1979 national middle school mathematics olympiads in the people's republic of china

10
1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA Author(s): JERRY P. BECKER Source: The Mathematics Teacher, Vol. 75, No. 2 (February 1982), pp. 161-169 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27962819 . Accessed: 12/09/2014 17:15 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 74.64.97.109 on Fri, 12 Sep 2014 17:15:53 PM All use subject to JSTOR Terms and Conditions

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Page 1: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OFCHINAAuthor(s): JERRY P. BECKERSource: The Mathematics Teacher, Vol. 75, No. 2 (February 1982), pp. 161-169Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962819 .

Accessed: 12/09/2014 17:15

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 74.64.97.109 on Fri, 12 Sep 2014 17:15:53 PMAll use subject to JSTOR Terms and Conditions

Page 2: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE

PEOPLE'S REPUBLIC OF CHINA

By JERRY P. BECKER Southern Illinois University

Carbondale, IL 62901

Mathematics competitions were orga nized in four major cities of China in

1956?Beijing, Shanghai, Tianjin, and

Wuhan. Thereafter the competitions, whose objective was to identify mathe

matically talented middle school students,

grew and eventually came to involve mid

dle school and university classroom teachers on a fairly large scale. (The Chi nese middle school [5 years] is analogous to the American secondary school [6

years]. Students begin middle school at

[approximately] age 11.) However, the

competitions were terminated in 1964 just

prior to the Great Proletarian Cultural Revolution in China. Subsequently, with

changes taking place in educational policy in the mid-1970s, the competitions were

reinstituted and the Chinese organized and carried out competitions at the munic

ipal, provincial, and national levels in 1978. (For fuller information on the 1978

olympiads, see the Becker article in the

Bibliography.)

The author is indebted to several people for mak

ing this article possible. Ding Shi-sung of Beijing

University provided the 1979 competition problems, and Ding Er-sheng and Zhong Shan-ji of Beijing Normal University provided copies of the 1979 is sues of Shuxue Tongboa, the second of which con

tained information about the results of the 1979

competitions. To all three go the author's apprecia tion for their kind and generous consideration in

providing the information on which this article is based. To Amy Lin go the author's thanks for

assisting in the translation of material to which

reference is made here. Finally, appreciation is ex

pressed to Jing-pang Lee of the Southern Illinois

University Mathematics Department for checking the accuracy of the translations.

The 1978 competitions involved more

than 200 000 Chinese youth from nine

municipalities and provinces and culmi

nated in prizes for fifty high scorers in

Beijing on 19 June 1978. Prizes included

admission to first-choice universities for

the fifty students who distinguished them

selves by performing well at all levels of

the competitions. The competitions were

reinstituted with support and leadership of

mathematicians and mathematics and sci ence organizations in various parts of the

country. Prominent among those encour

aging the contests were Hua Lo-keng (di rector of the Institute of Mathematics in

Beijing), Tuan Hsio-fu (chairman of the

Mathematics Department of Beying

University), and Wang Shou-ren (vice

president of the National Middle School Mathematics Association). Other repre sentatives from Tsinghua University,

Beijing Teacher's University, Beijing In

stitute of Technology, and Beijing Normal

College also participated in setting the

papers for the National Mathematics

Olympiads.

The 1979 Competitions As in the 1978 competitions, the 1979

olympiads began with two rounds of con tests at the municipality and provincial levels in early 1979. High scorers at the

first level advanced to the next level of

competition, and high scorers at the sec

ond level then advanced to two rounds at

the national level. Posing contest prob lems at the national level was the respon

sibility of the National Middle School Mathematics Contest Committee. In set

ting problems the national committee was

guided by the general principle that prob lems used in the competition should be

consistent with the educational policy of

February 1982 161

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Page 3: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

the party to facilitate quality and quantity of education, construction of the Four

Modernizations (Science and Technology,

Agriculture, Industry, and National De

fense) and socialism, and strengthening the process of selection of superior youth. The committee tried to select talented

youth with knowledge and basic training who can apply knowledge, analyze prob lems, and solve problems (National Mid

dle School Mathematics Contest Commit tee 1979).

