ข้อสอบ EMIC 2012
 

ไต้หวัน Chiuchang นี่เว็บไซต์ระดับชั้นหนึ่งจริง ๆ :happy:

สอบเสร็จ เด็กกลับถึงบ้าน ข้อสอบ เฉลย อัพรวดเร็วฉับไว :great:

Time limit: 90 minutes


Instructions:
 Do not turn to the first page until you are told to do so.
 Write down your name, your contestant number and your team's name on the answer sheet.
 Write down all answers on the answer sheet. Only Arabic NUMERICAL answers are needed.
 Answer all 15 problems. Each problem is worth 10 points and the total is 150 points. For problems involving more than one answer, full credit will be given only if ALL answers are correct, no partial credit will be given. There is no penalty for a wrong answer.
 Diagrams shown may not be drawn to scale.
 No calculator or calculating device is allowed.
 Answer the problems with pencil, blue or black ball pen.
 All papers shall be collected at the end of this test.

1. In how many ways can 20 identical pencils be distributed among three girls so that each gets at least 1 pencil?


2. On a circular highway, one has to pay toll charges at three places. In clockwise order, they are a bridge which costs $1$ to cross, a tunnel which costs $3$ to pass through, and the dam of a reservoir which costs $5$ to go on top. Starting on the highway between the dam and the bridge, a car goes clockwise and pays toll-charges until the total bill amounts to $130.$ How much does it have to pay at the next place if he continues?


3. When a two-digit number is increased by 4, the sum of its digits is equal to half of the sum of the digits of the original number. How many possible values are there for such a two-digit number?


4. In the diagram below, OAB is a circular sector with OA = OB and $\angle AOB = 30^\circ $ . A semicircle passing through A is drawn with centre C on OA, touching OB at some point T. What is the ratio of the area of the semicircle to the area of the circular sector OAB?



5. ABCD is a square with total area 36 $cm^2$. F is the midpoint of AD and E is the midpoint of FD. BE and CF intersect at G. What is the area, in $cm^2$, of triangle EFG?

6. In a village, friendship among girls is mutual. Each girl has either exactly one friend or exactly two friends among themselves. One morning, all girls with two friends wear red hats and the other girls all wear blue hats. It turns out that any two friends wear hats of different colours. In the afternoon, 10 girls change their red hats into blue hats and 12 girls change their blue hats into red hats. Now it turns out that any two friends wear hats of the same colour. How many girls are there in the village? (A girl can only change her hat once.)

7. The diagram below shows a 7 × 7 grid in which the area of each unit cell (one of which is shaded) is 1 $cm^2$. Four congruent squares are drawn on this grid. The vertices of each square are chosen among the 49 dots, and two squares may not have any point in common. What is the maximum area, in $cm^2$, of one of these four squares?

8. The sum of 1006 different positive integers is 1019057. If none of them is greater than 2012, what is the minimum number of these integers which must be odd?

9. The desks in the TAIMC contest room are arranged in a 6 × 6 configuration. Two contestants are neighbours if they occupy adjacent seats along a row, a column or a diagonal. Thus a contestant in a seat at a corner of the room has 3 neighbours, a contestant in a seat on an edge of the room has 5 neighbours, and a contestant in a seat in the interior of the room has 8 neighbours. After the contest, a contestant gets a prize if at most one neighbour has a score greater than or equal to the score of the contestant. What is maximum number of prize-winners?

10. The sum of two positive integers is 7 times their difference. The product of the same two numbers is 36 times their difference. What is the larger one of these two numbers?

11. In a competition, every student from school A and from school B is a gold medalist, a silver medalist or a bronze medalist. The number of gold medalist from each school is the same. The ratio of the percentage of students who are gold medalist from school A to that from school B is 5:6. The ratio of the number of silver medalists from school A to that from school B is 9:2. The percentage of students who are silver medalists from both school is 20%. If 50% of the students from school A are bronze medalists, what percentage of the students from school B are gold medalists?

12. We start with the fraction $\frac{5}{6}$ . In each move, we can either increase the numerator by 6 or increases the denominator by 5, but not both. What is the minimum number of moves to make the value of the fraction equal to $\frac{5}{6}$ again?

13. Five consecutive two-digit numbers are such that 37 is a divisor of the sum of three of them, and 71 is also a divisor of the sum of three of them. What is the largest of these five numbers?

