FRANK SWETZ AND YING-KING YU

M A T H E M A T I C A L O L Y M P I A D S IN T H E P E O P L E ' S

R E P U B L I C OF C H I N A

1. THE R E I N S T I T U T I O N OF EXAMINATIONS

China, the land where for thousands of years, intellectual excellence was sought

out and rewarded by a system of state sponsored examinations, is once again reverting to a similar practice for encouraging its young scholars. Academic and

scholastic examinations were degraded and finally abolished in the wake of

anti-intellectualism spawned by the Great Proletarian Cultural Revolution. Since

the death of Mao Tse-tung, the excesses of the Cultural Revolution have been

officially denounced and its leaders, the notorious 'Gang of Four' [ 1 ], publicly

repudiated. In attempting to rectify the damage done to their country's national

development by the reforms of the Cultural Revolution, China's leaders have

proclaimed a national campaign to promote the 'four modernizations': modern-

ization of agriculture, industry, national defence and science and technology

and thereby advance their nation along the road of industrialization and

technological self-sufficiency. Basic to the success of this campaign is an acknowledged need to improve the climate of scientific investigation and edu-

cation, particularly mathematics education. In recent months, many reforms

intended for this purpose have taken place and while many of these are fairly

dramatic in the light of recent Chinese educational history, none have had so

great an impact on China's youth as the resumption of systems of selective

examinations. University entrance examinations were reinstituted in December of 1977 [2]. The examination process is highly competitive and the exami-

nations, themselves, quite rigorous. All prospective university candidates must

sit for examinations in mathematics. In order to promote an interest in the science, the Outline National Plan for the Development of Science and Tech-

nology formulated at the October, 1977 meeting of the National Planning Conference on Natural Science held in Peking, urged the establishment of a national system of contests devoted to science studies - olympiads. The first such contest, of this new system has been held. It was a mathematical olympiad.

2. MATHEMATICAL OLYMPIADS

Mathematical Olympiads were first conducted in the People's Republic of

China in 1956 [3]. These early contests were inspired by Soviet experiences [4] and were regional in character, initially being held in Peking, Shanghai,

Educational Studies in Mathematics 10 (1979) 435-442. 0013-2954/79/0104-0435500.80 Copyright �9 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

436 FRANK SWETZ AND YING-KING YU

Tientsin and Wuhan. The guiding force in organizing and conducting the competitions was Hua Lo-keng, Director of the Institute of Mathematics of the Chinese Academy of Science and a mathematician of international renown [5]. The results of this first Chinese mathematical olympiad were

well documented and published as a guide to promote the spread of such competitions throughout China [6]. In the following years, the conducting of mathematical olympiads in the PRC was sporadic and the practice did not

gain the wide acceptance hoped for. Available information indicates that the last competition of this series was probably given in Peking in 1964. With the advent of the Cultural Revolution these 'elitist' examinations became anathema.

Implementation of the National Plan's recommendation for a new system of olympiads took place in May, 1978. Under the joint sponsorship of the

Ministry of Education and the Scientific and Technical Association of China, mathematical competitions were held in Peking, Shanghai, Tientsin and the provinces of Anhwei, Kwangtung, Liaoning, Szechwan and Shensi. Preliminary screening examinations were compiled by local committees comprised of teachers, scientists and university professors. Contest questions were confined to the mathematical content of the senior middle (upper secondary), school curriculum. Required exercises were supposed to probe the participants' basic knowledge as well as testing their problem solving ability. Each examination was comprised of two rounds. The questions from the Peking examinations are as follows [7] :

1.

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3.

4.

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The 19 78 Peking High School Mathematics Competition

The First Examination

Simplify x/x 2 -- 2x + 1 + x/x 2 -- 6x + 9 where ~ < x < ~.

(6 points)

Simplify log26.1 g ~ + 1 g ~s. (10 points)

Factorise the following:

4(x 2 + 3x + 1) 2 -- (x 2 + x -- 4) 2 -- (x 2 + 5x + 6) 2. (12 points)

Given that the perimeter of a right triangle is 2 + @ and the median to the hypotenuse has length l, find the area of this triangle.

(12 points)

Let D be the mid-point of BC in A ABC. A line passing through D meets AC at E and the extension of AB at F. Prove that AE:EC =

A F :BF. (15 points)

C H I N E S E M A T H E M A T I C A L O L Y M P I A D S 437

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7.

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4.

