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COLETANIA OLIMP IADAS DE MATEMATICA

IMO

INTERNATIONALMATHEMATICAL OLYMPIAD

1959 - 1985

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IMO - INTERNATIONAL MATHEMATICAL OLYMPIAD

Sumario

1 First International Olympiad, 1959 61.1 1959/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 1959/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 1959/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 1959/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 1959/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 1959/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Second International Olympiad, 1960 82.1 1960/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 1960/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 1960/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 1960/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 1960/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 1960/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 1960/7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Third International Olympiad, 1961 103.1 1961/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 1961/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 1961/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 1961/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 1961/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.6 1961/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Fourth International Olympiad, 1962 114.1 1962/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 1962/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 1962/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 1962/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 1962/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.6 1962/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.7 1962/7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Fifth International Olympiad, 1963 135.1 1963/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 1963/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 1963/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 1963/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.5 1963/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6 1963/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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6 Sixth International Olympiad, 1964 146.1 1964/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 1964/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 1964/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 1964/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.5 1964/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.6 1964/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Seventh Internatioaal Olympiad, 1965 157.1 1965/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 1965/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 1965/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 1965/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 1965/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.6 1965/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Eighth International Olympiad, 1966 168.1 1966/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 1966/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3 1966/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.4 1966/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.5 1966/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.6 1966/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

9 Ninth International Olympiad, 1967 17

9.1 1967/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.2 1967/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.3 1967/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.4 1967/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.5 1967/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.6 1967/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10 Tenth International Olympiad, 1968 1910.1 1968/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.2 1968/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10.3 1968/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.4 1968/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.5 1968/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.6 1968/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

11 Eleventh International Olympiad, 1969 2111.1 1969/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.2 1969/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.3 1969/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.4 1969/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.5 1969/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

11.6 1969/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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12 Twelfth International Olympiad, 1970 2212.1 1970/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.2 1970/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.3 1970/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.4 1970/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.5 1970/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.6 1970/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

13 Twelfth International Olympiad, 1970 2413.1 1970/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.2 1970/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.3 1970/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.4 1970/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.5 1970/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13.6 1970/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

14 Fourteenth International Olympiad, 1972 2614.1 1972/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.2 1972/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.3 1972/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.4 1972/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.5 1972/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.6 1972/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

15 Fifteenth International Olympiad, 1973 27

15.1 1973/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.2 1973/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.3 1973/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.4 1973/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.5 1973/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.6 1973/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

16 Sixteenth International Olympiad, 1974 2916.1 1974/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.2 1974/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

16.3 1974/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.4 1974/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.5 1974/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.6 1974/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

17 Seventeenth International Olympiad, 1975 3017.1 1975/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.2 1975/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3 1975/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.4 1975/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.5 1975/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

17.6 1975/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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18 Eighteenth International Olympiad, 1976 3118.1 1976/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.2 1976/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.3 1976/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.4 1976/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.5 1976/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.6 1976/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

19 Nineteenth International Olympiad, 1977 3219.1 1977/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.2 1977/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.3 1977/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.4 1977/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.5 1977/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

19.6 1977/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

20 Twentieth International Olympiad, 1978 3320.1 1978/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.2 1978/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.3 1978/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.4 1978/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.5 1978/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.6 1978/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

21 Twenty-first International Olympiad, 1979 34

21.1 1979/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3421.2 1979/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3421.3 1979/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3421.4 1979/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3421.5 1979/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3421.6 1979/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

22 Twenty-second International Olympiad, 1980 36

23 Twenty-second International Olympiad, 1981 3723.1 1981/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.2 1981/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.3 1981/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.4 1981/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.5 1981/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.6 1981/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

24 Twenty-third International Olympiad, 1982 3824.1 1982/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3824.2 1982/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3824.3 1982/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

24.4 1982/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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24.5 1982/50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3824.6 1982/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

25 Twenty-fourth International Olympiad, 1983 40

25.1 1983/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.2 1983/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.3 1983/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.4 1983/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.5 1983/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.6 1983/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

26 Twenty-fifth International Olympiad, 1984 4126.1 1984/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4126.2 1984/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

26.3 1984/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4126.4 1984/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4126.5 1984/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4126.6 1984/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

27 Twenty-sixth International Olympiad, 1985 4227.1 1985/1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.2 1985/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.3 1985/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.4 1985/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.5 1985/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

27.6 1985/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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1 First International Olympiad, 1959

1.1 1959/1.

