1 to Fig. 5 and is cut.
(1) When P and Q are opened in Fig.5, two figures are called S and T, respectively.
Draw the figures S and T on the original square paper as a solid line.
Moreover, draw all the fold lines by a dotted line.
(2) Find the ratio of the area of the figure S and the area of the figure T.
(3) Cut Q along the line which connects the central point M of CF and the point E.
Into how many portions is the figure T divided at this time?
Moreover, find the ratio of the area of the smallest portion among divided portions and the area of the figure T.
Passing mark : 70 %
Answer : End of the problem
Problem 1
(1) 33/14 / ｛2.25 × (17/7 - 11/9) - 62/35｝- (11/6 / 11/2 - 1/2 × 1/3) / 4/3 =
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(2)
Figure shows a rectangle 6 cm in vertical length and 10 cm in width.
Find the area of a shadow area.
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(3)
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.
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(4)
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.
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Problem 2
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?
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Problem 3
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.
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Problem 4
Square paper is folded in order of Fig. 1 to Fig. 5 and is cut.
(1) When P and Q are opened in Fig.5, two figures are called S and T, respectively.
Draw the figures S and T on the original square paper as a solid line.
Moreover, draw all the fold lines by a dotted line.
(2) Find the ratio of the area of the figure S and the area of the figure T.
(3) Cut Q along the line which connects the central point M of CF and the point E.
Into how many portions is the figure T divided at this time?
Moreover, find the ratio of the area of the smallest portion among divided portions and the area of the figure T.
<Answer>
Problem 1
(1) 33/14 / ｛2.25 × (17/7 - 11/9) - 62/35｝- (11/6 / 11/2 - 1/2 × 1/3) / 4/3 =
Answer
19/8
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(2)
Figure shows a rectangle 6 cm in vertical length and 10 cm in width.
Find the area of a shadow area.
Answer
29 cm2
Solution
3 × 6 / 2 + 4 × 10 / 2 = 9 + 20 = 29 cm2
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(3)
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.
Answer
9524 yen
Solution
10000 ≦ X × 1.05 < 10001
10000 / 1.05 = 9523.8
10001 / 1.05 = 9524.7
Then it is 9524.
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(4)
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.
Answer
1111
Solution
When it is divided by 3, remainder is 1. → 1, 4, 7, 10, 13,-----
When it is divided by 4, remainder is 3 → 3, 7 ------
Least is 7.
7 + 12 × 17 = 211
7 + 12 × 18 = 223
7 + 12 × 166 = 1999
7 + 12 × 167 = 2011
(18 ; 166) / 2 = 92
Therefore 7 + 12 × 72 = 1111.
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Problem 2
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?
Answer
(1) A→B→C→A
A→B→D→A
A→C→B→A
A→C→D→A
A→D→B→A
A→D→C→A
B : 7 ways, C : 7 ways , D : 7 ways
(2) 21 ways
(3) 60 ways
Solution
(1)
As for C and D, same as B above.
(2)
In order for P to be vertex A in 4 seconds, P should be B or C or D in 3 seconds.
Then 7 + 7 + 7 = 21 ways
(3)
In order for P to be vertex A in 5 seconds, P should be B or C or D in 4 seconds.
According to the above, 20 + 20 +20 = 60 ways.
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Problem 3
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.
Answer
(1) 6 ways
(2) 10:53
(3)
Solution
(1)
According to 0<A<B<C<D<E<F, 1≦A, 2≦B, 3≦C, 4≦D.
According to A + B + C + D = 15, starting from A=1, B=2 and make table below.
(2)
Q + S = A + B + C + D + E + F = 63 → X
X - P = E + F = 48
S - 48 = D = 8
According to (1), A = 1, B = 2, C = 4.
B + D = 10, A + C + E + F = 53
(1) 33/14 / ｛2.25 × (17/7 - 11/9) - 62/35｝- (11/6 / 11/2 - 1/2 × 1/3) / 4/3 =
Answer
19/8
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(2)
Figure shows a rectangle 6 cm in vertical length and 10 cm in width.
Find the area of a shadow area.
Answer
29 cm2
Solution
3 × 6 / 2 + 4 × 10 / 2 = 9 + 20 = 29 cm2
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(3)
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.