To fulfill its responsibilities, the com

mittee analyzed contemporary Chinese

teaching practices. It observed that (1) because of the disturbance of the Gang of

Four, student knowledge is not adequate (since removal of the Gang of Four, the

discipline of teaching has improved); (2) in contemporary society there are many dif

ficult and impractical problems?some students and teachers bury themselves in

the middle of these problems, which re

quires an unwarranted amount of time and

energy. The subsequent influence on stu

dent learning of basic knowledge is harm

ful to the physical and mental health and

development of students; (3) for historical reasons national cultural development is not balanced?there needs to be coordina tion at the national level to provide the

proper balance between urban and rural

development (National Middle School Mathematics Contest Committee 1979). The national committee also considered and analyzed problems posed by local and

provincial committees in their respective competitions. Further, the committee

took into consideration the 1979 Prospec tus concerning the National Admissions

Test. (The Prospectus is set by the Minis

try of Education in Beijing and is con

cerned with preparation of students for

the annual, nationwide competitive uni

versity entrance examinations that were

reinstituted in China in 1978. These ex

aminations, different from the Mathemati

cal Olympiad Competitions, are aimed at

selecting the most promising youth throughout China for advanced training.

Mathematics is one of eight subjects cov

ered in the testing program. [For fuller

information, see the Barendsen and Hu

articles in the Bibliography.]) With this background, two sets of contest problems were decided (tables 1,2).

Comments on the Problems

Paper I

The first set of ten problems emphasizes basic knowledge and degree of mastery,

reflecting the mathematics standard of the

senior middle school. These problems are

intended to detect whether students have

been studying diligently. For example, one problem (#7) is based on a theorem

from student textbooks, the theorem on

three perpendiculars. Two problems (#3, #9) are aimed at determining whether

students have the bad habit of taking

things for granted without carefully con

sidering the problem (e.g., in #3, find the

smallest circle to cover a triangle when

the triangle is obtuse; another problem [#9] has two possible cases and two solu

tions). To determine whether students

have learned basic ideas and are able to

synthesize their learning, one problem (#4) is a geometry problem but requires an indirect proof in its solution; and an

other problem (#8) requires the student to

synthesize algebra, geometry, and trigo nometry in formulating a solution (Nation al Middle School Mathematics Contest

Committee 1979).

Paper II

The second set of seven problems empha sizes application of knowledge and analy sis of problems, in addition to finding solutions. The problems not only measure

mastery of mathematical knowledge but

are aimed at helping to determine which

students have high potential in furthering their study of mathematics. Two problems (#1 and #3) both involve two variables.

The aim here is to determine if students can apply knowledge of one-variable func

tions to solve multiple-variable function

problems. The latter occur, of course, in

undergraduate mathematic s programs.

162 Mathematics Teacher

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Page 4: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

TABLE 1

PROBLEMS OF THE 1979 NATIONAL MATHEMATICS COMPETITION

Paper I: 120 minutes, total points = 80

(Solutions to selected problems are given in the Appendix.)

1. Prove that

sin 30 = 4 sin sin + ej sin + 0j . (4 points)

2. The equations of two asymptotes of a hyperbola are + y = 0 and -

y =

0, and the distance between its two vertices is 2. Find the equation of this hyperbola. (4 points)

3. Angle A of a triangle ABC is an obtuse angle. Find a circle with minimum area that can cover triangle ABC completely. (4 points)

4. Prove that in a circle, any two nondiametrical chords cannot bisect each other. (8 points)

5. Solve the system of equations: -

y + =

1, y

- + u = 2, -

u + = 3,

u ?

+ = 4,

? + y

= 5.

6. Solve the equation:

5x2 + -

V5x2 - 1 -2 = 0

(8 points)

(8 points)

7. State and prove the theorem on three perpendiculars in solid geome try. (8 points)

8. In a triangle ABC, the angles A, B, an4 C form an arithmetical

progression, and the inverses of the corresponding sides a, b, and c also form an arithmetical progression. Find the values of A, B, and C.