Prove that the fraction 21n+414n+3 is irreducible for every natural number n.

1.2 1959/2.

For what real values ofxis(x +

2x 1) +

(x 2x 1) =A,

given (a) A=

2, (b) A= 1, (c) A= 2, where only non-negative real numbersare admitted for square roots?

1.3 1959/3.

Let a, b, c be real numbers. Consider the quadratic equation in cos x:

a cos2 x + b cos x + c= 0.

Using the numbers a,b, c, form a quadratic equation in cos 2x, whose roots arethe same as those of the original equation. Compare the equations in cos x andcos2xfor a= 4, b= 2, c=1.

1.4 1959/4.Construct a right triangle with given hypotenuse c such that the median drawn tothe hypotenuse is the geometric mean of the two legs of the triangle.

1.5 1959/5.

An arbitrary point M is selected in the interior of the segment AB. The squaresAMCDandMBEFare constructed on the same side ofAB, with the segmentsAMand MB as their respective bases. The circles circumscribed about these squares,with centers P and Q, intersect at Mand also at another point N. Let N denote

the point of intersection of the straight lines AF and BC.(a) Prove that the points N and N coincide.(b) Prove that the straight lines MNpass through a fixed point S independent

of the choice ofM .(c) Find the locus of the midpoints of the segments P Q as Mvaries between A

and B.

1.6 1959/6.

Two planes, P and Q,intersect along the line p. The point A is given in the plane

P, and the point C in the plane Q; neither of these points lies on the straight linep. Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a

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circle can be inscribed, and with vertices B and D lying in the planes P and Qrespectively.

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2 Second International Olympiad, 1960

2.1 1960/1.

Determine all three-digit numbers Nhaving the property that Nis divisible by 11,and N/11 is equal to the sum of the squares of the digits ofN.

2.2 1960/2.

For what values of the variable xdoes the following inequality hold:

4x2

(1 1 + 2x)2

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2.7 1960/7.

An isosceles trapezoid with basesaand c and altitude h is given.(a) On the axis of symmetry of this trapezoid, find all points P such that both

legs of the trapezoid subtend right angles at P.(b) Calculate the distance ofP from either base.(c) Determine under what conditions such points P actually exist. (Discuss

various cases that might arise.)

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3 Third International Olympiad, 1961

3.1 1961/1.

Solve the system of equations:

x + y+ z = a

x2 + y2 + z2 = b2

xy = z2

where a and b are constants. Give the conditions that a and b must satisfy sothat x, y,z(the solutions of the system) are distinct positive numbers.

3.2 1961/2.

Let a, b, c be the sides of a triangle, and T its area. Prove: a2

+ b2

+ c2

43T.Inwhat case does equality hold?

3.3 1961/3.

Solve the equation cosn x sinn x= 1,where nis a natural number.

3.4 1961/4.

Consider triangle P1P2P3 and a point P within the triangle. Lines P1P, P2P, P3Pintersect the opposite sides in points Q1, Q2, Q3 respectively. Prove that, of thenumbers

P1P

P Q1,P2P

P Q2,P3P

P Q3at least one is2 and at least one is2.

3.5 1961/5.

Construct triangle ABC if AC = b,AB = c and AMB = , where M is themidpoint of segment BCand

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4 Fourth International Olympiad, 1962

4.1 1962/1.

Find the smallest natural number n which has the following properties:(a) Its decimal representation has 6 as the last digit.(b) If the last digit 6 is erased and placed in front of the remaining digits, the

resulting number is four times as large as the original number n.

4.2 1962/2.

Determine all real numbers xwhich satisfy the inequality:

3 x x + 1 > 1

2.

4.3 1962/3.