14. ABCD is a square. M is the midpoint of AB and N is the midpoint of BC. P is a point on CD such that CP = 4 cm and PD = 8 cm, Q is a point on DA such that DQ = 3 cm. O is the point of intersection of MP and NQ. Compare the areas of the two triangles in each of the pairs (QOM, QAM), (MON, MBN), (NOP, NCP) and (POQ, PDQ). In $cm^2$, what is the maximum value of these four differences?


15. Right before Carol was born, the age of Eric is equal to the sum of the ages of Alice, Ben and Debra, and the average age of the four was 19. In 2010, the age of Debra was 8 more than the sum of the ages of Ben and Carol, and the average age of the five was 35.2. In 2012, the average age of Ben, Carol, Debra and Eric is 39.5. What is the age of Ben in 2012?

แปะเก็บไว้ก่อน เว็บนั้นเดี๋ยวก็โดนลบ


Team:

1. Each of the nine circles in the diagram below contains a different positive integer. These integers are consecutive and the sum of numbers in all the circles on each of the seven lines is 23. The number in the circle at the top right corner is less than the number in the circle at the bottom right corner. Eight of the numbers have been erased. Restore them.


Team:
2. A clay tablet consists of a table of numbers, part of which is shown in the diagram below on the left. The first column consists of consecutive numbers starting from 0. In the first row, each subsequent number is obtained from the preceding one by adding 1. In the second row, each subsequent number is obtained from the preceding one by adding 2. In the third row, each subsequent number is obtained from the preceding one by adding 3, and so on. The tablet falls down and breaks up into pieces, which are swept away except for the two shown in the diagram below on the right in magnified forms, each with a smudged square. What is the sum of the two numbers on these two squares?

Team
3. In a row of numbers, each is either 2012 or 1. The first number is 2012. There is exactly one 1 between the first 2012 and the second 2012. There are exactly two 1s between the second 2012 and the third 2012. There are exactly three 1s between the third 2012 and the fourth 2012, and so on. What is the sum of the first 2012 numbers in the row?


Team
4. In a test, one-third of the questions were answered incorrectly by Andrea and 7 questions were answered incorrectly by Barbara. One fifth of the questions were answered incorrectly by both of them. What was the maximum number of questions which were answered correctly by both of them?

Team:
5. Five different positive integers are multiplied two at a time, yielding ten products. The smallest product is 28, the largest product is 240 and 128 is also one of the products. What is the sum of these five numbers?

Team
6. The diagram below shows a square MNPQ inside a rectangle ABCD where AB - BC = 7 cm. The sides of the rectangle parallel to the sides of the square. If the total area of ABNM and CDQP is 123 cm2 and the total area of ADQM and BCPN is 312 cm2, what is the area of MNPQ in cm2?

Team
7. Two companies have the same number of employees. The first company hires new employees so that its workforce is 11 times its original size. The second company lays off 11 employees. After the change, the number of employees in the first company is a multiple of the number of employees in the second company. What is the maximum number of employees in each company before the change?


team
8. ABCD is a square. K, L, M and N are points on BC such that BK = KL = LM = MN =NC. E is the point on AD such that AE = BK. In degrees, what is the measure of $ \angle AKE+ \angle ALE+\angle AME+ \angle ANE+ \angle ACE$ ?

Team
9. The numbers 1 and 8 have been put into two squares of a 3×3 table, as shown in the diagram below. The remaining seven squares are to be filled with the numbers 2, 3, 4, 5, 6, 7 and 9, using each exactly once, such that the sum of the numbers is the same in any of the four 2×2 subtables shaded in the diagram below. Find all possible solutions.

Team
10. At the beginning of each month, an adult red ant gives birth to three baby black ants. An adult black ant eats one baby black ant, gives birth to three baby red ants, and then dies (Also, it is known that there are always enough baby black ants to be eaten.) During the month, baby ants become adult ants, and the cycle continues. If there are 9000000 red ants and 1000000 black ants on Christmas day, what was the difference between the number of red ants and the number of black ants on Christmas day a year ago?