Prove that sin 50~ + x /3 tg 10 ~ = 1. (15 points)

Let P be a point moving along the circle x 2 + y2 = 4, and Q the point with coordinates (4, 0). Find an equation for the locus of the

mid-point of PQ. (15 points)

An isosceles trapezoid is touched by a circle at its upper base and two sides. It is known that the length of the lower base is 6, the

height of the trapezoid is 5 and the radius of the circle is 1. Find

the value of the cosine of the angle formed by one side of the

trapezoid and its lower base. (15 points)

The Second Examination

A light beam is emitted from a point A(-- 3, 5) to the straight

line 1 : 3x -- 4y + 4 = 0 and is then reflected to a point B(2, 15).

Find the total length of the light beam from A to B. (15 points)

Let a and b be any two given integers. Prove that both the equations

x 2 + 1 0 a x + 5 b + 3 = 0 and x 2 + l O a x + 5 b - 3 = 0 have no

integer solutions.

From the given arithmetic sequence

3 , 5 , 7 , 9 , 11, 13, 15 . . . .

we want to construct another sequence

3 , 7 , 1 5 . . . . .

(15 points)

(1)

(2)

The rule to form sequence (2) is the following: The first term ba,

is 3; the second term b2 is the third term of (1), that is, 7; the third term b3 is the seventh term o f ( l ) , that is, 1 5 ; . . . ; the nth term bn

is the b~_ 1 th term of (I) . Write down the fourth term and the fifth term of (2). Find an expression for the general term and prove its

correctness. (20 points)

The picture overleaf shows the map of a factory district. There is a main road (thick line) passing through this district. Seven factories

are scattered on both sides of the road and are connected to the

main road by small roads (thin lines). Now we want to build a long distance bus stop somewhere on the big road such that the sum of

the distances between the bus stop and the factories (along big and small roads) is a minimum. (a) Where should the bus stop be built?

(b) Prove your conclusion.

438 FRANK SWETZ AND YING-KING YU

A 1

A 3

A 4 ~.~ ~ A 5

P

Fig. 1.

.

(c) If an additional factory is built at the site P and a small road

is also built along the dotted line, where should the bus stop

be built now? (20 points)

Let QOP be a right angle. Find a point A on OP, a point B on OQ and a point C inside the right angle such that BC+ CA is a

constant L and the area of the quadrilateral ACBO is maximum.

(30 points)

(Hint: Start by considering the areas of BOA and BAC.)

Of the 200 000 contestants who entered the competitions nationwide, 350 advanced to the finals held on May 21. The f'mal, a unified examination also

consisted of two qualifying rounds and was held at regional centers. Peking saw

fifty of its students enter the final round of this competition. Several hundred

of that city's students had been eliminated by the previous three examinations they had taken. The questions of the Chinese national mathematical olympiad are given below [8] :

CH IN E SE M A T H E M A T I C A L O L Y M P I A D S 439

The 1978 Selected Provinces and Cities High School

Mathematics Competition

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3.

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The First Examination

1 Giveny = logl/xA x- - -~ ' for what values o f x is:

0) y > 0 ; 0i) y<0?

Given tg x = 2V~ (180 ~ < x < 270~ find cos 2x and cos x/2.

An ellipse is centered at the origin with its loci on the x-axis. The

lines joining one focus and the two end points of the minor axis

are perpendicular to each other. The distance between the focus

and the closer end point of the major axis is ~ - - V ~ . Find an

equation for this ellipse.

Given the equation 2x 2 -- 9x + 8 = 0, find a quadratic equation

such that one of its roots is the reciprocal of the sum of the two

roots of the given equation while the other root is the square of

their difference.

Four spheres with radii all equal to 1 are piled up on a table in

two levels such that three are in the lower level and one in the

upper level and each is in contact with the others. Find the height

of the highest point of the sphere in the upper level with respect

to the table.

Let M be the mid-point o f a line segment AB. Let CD be another

segment meeting A B at C. The mid-points of CD and BD are N

and P respectively. Let Q be the mid-point of MN. Prove that the

straight line PQ bisects the segment AC.

Prove that n(n -- 1) k-1 can be expressed as the sum o f n successive

even integers if n and k are integers and n > 1 and k > 2.

Given an acute triangle with its vertices on the circumference of a

unit circle, prove that the sum of the cosines o f the three angles

is less than one half of the perimeter o f the triangle.

Given the straight line 11 : y = 4x and the point P(6, 4), find a point

Q on the line ll such that the area of the triangle formed from l~, the x-axis and the line PQ in the first quadrant

is a minimum.

440

10.

FRANK SWETZ AND YING-KING YU

Find the integer solution to the system of equations

x + y + z = 0

x 3 + y 3 + z 3 = _ 1 8 .