Consider the cubeABCDABCD (ABCDandABCD are the upper and lowerbases, respectively, and edgesAA, BB , CC, DD are parallel). The point Xmovesat constant speed along the perimeter of the square ABCDin the direction ABCDA,and the point Ymoves at the same rate along the perimeter of the square B CCBin the direction B CCBB . PointsXandYbegin their motion at the same instantfrom the starting positions A and B , respectively. Determine and draw the locus ofthe midpoints of the segments XY.

4.4 1962/4.

Solve the equation cos2 x + cos2 2x + cos2 3x= 1.

4.5 1962/5.

On the circle K there are given three distinct points A,B,C. Construct (usingonly straightedge and compasses) a fourth point D on Ksuch that a circle can beinscribed in the quadrilateral thus obtained.

4.6 1962/6.

Consider an isosceles triangle. Let r be the radius of its circumscribed circle and the radius of its inscribed circle. Prove that the distance d between the centers ofthese two circles is

d=

r(r 2).

4.7 1962/7.

The tetrahedron SABChas the following property: there exist five spheres, each

tangent to the edges SA, SB,SC, BCCA, AB,or to their extensions.(a) Prove that the tetrahedron SABCis regular.

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(b) Prove conversely that for every regular tetrahedron five such spheres exist.

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5 Fifth International Olympiad, 1963

5.1 1963/1.

Find all real roots of the equationx2 p + 2

x2 1 =x,

where pis a real parameter.

5.2 1963/2.

Point A and segment BCare given. Determine the locus of points in space whichare vertices of right angles with one side passing through A, and the other sideintersecting the segment BC.

5.3 1963/3.

In an n-gon all of whose interior angles are equal, the lengths of consecutive sidessatisfy the relation

a1a2 an.Prove that a1=a2= = an.

5.4 1963/4.

Find all solutions x1, x2, x3, x4, x5 of the systemx5+ x2 = yx1

x1+ x3 = yx2

x2+ x4 = yx3

x3+ x5 = yx4

x4+ x1 = yx5,

where y is a parameter.

5.5 1963/5.Prove that cos

7 cos 2

7+ cos 3

7 = 1

2.

5.6 1963/6.

Five students, A, B,C,D,E, took part in a contest. One prediction was that thecontestants would finish in the order ABCDE. This prediction was very poor. In factno contestant finished in the position predicted, and no two contestants predicted tofinish consecutively actually did so. A second prediction had the contestants finishingin the order DAECB. This prediction was better. Exactly two of the contestants

finished in the places predicted, and two disjoint pairs of students predicted to finishconsecutively actually did so. Determine the order in which the contestants finished.

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6 Sixth International Olympiad, 1964

6.1 1964/1.

(a) Find all positive integers nfor which 2n 1 is divisible by 7.(b) Prove that there is no positive integer nfor which 2n + 1 is divisible by 7.

6.2 1964/2.

Supposea, b, c are the sides of a triangle. Prove that

a2(b + c a) + b2(c + a b) + c2(a + b c)3abc.

6.3 1964/3.

A circle is inscribed in triangleABCwith sidesa,b, c.Tangents to the circle parallelto the sides of the triangle are constructed. Each of these tangents cuts off a trianglefrom ABC. In each of these triangles, a circle is inscribed. Find the sum of theareas of all four inscribed circles (in terms ofa, b, c).

6.4 1964/4.

Seventeen people correspond by mail with one another - each one with all the rest.In their letters only three different topics are discussed. Each pair of correspondentsdeals with only one of these topics. Prove that there are at least three people who

write to each other about the same topic.

6.5 1964/5.

Suppose five points in a plane are situated so that no two of the straight lines joiningthem are parallel, perpendicular, or coincident. From each point perpendiculars aredrawn to all the lines joining the other four points. Determine the maximum numberof intersections that these perpendiculars can have.

6.6 1964/6.

In tetrahedronABCD,vertexD is connected withD0the centroid of ABC.Linesparallel to DD0 are drawn through A, B and C. These lines intersect the planesBCD,CAD and ABDin pointsA1, B1 andC1, respectively. Prove that the volumeof ABCD is one third the volume of A1B1C1D0. Is the result true if point D0 isselected anywhere within ABC?

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7 Seventh Internatioaal Olympiad, 1965

7.1 1965/1.

Determine all values xin the interval 0x2 which satisfy the inequality2cos x

1 + sin 2x 1 sin2x 2.7.2 1965/2.

