Math New Drill : KIS13-0704-L1

Odd integer A is a multiple of 3. 
When 11573 is divided by A, remainder is 23 and when 6940 is divided by A, the remainder is 10. 
Find all integers applicable to these conditions. 

Math New Drill : KIS13-0705-L1

The sum of the three numbers out of four becomes 180, 194, 206, 215 respectively. 
Find the largest number in four.

Math problem : KKK.16 Rectangular prism and triangular pyramid

There is a rectangular prism as shown in a figure.
I is a point on the side FG. AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.
The intersection of the plane which passes along the three points A, H, and I, and the side BF is sets to J.
The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.



Answer the following questions.

(1) Find the length of FJ and FK.

(2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.

< Cautions > Triangle AEH is already drawn. Draw not to overflow the frame in which the answer sheet is defined.

(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.

Math problem : III.7 Removing cards based on the rule

N sheets of card in which 1, 2, 3, ...., N is written each are clockwise arranged in the small order of the number circularly.

When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.

<Rule 1>
The card in which 1 is written is removed first.

<Rule 2>
When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.

This operation is repeated until only one card remains.

For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains.
(x mark expresses the removed card.)



Answer the following questions.

(1) Answer the number written to the card which remains at the end in case of N = 8.

(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain.

The card in which 2 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 16.

Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64.



(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain.

The card in which 7 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 35.

(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain.

Answer the number written to the card which remains at the end in case of N = 100.

(5) Answer the number written to the card which remains at the end in case of N = 2009.

Math Problem : KKK.13 Shadow of block by streetlight

As shown in a figure, there is a concrete block of a rectangular prism and a streetlight which stands at right angles from point A on the level ground.

When the light of the streetlight is on, express the portion (except for the ground on which the concrete block is put) of the shadow of the concrete block made on the surface of the ground which is seen from the right above by drawing slash in the figure of the answer.

Moreover, find the area of the slash portion.

All the measure of the number in a figure are set to meter.

Math Problem : III.3 Table of integers written clockwise

The number of integer from 0 is written down clockwise in whorl to small order in a grid as shown in a figure as 0, 1, 2, -----, 26, 27, --------.

It is considered eight directions which added slanting direction to the direction of vertical and horizontal.

For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.

Answer the following questions.

(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further.

(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.

(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.

Math Problem : JJ.55 Make right hexagon of parallelograms and trapezoids

As shown in a figure, two kinds of parallelograms A and B are made of using four equilateral triangles of which one side is 1 cm and trapezoid C is made of using three equilateral triangles of which one side is 1 cm in large numbers, respectively.

When these parallelograms and trapezoids are spread out inside of the right hexagon of which one side is 4 cm with, a total of 26 pieces were used.

In this case, find the number of the trapezoid C used.

Math Problem : FF.5 Distribution of seeds of flower

In a certain class of 40 students, it decided to distribute same number of the seeds of flower to applicants.

When distributed to the applicants at first, there is no seed remained.

However, since number of applicants was increased by three later, when redistributed, 18 seeds remained and it was not able to distribute one more to applicants.

At this time, the teacher said, “Although number of seeds remained is not enough for three persons, it is enough for two persons.”

Find the number of applicants at first.

GG.31 1/12 = 1/x + 1/y

Find all the groups of the integer x and y applicable to the formula of 1/12 = 1/x + 1/y.

Noted, suppose x =< y.

GG.28 Sum of three integers is 1000

There are three integers A, B and C which are A + B + C = 1000.
B/A is calculated to the first decimal place and it is 7 when rounding off the first decimal place of the quotient.
Moreover, when C is divided by B, the quotient is 2 and remainder is 16.

Answer the following questions.

(1) When B/A is calculated, it can be divided at the first decimal place exactly and the quotient is 6.5.
In this case, find A, B and C, respectively.

(2) In other than (1), find all the numbers considered as a combination of A, B and C.

JJ.39 Rotate triangle around a rectangle

There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.

Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.

The point A is at point P and point B is on the side PS at first.

① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.

② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.

③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.

In this case, find the surrounding length of the portion by which triangle ABC moved.

The answer should round off the 2nd decimal place and should be found to the 1st decimal place.

EEE.2 Two points moving in a square

There is a square ABCD whose length of one side is 10 cm. E, F, G, and H of the figure are the middle points of square side, respectively.

The point O is the intersection of EF and GH.

From the point A, the point P starts from point A at 1 cm/s and moves on the line of the figure as A - E - O - F - C - G - O - H - A.