เด็กๆของเราได้เหรียญน้อย คงเพราะอ่อนเรขา

ไม่เหมือนผู้ใหญ่ ที่ชอบเลขา :haha:

เอาเรขามาเฉลยก่อนก็แล้่วกัน

Attachment 9842

ให้รัศมีครึ่งวงกลมเท่ากับ r (= CA = CT)

พื้นที่ครึ่งวงกลม = $\frac{1}{2} \pi r^2$

สามเหลี่ยม OCT โดยตรีโกณฯ จะได้ OC = 2r

พื้นที่ sector OAB = $\frac{30}{360} \pi (3r)^2$

$\frac{area \ of \ semicircle}{area \ of \ sector \ OAB} = \frac{\frac{1}{2} \pi r^2}{\frac{30}{360} \pi (3r)^2} = \frac{2}{3}$

Attachment 9843

ลาก PQ ขนาน AD และผ่านจุด G

สามเหลี่ยม BGC คล้ายสามเหลี่ยม EFG (มมม.) จะได้

FE : BC = 1 : 4 = FG : GC = DQ : QC

พื้นที่สี่เหลี่ยม ADQP : พื้นที่สี่เหลี่ยม ABCD = 1 : 5

พื้นที่สี่เหลี่ยม ADQP = $\frac{1}{5} \times 36 = 7.2 \ $ตารางเซนติเมตร

$4x = \frac{1}{2} \times 7.2 = 3.6 \ $ตารางเซนติเมตร

$x = 0.9 \ $ตารางเซนติเมตร

Attachment 9846

พื้นที่สามเหลี่ยมที่คำนวนได้ดังรูป

พื้นที่สามเหลี่ยมที่ไม่ทราบค่าเท่ากับ W, X, Y และ Z ตามลำดับตามรูป

พื้นที่สี่เหลี่ยมคางหมู AMPD = $\frac{1}{2} (6+8) \times 12 = 84 \ \to \ w+z = 45$

พื้นที่สี่เหลี่ยมคางหมู CDQM = $\frac{1}{2} (3+6) \times 12 = 54 \ \to \ z+y = 30$

พื้นที่สี่เหลี่ยมคางหมู BCPM = $\frac{1}{2} (6+4) \times 12 = 60 \ \to \ y+x = 30$

พื้นที่สี่เหลี่ยมคางหมู ABNQ = $\frac{1}{2} (9+6) \times 12 = 90 \ \to \ w+x = 45$

จะได้ว่า $ \ X = Z \ \to \ QM // PN \ \to \ $ สามเหลี่ยม $ \ OPN \ $ คล้ายสามเหลี่ยม $ OMQ \ \to QO : ON = 3 :2$

$Z = 18, \ Y = 12, \ X =18, \ W =27$


(QOM, QAM) = (27, 27)
(MON, MBN) = (18, 18)
(NOP, NCP) = (12, 12)
(POQ, PDQ) = (18, 12)

The maximum value of these four differences is 6

Attachment 9847

มุมแย้ง + เส้นขนาน จะได้มุมที่เท่ากันดังรูป

$ \angle AKE+ \angle ALE+\angle AME+ \angle ANE+ \angle ACE = \angle BAC = \angle CAD = 45^\circ $

Attachment 9848

เนื่องจาก PQMN เป็นสี่เหลี่ยมจัตุรัส จะเลื่อนไปตรงไหน พื้นที่แรเงาก็ยังมีพื้นที่เท่าเดิม

เลื่อน PQMN มาอยู่กึ่งกลางของสี่เหลี่ยม ABCD เกิดรูปสมมาตร สี่เหลี่ยมคางหมู 4 รูป เขียว 2 และเหลือง 2

พื้นที่สี่เหลี่ยมคางหมูเขียว 2 รูป 312 = $ 2 \times \frac{1}{2} \left( \frac{x+7-y}{2}\right) \left(x+y \right)$

$x^2-y^2+7x+7y = 624$

พื้นที่สี่เหลี่ยมคางหมูเหลือง 2 รูป 123 = $ 2 \times \frac{1}{2} \left( x+7+y \right) \left( \frac{x-y}{2} \right)$

$x^2-y^2+7x-7y = 246$

จากสองสมการข้างต้น จะได้ $y = 27$

พื้นที่สี่เหลี่ยมจัตุรัส $PQMN = y^2 = 27^2 = 729 \ $ตารางเซนติเมตร

link เฉลย
บุคคล http://www.taimc2012.org/file/Primar...vidual_Sol.pdf
ทีม http://www.taimc2012.org/file/Primary_Team_Sol.pdf

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ขอบคุณครับ คุณ gon
ข้อสอบปี 2012 หลายข้อแปลไม่ถูกเลยครับ อย่างประเภทบุคคล ข้อ 6 ข้อ 9