Consider the system of equations

a11x1+ a12x2+ a13x3 = 0

a21x1+ a22x2+ a23x3 = 0

a31x1+ a32x2+ a33x3 = 0

with unknowns x1, x2, x3. The coefficients satisfy the conditions:(a) a11, a22, a33 are positive numbers;(b) the remaining coefficients are negative numbers;(c) in each equation, the sum of the coefficients is positive.Prove that the given system has only the solution x1 = x2=x3 = 0.

7.3 1965/3.

Given the tetrahedronABCDwhose edgesAB andC Dhave lengthsaandbrespec-

tively. The distance between the skew lines AB andC Dis d, and the angle betweenthem is . Tetrahedron ABCD is divided into two solids by plane , parallel tolines AB and CD. The ratio of the distances of from AB and CD is equal to k.Compute the ratio of the volumes of the two solids obtained.

7.4 1965/4.

Find all sets of four real numbers x1, x2, x3, x4 such that the sum of any one and theproduct of the other three is equal to 2.

7.5 1965/5.Consider OAB with acute angle AOB. Through a point M= O perpendicularsare drawn to OAandOB, the feet of which are P and Q respectively. The point ofintersection of the altitudes of OP QisH. What is the locus ofHifMis permittedto range over (a) the side AB ,(b) the interior of OAB?

7.6 1965/6.

In a plane a set of n points (n 3) is given. Each pair of points is connectedby a segment. Letd be the length of the longest of these segments. We define a

diameter of the set to be any connecting segment of length d. Prove that the numberof diameters of the given set is at most n.

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8 Eighth International Olympiad, 1966

8.1 1966/1.

In a mathematical contest, three problems, A,B,C were posed. Among the par-ticipants there were 25 students who solved at least one problem each. Of all thecontestants who did not solve problem A, the number who solved B was twice thenumber who solved C. The number of students who solved only problem A was onemore than the number of students who solved A and at least one other problem. Ofall students who solved just one problem, half did not solve problem A.How manystudents solved only problem B ?

8.2 1966/2.

Leta, b, cbe the lengths of the sides of a triangle, and, , , respectively, the anglesopposite these sides. Prove that if

a + b= tan

2(a tan + b tan ),

the triangle is isosceles.

8.3 1966/3.

Prove: The sum of the distances of the vertices of a regular tetrahedron from thecenter of its circumscribed sphere is less than the sum of the distances of thesevertices from any other point in space.

8.4 1966/4.

Prove that for every natural number n, and for every real number x= k/2t(t =0, 1,...,n; k any integer)

1

sin2x+

1

sin4x+ + 1

sin2nx= cot x cot2nx.

8.5 1966/5.

Solve the system of equations

|a1 a2| x2 + |a1 a3| x3 + |a1 a4| x4 = 1|a2 a1| x1 + |a2 a3| x3 + |a2 a3| x3 = 1|a3 a1| x1 + |a3 a2| x2 = 1|a4 a1| x1 + |a4 a2| x2 + |a4 a3| x3 = 1

where a1, a2, a3, a4 are four different real numbers.

8.6 1966/6.

In the interior of sides BC,CA, AB of triangle ABC, any points K,L,M, respec-

tively, are selected. Prove that the area of at least one of the triangles AML,BKM,CLKis less than or equal to one quarter of the area of triangle ABC.

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9 Ninth International Olympiad, 1967

9.1 1967/1.

Let ABCD be a parallelogram with side lengths AB = a,AD = 1, and withBAD =. If ABD is acute, prove that the four circles of radius 1 with centersA,B,C,D cover the parallelogram if and only if

acos +

3sin .

9.2 1967/2.

Prove that if one and only one edge of a tetrahedron is greater than 1, then itsvolume is1/8.

9.3 1967/3.

Let k,m, n be natural numbers such that m+k+ 1 is a prime greater than n+ 1.Let cs= s(s + 1). Prove that the product

(cm+1 ck)(cm+2 ck) (cm+n ck)

is divisible by the product c1c2 cn.

9.4 1967/4.