(12 points)

9. Given a point P(3, 1) and two straight lines:

Lx\ + 2y + 3 = 0 L2: + 2y

- 7 = 0

Find the equation of a circle that passes through and is tangent to L\ and L2. (12 points)

10. In an acute triangle, the sides a, b, and c satisfy the inequality a> b> c. There are three squares with four vertices all lying on the boundary of the triangle. Which one has the largest area? Prove your conclusion.

(12 points)

February 1982 163

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Page 5: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

TABLE 2

PROBLEMS OF THE 1979 NATIONAL MATHEMATICS COMPETITION

Paper II: 180 minutes, total points = 120

(Solutions to selected problems are given in the Appendix.)

1. Let/(jc) = je2 - 6jc + 5. What is the set of points (x, y) in the plane that sat

isfy the inequalities: flx) + fly) ̂ 0 and flx) - fly) ̂ 0? Draw a figure.

(12 points) 2. Is the proposition "A quadrilateral with a pair of equa} opposite sides and

a pair of equal opposite angles is a parallelogram9 * true? If it is true, give a

proof. If it is not, give a counterexample and prove your conclusion.

(12 points)

3. Let 0 < a < ^, 0 < ? < ^. Prove that

cos2a sin2a sin2? c?s2? ~

For what values of a, ? does the equality hold? (1$ points) 4. Connect any two points on the boundary of a unit square with a simple

continuous curve withiri the square. If the curve bisects the area of the

square, prove that its length is greater than or equal to 1. (18 points)

5. Define a function fln) on the positive integers as:

? x nil when is even. ) = <

+ 3 when is odd.

(1) Prove that for any positive integer m, 1 or 3 must occur in the

sequence

a0 =

m, ax =

fla0), . . . , ak+x =

flak), . . .

(2) Give the characterizing property of an integer to ensure the occur rence of 1 and for the occurrence of 3 in the sequence above.

(24 points)

6. As in the figure, let circles and 02 intersect at points A and B. The chord BC of Oy meets circle 02 at E\ th? chord BD of 02 meets circle Oi at F. Prove that L?BA =

CCBA if and only if DF = C?. (18 points)

164 Mathematics Teacher

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Page 6: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

7. In a district a number of students participate in a mathematical contest,

and the mark of each student is an integer. The total sum of their marks is

8250. The marks of the first three are 88, 85, and 80, respectively. The

lowest mark is 30, and the number of students having the same mark does

not exceed three. What is the least number of students whose marks are

greater than or equal to 60 (including the first three)? (18 points)

The problems also serve to help determine

whether students can simplify and solve more difficult problems. In order to deter

mine whether students can handle the

essence of principles or rules, understand

problems, and understand and consider

the counter side of rules, one problem

(#2) requires students to exhibit a coun

terexample to deny the conclusion of the

problem. Three problems (#4, #5, and

#7) do not require a high level of mathe

matical knowledge but do require higher level skill in solving problems?here it can

be determined whether students have logi cal thinking ability and an ability at logical expression. One problem (#6) is aimed at

measuring whether students can analyze a

problem, think through a proof logically, and express that thinking in the form of a

proof. All the problems, it can be seen,

require some level of basic knowledge, but each also has its own approach to a

solution. Moreover, students should be

able to at least begin a solution and there

fore receive some credit; additionally,

only outstanding students should be able to exhibit well-thought-out, complete so

lutions (National Middle School Mathe

matics Contest Committee 1979).

Analysis off the Results of the 1979 Competition

A subcommittee consisting of people from the Institute of Mathematics (Acade

my of Sciences) and Beijing University undertook the task of scoring student pa

pers. Results appear in table 3 (National Middle School Mathematics Contest

Committee 1979). As indicated earlier, total points possi

ble equaled 200 (80 points on Paper I, 120

points on Paper II). A total of 1106 papers were scored, and as can be seen in the

table, 355 students scored above 120. In

order to examine the results more closely, an analysis of papers was done. See tables

4 and 5 (National Middle School Mathe matics Contest Committee 1979).

TABLE 3 Results of the 1979 Competition

Total Points Number of Students on the with Scores

Two Papers in the Interval

180-188 4 169-179 10 150-168 52 140-149 56 130-139 103 120-129 130 < 119 751

TABLE 4 Paper I

Percentage of Percentage of Problem Students Getting Students Getting Number Full Points No Points

1 963 33 2 68.5 3.3

3 39.5 36.8

4 79.6 4.9

5 87.4 8.9 6 49.2 15.9

7 28.0 6.1

8 54.0 0.3

9 57.1 9.9

10 41.2 19.8

Average score = 61.4 (percentiie = 76 percent).