Moreover, the point Q leaves the point A at the same time with the point P and turns around the circumference of the square ABCD clockwise with fixed speed quicker than the point P.

In this case, answer the following questions.

(1) Draw the graph of the situation of the point P while it is moving on the circumference of the square ABCD.

(You must not draw the graph while the point P is not moving on the circumference of the square ABCD.)

(2) After leaving, the point Q overlapped point P for the first time at the point of 1 cm from C on CF and this overlapping was in the 2nd round.

Find the speed of the point Q.

Moreover, find the point where P and Q overlaps at the 2nd time after starting.

Write the suitable number or character in underline parts.

The point is located at __X___ cm from ___Y___ on the side ___Z____.

DDD.8 Three points moving on a circumference

There are three points A, B, and C which move around on the one circumference with a fixed speed, respectively. 

A and B move to the same direction and C moves to the opposite direction of A and B.

These three points A, B, and C departed from the same point exactly at 1:00.

A and C met at 1:02 and B and C met for the first time after the start at 1:07.

Moreover, A returned to the original point for the first time at 1:02 30 seconds.

(1) Find the first time for B to return to the original point.

(2) Find the first time for A to catch up with B.

(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.

DDD.7 Hour hand, minute hand and second hand

Answer the following questions about the hour hand (short hand), the minute hand (long hand) and the second hand of the clock in 60 minutes from 10:10 to 11:10.

(1) Find the time when an hour hand and the minute hand overlap. 

The second only should answer using a fraction.

(2) Considering the case that any two of three hands do not have overlapped either among three hands, an hour hand, the minute hand and the second hand.

In this case, the center of the clock is divided into three angles as shown in a figure.

Find the number of times that two angles become equal among these from the time when it is found at (1) until 11:10 and also find the last time when two angles become equal.

As for the time, the second only should answer using a fraction.

JJJ.15 Tape winded around rectangular board

As shown in a figure, there is a translucent red tape in which the end is cut at 45 degrees and there is a line drawn in the center of the tape.

Moreover, as shown in a figure, there is a rectangular transparent board and the point P is set on the side BC so as to BP = 3cm.

As shown in Fig. 1, the end of the central line of the tape is put on the point P and the tape is winded around a board on the surface.

The central line of the tape crosses each side of the rectangle at 45 degrees.

As for the sample in Fig.1, the tape is winded to the point S and cut with scissors along the side.

In this case, the length of the central line of the winded tape becomes PQ + QR + RS.

In addition, the shadow area of Fig. 1 is a portion where the surface and the back of the board are covered by the tape and look deep red.

Fig.2 shows that the tape is winded partially to the middle in case of the rectangle with AB=4cm and AD=12cm.

Then, the tape continues to be winded further and when the central line of the tape overlaps with the point P again, it is stopped winding and the tape is cut with scissors along the side.

Answer the following questions about the case of Fig. 2.

Consider if necessary that in case the length of two sides forming a right angle of a right-angled isosceles triangle is 1 cm, the length of the 3rd side of the triangle is assumed to be 1.414 cm.

Moreover, the thickness of the tape and the board is assumed to be 0 cm.

(1) Write the central line of the tape winded around in the figure.

Noted that write the central line of the tape on the back with a dotted line.



(2) Find the length of the central line of the winded tape.

(3) Find the area of the portion covered on the tape among the surfaces of a board.

(4) Find the area of the portion which is covered by the tape both of the surface and the back of the board and looks deep red.

JJJ.12 Area ratio divided in parallelogram

The quadrangle ABCD of the figure is a parallelogram. E, P and G are points on the sides AB, CD and DA, respectively and AE = EB and AG=GD and DF : FC = 4 : 1.

Moreover, the point H is an intersection of EF and BG.

The point I is an intersection of EF and CG.

Answer the following questions.

(1) The intersection of the straight line which extended side DA and the straight line which extended FE is set to P.

Moreover, the intersection of the straight line which extended the side BC and the straight line which extended EF is set to Q.

In this case, Find PA : AD and BC : CQ.

(2) Find GH : HB and GI : IC.

(3) Find the ratio of the area of the triangle AHG, the triangle GHI and the triangle GID.

KK.6 Development view of cube

The development view of the cube of ①~⑥ is assembled.

The assembled cube is placed on a board so that when the face of K is looked at from A as shown in a figure, it may look right in any direction with left and right, up and down.

This cube is clockwise turned shown as an arrow.

In this case, write o to the development view which comes to look right from A in any direction with left and right, up and down in order of K, P, Q and R, and write × to the development view which comes not to look right.