Let A0B0C0 and A1B1C1 be any two acute-angled triangles. Consider all trianglesABCthat are similar to A1B1C1(so that verticesA1, B1, C1correspond to verticesA,B,C, respectively) and circumscribed about triangle A0B0C0 (where A0 lies onBC,B0 on CA,and AC0 on AB). Of all such possible triangles, determine the onewith maximum area, and construct it.

9.5 1967/5.

Consider the sequence{cn}, where

c1 = a1+ a2+ + a8c2 = a

21+ a

22+ + a28

cn = a

n1 + a

n2 + + an8

in which a1, a2, , a8 are real numbers not all equal to zero. Suppose that aninfinite number of terms of the sequence{cn} are equal to zero. Find all naturalnumbersn for which cn = 0.

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9.6 1967/6.

In a sports contest, there were m medals awarded on n successive days (n >1). Onthe first day, one medal and 1/7 of the remaining m 1 medals were awarded. Onthe second day, two medals and 1/7 of the now remaining medals were awarded; andso on. On the n-th and last day, the remaining nmedals were awarded. How manydays did the contest last, and how many medals were awarded altogether?

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10 Tenth International Olympiad, 1968

10.1 1968/1.

Prove that there is one and only one triangle whose side lengths are consecutiveintegers, and one of whose angles is twice as large as another.

10.2 1968/2.

Find all natural numbersxsuch that the product of their digits (in decimal notation)is equal to x2 10x 22.

10.3 1968/3.

Consider the system of equations

ax21+ bx1+ c = x2

ax22+ bx2+ c = x3

ax2n1+ bxn1+ c = xn

ax2n+ bxn+ c = x1,

with unknownsx1, x2, , xn, wherea, b, care real anda= 0.Let = (b1)24ac.Prove that for this system(a) if < 0,there is no solution,

(b) if = 0,there is exactly one solution,(c) if > 0, there is more than one solution.

10.4 1968/4.

Prove that in every tetrahedron there is a vertex such that the three edges meetingthere have lengths which are the sides of a triangle.

10.5 1968/5.Let fbe a real-valued function defined for all real numbers x such that, for somepositive constant a,the equation

f(x + a) =1

2+

f(x) [f(x)]2

holds for allx.(a) Prove that the function f is periodic (i.e., there exists a positive number b

such that f(x + b) =f(x) for allx).(b) For a = 1, give an example of a non-constant function with the required

properties.

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10.6 1968/6.

For every natural number n, evaluate the sum

k=0

n + 2

k

2k+1

=

n + 12

+

n + 24

+ + n + 2k

2k+1

+

(The symbol [x] denotes the greatest integer not exceeding x.)

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11 Eleventh International Olympiad, 1969

11.1 1969/1.

Prove that there are infinitely many natural numbers a with the following property:the number z=n4 + ais not prime for any natura1 number n.

11.2 1969/2.

Let a1, a2, , an be real constants, xa real variable, and

f(x) = cos(a1+ x) +1

2cos(a2+ x) +

1

4cos(a3+ x)

+

+ 1

2n1cos(a

n+ x).

Given thatf(x1) =f(x2) = 0, prove that x2 x1 = m for some integer m.

11.3 1969/3.

For each value of k = 1, 2, 3, 4, 5, find necessary and sufficient conditions on thenumber a >0 so that there exists a tetrahedron with k edges of length a, and theremaining 6 k edges of length 1.

11.4 1969/4.A semicircular arc is drawn on AB as diameter. Cis a point on other than Aand B, and D is the foot of the perpendicular from C to AB. We consider threecircles, 1, 2, 3, all tangent to the line AB. Of these, 1 is inscribed in ABC,while 2 and 3 are both tangent to CD and to , one on each side ofCD. Provethat 1, 2 and 3 have a second tangent in common.

11.5 1969/5.

Given n >4 points in the plane such that no three are collinear. Prove that there

are at leastn3

2

convex quadrilaterals whose vertices are four of the given points.

11.6 1969/6.

Prove that for all real numbers x1, x2, y1, y2, z1, z2, with x1 >0, x2 >0, x1y1 z21 >0, x2y2 z22 >0,the inequality

8

(x1+ x2) (y1+ y2) (z1+ z2)2 1

x1y1 z21+

1

x2y2 z22is satisfied. Give necessary and sufficient conditions for equality.