TABLE 5 Paper II

Percentage of Percentage of Problem Students Getting Students Getting Number Full Points No Points

1 ??6 3?3 2 17.4 32.4

3 24.0 51.7

4 6.8 28.2

5 1.8 45.2

6 37.9 34.2 7 18.6 25.4

Average score = 43.3 (percentiie = 36 percent).

February 1982 165

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Page 7: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

Comments on each problem of Paper I

1. Students have good knowledge of

simple trigonometric identities.

2. One-fourth of the students did not

know the equation of a hyperbola with the

y-axis as its real axis. Many students did not consider the whole problem.

3. More than one-half of the students assumed that the circumscribed circle is

the answer or did not prove what they claimed.

4. Most students handled the counter

example proof well.

5. Many computational errors were

made by students?too much careless ness.

6. Most students did not check their

solutions or committed errors in computa tion.

7. More than 70 percent of the students were unable to provide the statement and

proof of the theorem on three perpendicu lars with full details. Some students could not distinguish between the theorem and

its converse.

8. Most students received good credit,

indicating synthesizing ability. 9. A majority of students completed

the problem, but more basic knowledge of

analytic geometry is needed.

10. Half the students were able to de termine the area of squares, but many

were not able to compare the size of two

fractions.

Comments on each problem of Paper II

1. Students were not good at applying knowledge of one variable to problems of two variables. A majority of students

could not determine a plane region repre

senting the solution set of a system of inequalities.

2. One-third of the students misjudged the problem; others judged the problem correctly but could not exhibit a counter

example. This indicates that more empha sis should be placed on mathematical pre cision and completeness.

3. Most students were not good at

simplifying the problem, indicating low

problem-solving ability. 4. Students did not consider all three

possible cases. More emphasis needs to

be placed on completeness in solving

problems. 5. A majority of students could not

understand the two parts or understood the conclusions but could not explain the

underlying theory or rules. There is low

student ability in symbolic thinking and expression.

6. Students showed ability to prove the statement, but many could not orga nize their proofs in the best manner.

7. A majority of students could not

analyze the problem and think deductive

ly.

General Observations

The Chinese were pleased with many of

the results of the 1979 mathematical com

petitions. Some teachers commented that

"these kinds of contest problems can pro mote the combination of quality and quan

tity of teaching and learning. We are hap py we could be involved, and the

problems and results show the direction of

teaching" (National Middle School Math ematics Contest Committee 1979). Other

teachers commented that "this year's na

tional mathematics contest was rooted in

the reality of practice in the middle schools?it is energetic and lively. It can

facilitate or promote the correction of

contemporary fate of working difficult

problems. If we can insist on doing this

several years, the hope to promote the

quality and quantity of teaching and pro ducing the talented student is higher" (National Middle School Mathematics Contest Committee 1979).

Contest results were seen as an indica

tion that the teaching of mathematics has improved. Results from 1979 show a sig nificant number of students better able to

handle geometric proofs than in 1978 (11 percent more students got full credit on

such problems in 1979 over 1978 [38 per cent in 1979 and 27 percent in 1978]), and

166 Mathematics Teacher

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Page 8: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

the number of students receiving no credit

dropped from 58 percent in 1978 to 34

percent in 1979. Good performance was noted on problems requiring technical

knowledge, and students commonly com

pleted such problems with no "messing around." Student ability to apply knowl

edge of algebra, trigonometry, and geome try and then to synthesize it is improving. For example, on problem #8 of Paper I

(the same level as 1978), 598 (or 54 per cent) students received full credit, where as only 3 received no credit at all. Similar

ly, on problem #10, 41 percent of the students received full credit. Many stu dents exhibit an ability to think broadly and give a creative method of solving problems. For example, on problem #2 of

Paper II, there is a degree of difficulty requiring a counterexample in the solu tion. Quite a number of students did very

well giving, for example, tetrahedra in

space, concave quadrilaterals, partitions of a plane figure, and exhibiting all possi ble cases as their solution (National Mid dle School Mathematics Contest Commit tee 1979).