Answer
9524 yen
Solution
10000 ≦ X × 1.05 < 10001
10000 / 1.05 = 9523.8
10001 / 1.05 = 9524.7
Then it is 9524.
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(4)
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.
Answer
1111
Solution
When it is divided by 3, remainder is 1. → 1, 4, 7, 10, 13,-----
When it is divided by 4, remainder is 3 → 3, 7 ------
Least is 7.
7 + 12 × 17 = 211
7 + 12 × 18 = 223
7 + 12 × 166 = 1999
7 + 12 × 167 = 2011
(18 ; 166) / 2 = 92
Therefore 7 + 12 × 72 = 1111.
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Problem 2
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?
Answer
(1) A→B→C→A
A→B→D→A
A→C→B→A
A→C→D→A
A→D→B→A
A→D→C→A
B : 7 ways, C : 7 ways , D : 7 ways
(2) 21 ways
(3) 60 ways
Solution
(1)
As for C and D, same as B above.
(2)
In order for P to be vertex A in 4 seconds, P should be B or C or D in 3 seconds.
Then 7 + 7 + 7 = 21 ways
(3)
In order for P to be vertex A in 5 seconds, P should be B or C or D in 4 seconds.
According to the above, 20 + 20 +20 = 60 ways.
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Problem 3
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.
Answer
(1) 6 ways
(2) 10:53
(3)
Solution
(1)
According to 0<A<B<C<D<E<F, 1≦A, 2≦B, 3≦C, 4≦D.
According to A + B + C + D = 15, starting from A=1, B=2 and make table below.
(2)
Q + S = A + B + C + D + E + F = 63 → X
X - P = E + F = 48
S - 48 = D = 8
According to (1), A = 1, B = 2, C = 4.
B + D = 10, A + C + E + F = 53
(3)
E = 21 - 1- 4 = 16, F = 42 - 2 - 8 =32
2009 / 60 = 33 hours and 29 minutes
10 : 53 + 33 : 29 = 44 : 22
44 : 22 - 24 : 00 = 20 : 22
20 = 4 + 16 = C + E
22 = 2 +4 + 16 = B + C + E
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Problem 4
Square paper is folded in order of Fig. 1 to Fig. 5 and is cut.
(1) When P and Q are opened in Fig.5, two figures are called S and T, respectively.
Draw the figures S and T on the original square paper as a solid line.
Moreover, draw all the fold lines by a dotted line.
(2) Find the ratio of the area of the figure S and the area of the figure T.
(3) Cut Q along the line which connects the central point M of CF and the point E.
Into how many portions is the figure T divided at this time?
Moreover, find the ratio of the area of the smallest portion among divided portions and the area of the figure T.
Answer
(1)
(2)
3 : 7
(3)
5 pieces
1 : 56
Solution
(2)
Assuming S = 6, T = 1 + 1 + 3 + 3 + 3 + 3 = 14.
6 : 14 = 3 : 7.
(3)
5 pieces as the figure below.
Assuming △DEF = 1, ④ is 1/4.
As T = 14, 1/4 : 14 = 1 : 56.
E = 21 - 1- 4 = 16, F = 42 - 2 - 8 =32
2009 / 60 = 33 hours and 29 minutes
10 : 53 + 33 : 29 = 44 : 22
44 : 22 - 24 : 00 = 20 : 22
20 = 4 + 16 = C + E
22 = 2 +4 + 16 = B + C + E
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Problem 4
Square paper is folded in order of Fig. 1 to Fig. 5 and is cut.
(1) When P and Q are opened in Fig.5, two figures are called S and T, respectively.
Draw the figures S and T on the original square paper as a solid line.
Moreover, draw all the fold lines by a dotted line.
(2) Find the ratio of the area of the figure S and the area of the figure T.
(3) Cut Q along the line which connects the central point M of CF and the point E.
Into how many portions is the figure T divided at this time?
Moreover, find the ratio of the area of the smallest portion among divided portions and the area of the figure T.
Answer
(1)
(2)
3 : 7
(3)
5 pieces
1 : 56
Solution
(2)
Assuming S = 6, T = 1 + 1 + 3 + 3 + 3 + 3 = 14.
6 : 14 = 3 : 7.
(3)
5 pieces as the figure below.
Assuming △DEF = 1, ④ is 1/4.
As T = 14, 1/4 : 14 = 1 : 56.
