Time : 50 minutes, Passing mark : 70%
Problem 1
Find X.
(2003 + X ) × 1/4 × 1/5 × 1/6 × 1/8 + 7/10 = 17/6
Problem 2
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.
Problem 3
The ratio of the amount of the money that Taro and Jiro had was 7 : 4.
When Taro gave 150 yen to Jiro, the ratio of the money that Taro and Jiro had became 8 : 5.
Find the amount of the money Taro had first.
Problem 4
As for a price of an apple, a persimmon, and an orange, the price of apple is highest and that of orange is cheapest.
When I purchase five pieces in all including at least one of each fruit, the total amount differs in six kinds of 580 yen, 600 yen, 620 yen, 640 yen, 660 yen, and 700 yen depending on the combination of five pieces.
Find the sum total when I purchase one apple, one persimmon, and one orange.
Moreover, find the price of one persimmon.
Problem 5
I plan to distribute notes in a certain class.
Supposing I distribute four notes to to every boy and six to every girl, seven notes will remain.
Supposing I distribute six to every boy and four to every girl, nine notes fall short.
Supposing I distribute five to every boy and seven to every girl, 33 notes fall short.
Find the number of boys in this class.
Moreover, find the number of notes.
Problem 6
Yesterday I could exchange some amount of Japanese yen I had into 100 US dollars.
Today the Japanese yen of the same amount as yesterday can be exchanged for 1.02 times as many dollars, but I exchanged 12750 yen into 100 US dollars.
Find the amount of money of the Japanese yen which it had yesterday.
Problem 7
When integer of 6 digits 5ABC15 becomes a multiple of 999, find the integer ABC of 3 digits.
Problem 8
When doing some shopping, the consumption tax is 5% of sales price and less than 1 yen is omitted.
For example, if sales price is 219 yen, 10 yen consumption tax will be cost and 229 yen will be paid.
When sales price is 220 yen, 231 yen will be paid.
Therefore, 230 yen does not emerge with the amount of money including the consumption tax.
Find an amount of money nearest to 1000 yen among those which do not emerge with the amount of money of including consumption tax.
Problem 9
I arrange four number of 1,2,3,4 to one line.
For example, I arrange it as "2,4,1,3".
Considering a set of two numbers, there are three cases that big number is on the left side than small number which is "2 and 1", "4 and 1", and "4 and 3".
This case is called that “the number of inverse order of “2,4,1,3” is 3.”
The table shows the number of inverse order and arrangement ways of each number of inverse order about all 24 ways of arrangement of 1,2,3,4.
When I arrange five numbers of 1,2,3,4,5 to one line, I think about the number of inverse order in the same way as a case of the number of four.
Answer the following questions.
(1) What is the number of inverse order when I arrange it as "5,2,4,1,3"?
(2) In reference to an upper table, considering 120 arrangement ways of 1,2,3,4,5, how many ways are there the number of inverse order is 5?
Problem 10
In a figure, a quadrangle ABCD is a parallelogram.
(length of AE) : (length of ED) = 2 : 3.
The area of the triangle ABE is 80 cm2.
Find the area of triangle CDF.
Problem 11
I make a large equilateral triangle by arranging small equilateral triangle 1cm of the length of one side side by side with no gap nor overlapping.
How many small equilateral triangles do I need to make a large equilateral triangle of 20cm of one side?
Problem 12
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.
Problem 13
The shadow area of a figure is a figure which is a combination of three equilateral triangles with 6 cm one side cm and three sectors 6 cm in radius.
When a circle 1 cm in radius takes one round touching the circumference of this figure, find the area of the portion along which this circle passes.
Pi is assumed to be 3.14.
Problem 14
There is a big wall which stands vertically to the ground.
The bottom of the square pillar of Fig.2 is a trapezoid as shown in Fig.1.
The height is 30 cm.
It is placed as the side CD is parallel to the wall and the distance to the wall is 10 cm.
The angle to look up at the sun from the ground is 45 degrees.
The shadow area of Fig.2 expresses the shadow made on the ground of this quadratic prism.
Find the area of the whole shadow made to the wall by this quadratic prism.
Problem 15
Solid O-ABCD in a figure is a right quadrangular pyramid.
The bottom is a square and all the length of OA, OB, OC and OD is equal.
Moreover, E and F divide OB and OD into the ratio of 3 : 1, respectively.
The intersection of the plane which passes along three point A, E, F and the side OC is set to G.
Find OG : GC.
Moreover, in two portions of the right quadrangular pyramid divided by this plane, find the volume ratio of the volume of a portion containing O and the volume of the right quadrangular pyramid.