But if there is a source of optimism among the Chinese for the improvement of mathematics teaching, there is also concern about the weaknesses of students on contest problems: (1) More emphasis must be placed on basic knowledge; for

example, as many as 72 percent of the students could not express or prove the theorem on three perpendiculars, a fam ous theorem of three-dimensional geome try that is included in student textbooks. Some students paid little heed to the Pro

spectus and textbooks and solved only difficult problems in preparation for the contests. The result was that such stu dents were able to solve some difficult problems, but they also received no points on problems involving mostly basic knowledge. (2) There is a need for stu dents to be more careful, serious, and

precise in solving problems. Many stu dents could not distinguish between a

theorem and its converse, could not judge the correctness of a solution, or failed to

solve some problems completely (e.g., failed to check for extraneous roots or to see erroneous deductions). Too many stu dents seem to be satisfied with finding a

solution, as opposed to being careful, seri

ous, and precise in completely solving problems. Textbooks and teaching need to be improved in this respect, and stu dents need help to consider the counter

parts of a problem. (3) There is a need for

improvement in logical deductive thinking and expression. Problem #5 of Paper II was only a little symbolic, but the solution was easy to find. However, students did not do well in writing down their solu

tions, even when they recognized the

conclusion?they could not express their solutions succinctly in mathematical lan

guage. In some cases, students wrote down much, but it was confusing. More than 90 percent of the students failed to reach a solution to problem #7 on Paper II. Though basically an arithmetic prob lem, students didn't seem to have an

ability to think logically and come to a full solution. In working to improve symbolic thinking and logical expression, more em

phasis needs to be placed on language knowledge, particularly since mathemat ics is a language using symbols to express and reflect ideas. At the same time, in struction needs to emphasize basic mathe

matical knowledge more, since deductive

thinking involves a process of judging situations and applying concepts (Nation al Middle School Mathematics Contest Committee 1979).

Conclusion

The Chinese are enthusiastic about the National Mathematics Competition and its value in identifyng talented youth. But

whereas they feel their problem posing is good and there is improvement in teach

ing, there still remain weaknesses that need to be overcome. In their words, "There is still some distance from the international standard. We should gradu ally promote the standard" (National Middle School Mathematics Contest Committee 1979). It appears that this is

February 1982 167

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Page 9: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

exactly what the Chinese are striving to do. They are confident that with the coop eration of teachers and students, the qual ity and quantity of instruction in mathe

matics will continue to improve.

BIBLIOGRAPHY Barendsen, Robert D., ed. The 1978 National Col

lege Entrance Examination in The People's Re

public of China. Washington, D.C.: Office of Education, Department of Health, Education, and

Welfare, July 1979. Becker, Jerry P., and Kathy C. Hsi. "Mathematical

Olympiad Competitions in the People's Republic of China." Mathematics Teacher 74 (September 1981):421-33.

Becker, Jerry P., ed. Chinese Mathematical Educa tion-^ Report of a Study Visit to the People's

Republic of China. Salem, Oreg.: Mathematics

Learning Center (695 Summer Street, N.E., Sa

lem, OR 97301), January 1980. DeFrancis, John. "Mathematics Competitions in

China." American Mathematical Monthly 67 (Oc tober 1960):756-62.

Hu, C. T. "China's New Examination System: A Crucial Aspect of Education in the 'New Period'

"

(monograph). New York: Teachers College, Co lumbia University, in press.

National Middle School Mathemaics Contest Com mittee. "Analysis of the 1979 National Middle School Mathematics Contest." Shuxue Tongboa, no. 2 (1979), pp. 4-6, 29-32.

Swetz, Frank. "The Chinese Mathematical Olympi ads: A Case Study." American Mathematical

Monthly 79 (October 1972):899-904.

APPENDIX Solutions to Selected Problems

Paper I

Number 4

Solution: Refer to figure 1.

Fig. 1. Problem 4, Paper I

Given: AC, BD are two chords of circle 0, and they are not diameters. They intersect at P.