The Second Examination

1. The extensions of a pair of opposite sides of a quadrilateral meet

at one point. Similarly, the other pair of opposite sides meet at another point. The line segment joining these two points is paraUel to one of the diagonals. Prove that this line segment is bisected by the extension of the remaining diagonal.

2. (1) Factorize: x12+x9+x6+x3+ 1.

(2) Prove that for any angle 0,

5 + 8 cos 0 + 4 cos 20 + cos 30/>0.

3. Let R be the region bounded by the triangle with vertices A(4, 1), B(-- 1, -- 6) and C(-- 3, 2) (including the interior and the boundary of the triangle). Find the minimum and maximum values of4x - 3y as (x, y) varies within R (must provide proof).

4. Let ABCD be a given quadrilateral and E, F, G and H be the mid- points of sides AB, BC, CD and DA respectively. Prove:

Area of ABCD ~EG. HF ~ �89 (AB + CD) x �89 (AD + BC)

5. Ten persons, each with one bucket, want to fetch water from a tap.

Suppose it takes Ti minutes to Fill the ith person's bucket and the Ti's are distinct values.

(i) If there is only one tap available, how. should the people take

turns so that the total time required in waiting and filling the buckets is minimum? What is this total time? (Must include proof.)

(ii) If there are two taps available, how should the turns be arranged? What is the minimum time required in this case? (Must include proof.)

6. Given a square with a side of length l, find an inscribed equilateral triangle with maximum area and one with minimum area. Also f'md the values of these two areas. (Must include proof.)

The Olympiad Contest Committee that organised and conducted this final competition was once again headed by Hua Lo-keng [9]. Fifty seven winners

CHINESE MATHEMATICAL OLYMPIADS 441

emerged from the competitions, of these: five students received first place

honors; twenty, second place and thirty two, third place. The best overall

performance was given by Li Chun, a 17 year old student from Shanghai, while second place was taken by Yen Yung, a 15 year old from Peking, third place

was also taken by a Peking student.

In early June, all winners of the competition were feted at a ceremony attended by educators and government officials and held in the Music Hall of

Chungshan Park in Peking. Vice Premier, Fang Yi awarded prizes to the student

scholars. Schools of the five highest scoring contestants were awarded special

silk banners in recognition of their excellence in mathematics teaching. A

number of top performers were exempted from further university entrance

examinations thus winning automatic entry to the universities of their choice.

All winners were acknowledged as national heroes and worthy models of emulation for Chinese youth.

3. CONCLUSIONS

This new series of Chinese olympiads is admittedly experimental. Chinese educators are studying the results of these contests. Building upon this acquired

experience, they hope to improve and expand such competitions in the future. New standards of mathematical excellence are now being established in the PRC and future mathematical olympiads will accordingly be more demanding.

It appears evident that in the near future, a team from the People's Republic

of China will participate in the International Mathematical Olympiad [10].

Pennsylvania State University

BIBLIOGRAPHY

[1] The group is comprised of the leader, Chiang Ching, (Madame Mao), Wang Hung- wen, Chang Chun-chiao and Yao Wen-yuan.

[2] See 'University Entrance Examinations, 1977', China Reconstructs (April, 1978), pp. 9-13; 'College Entrance Exams', Peking Review (January 13, 1978), pp. 30.

[3] For a discussion of this early series of Chinese olympiads see Frank Swetz, Math- ematics Education in China: Its Growth and Development (Cambridge, Ma: MIT Press, 1974), pp. 246--257.

[4] Tuan Hsueh-fu, 'Learn from Russia to Have Mathematical Competitions', Shuxue Tongbao (January, 1956), pp. 3-5 (in Chinese).

[5] See Stephen Salaff, 'A Biography of Hua Lo-keng', 1sis 63 (June, 1972), 143-183. [6] Shanghai shih, 1956-57 nien Chung hsueh.sheng Shu-hsueh Ching-sai his-t'i pien-

hui [Compilation of Problems from 1956-1957 Mathematics Competitiotts for Middle School Students in Shanghai Municipality[ New Knowledge Press, Shanghai, 1958.

[7] Obtained through private correspondence. [8] 'The Questions on the First and Second Examinations of the 1978 Selected Provinces

442 F R A N K S W E T Z AND Y I N G - K I N G YU

and Cities Middle School Mathematics Competition', Kuang-ming Jih-pao (June 6, 1978), pp. 3.

[9] 'Middle School Mathematics Contest', China Reconstructs 27 (September, 1978) 30-33.

[10] The PRC received an invitation to participate in the 1978 IMO but declined.