Show: AC, BD cannot bisect each other. Proof: Assume is the midpoint of AC and BD. Connect O and P, then OP 1 AC (the line from the midpoint of a chord to the center is perpendicular to that chord) and OP X BD. But a straight line cannot be perpendicular to two intersecting lines. .?.

cannot be the midpoint of AC and BD at the same time.

Number 5

Solution: By (2) + (3) -

(5), we get = 0 (3) + (4)

- (1), we get y = 6

Take = 0, y

= 6 in (1), we get = 7

- 7, y

= 6 in (2), we get u - 3 jc = 0, u = 3 in (4), we get

= ~1

.?. The solution of this equation system is = 0,

y = 6, =

7, ? = 3,

= ~1.

Number 7

Solution: Refer to figure 2.

Statement: If there is a line that is perpendicular to the projection of a slope line in the plane, then it is perpendicular to that slope line.

Given: As in the figure, AC, AB are, respectively, a vertical and slope line of plane N. BC is the projec tion of AB to plane N, DE is in N, and DE 1 BC. Show: DE 1 AB. Proof: If AC 1 plane and DE is in N, then DE 1 AC (if a line is perpendicular to a plane, then it is perpendicular to all the lines in the plane); and if DE ? BC (known), then DE is normal to the plane determined by AC and BC. (If a line is perpendicular to two intersecting lines, it is perpendicular to the

plane determined by those two lines.) If A? is in the plane determined by AC and BC, hence DE 1 AB (by the same reason of DE 1 AC).

Paper II

Number 1

Solution:

Fig. 2. Problem 7, Paper I

168 Mathematics Teacher

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Page 10: 1979 NATIONAL MIDDLE SCHOOL MATHEMATICS OLYMPIADS IN THE PEOPLE'S REPUBLIC OF CHINA

) + fly) = 2 + y2

- 6* - + io =

(je -

3)2 + (y -

3)2 - 8.

A*) - fty) =

2 - y2

- & + 6y =

( -

y)(x + y -

6).

See figure 3.

Fig. 3. Problem 1, Paper II

All points that satisfy ) + fly) < 0 will be inside (jc - 3)2 + (y

- 3)2 = 0 or on the circle./W -fly) > 0

is equivalent to

~- y ? 0

+ y - 6 s: 0

jc - y < 0

jt + y - 6 < 0

The points that satisfy jc - y ̂ 0 are on line jc = y or below it.

The points that satisfy jc + y- 6s0areon line jc +

y - 6 = 0 or above it.

The points that satisfy jc - y ̂ 0 are ?n line jc - y =

0 or above it.

The points that satisfy jc + y - 6 ^ 0 are on line +

y -

6 = 0 or below it.

Hence all points in sector AB, PCD and their boundary (as shaded in the figure) can satisfy

Fig. 4. Problem 2, Paper II

Number 5

Solution: (1) If there is an ak > 3, then *+1 < ak or a*+2 < ak

since if a* is even,

2~ *'

if ak is odd,

a*+2 = < a*

so this sequence will go to 3 or 2 or 1. If one term goes to 2, the next will be 1, so it

must go to 1 or 3.

(2) (i) If m is an even multiple of 3, then we can get the odd multiple of 3 after some steps; if m is an odd triple, the next will be a multiple of 3. Hence we know all terms of the sequence are

triples when m is a triple and this sequence must go to 3.

(ii) If m is not a triple but an even number, then the next term mil must not be a triple; if m is odd, (m + 3) is not a triple either. So we know this sequence must go to 1 when m is not a triple because all of them are not multiples of 3.

flx)+fly)^0

[flx)-fly)^o. Number 2

Solution: It's wrong. Refer to figure 4.

Counterexample: For any regular triangle ABC, take D in the bottom side BC such that BD > DC. At D, make Ll - L? (as in the figure) and take DE = AC. Connect A and E, then quadrilateral ABDE satisfies those two conditions. Since from ?1ADC = ADAE, we know LE = LC = LB and DE = AC = AB (by assumption and some work we have done). Quadri lateral ABDE satisfies those given conditions, but

AE - DC < BD; therefore ABDE is not a parallelo gram.

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