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12 Twelfth International Olympiad, 1970

12.1 1970/1.

Let M be a point on the side AB of ABC. Let r1, r2 and r be the radii of theinscribed circles of triangles AMC,BMCand ABC.Letq1, q2 andqbe the radii ofthe escribed circles of the same triangles that lie in the angle ACB. Prove that

r1q1

r2q2

=r

q.

12.2 1970/2.

Leta, band n be integers greater than 1,and leta and b be the bases of two numbersystems. An1 andAn are numbers in the system with base a, andBn1 and Bnare

numbers in the system with base b; these are related as follows:

An = xnxn1 x0, An1 = xn1xn2 x0,Bn = xnxn1 x0, Bn1 = xn1xn2 x0,xn = 0, xn1= 0.

Prove:An1

An b.

12.3 1970/3.

The real numbers a0, a1,...,an,...satisfy the condition:

1 =a0a1a2 an .

The numbersb1, b2,...,bn,...are defined by

bn=n

k=1

1 ak1

ak

1

ak.

(a) Prove that 0

bn c for large enough n.

12.4 1970/4.

Find the set of all positive integers n with the property that the set{n, n+ 1, n+2, n + 3, n + 4, n + 5} can be partitioned into two sets such that the product of thenumbers in one set equals the product of the numbers in the other set.

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12.5 1970/5.

In the tetrahedron ABCD,angleBDC is a right angle. Suppose that the foot Hofthe perpendicular from D to the plane ABCis the intersection of the altitudes of

ABC. Prove that

(AB+ BC+ CA)2 6(AD2 + BD2 + CD2).

For what tetrahedra does equality hold?

12.6 1970/6.

In a plane there are 100 points, no three of which are collinear. Consider all possibletriangles having these points as vertices. Prove that no more than 70% of thesetriangles are acute-angled.



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13 Twelfth International Olympiad, 1970

13.1 1970/1.

Let M be a point on the side AB of ABC. Let r1, r2 and r be the radii of theinscribed circles of triangles AMC,BMCand ABC.Letq1, q2 andqbe the radii ofthe escribed circles of the same triangles that lie in the angle ACB. Prove that

r1q1

r2q2

=r

q.

13.2 1970/2.

Leta, band n be integers greater than 1,and leta and b be the bases of two numbersystems. An1 andAn are numbers in the system with base a, andBn1 and Bnare

numbers in the system with base b; these are related as follows:

An = xnxn1 x0, An1 = xn1xn2 x0,Bn = xnxn1 x0, Bn1 = xn1xn2 x0,xn = 0, xn1= 0.

Prove:An1

An b.

13.3 1970/3.

The real numbers a0, a1,...,an,...satisfy the condition:

1 =a0a1a2 an .

The numbersb1, b2,...,bn,...are defined by

bn=n

k=1

1 ak1

ak

1

ak.

(a) Prove that 0

bn c for large enough n.

13.4 1970/4.

Find the set of all positive integers n with the property that the set{n, n+ 1, n+2, n + 3, n + 4, n + 5} can be partitioned into two sets such that the product of thenumbers in one set equals the product of the numbers in the other set.



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13.5 1970/5.

In the tetrahedron ABCD,angleBDC is a right angle. Suppose that the foot Hofthe perpendicular from D to the plane ABCis the intersection of the altitudes of

ABC. Prove that

(AB+ BC+ CA)2 6(AD2 + BD2 + CD2).

For what tetrahedra does equality hold?

13.6 1970/6.

In a plane there are 100 points, no three of which are collinear. Consider all possibletriangles having these points as vertices. Prove that no more than 70% of thesetriangles are acute-angled.



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14 Fourteenth International Olympiad, 1972

14.1 1972/1.

Prove that from a set of ten distinct two-digit numbers (in the decimal system), itis possible to select two disjoint subsets whose members have the same sum.

14.2 1972/2.

Prove that if n 4, every quadrilateral that can be inscribed in a circle can bedissected into n quadrilaterals each of which is inscribable in a circle.

14.3 1972/3.

Let m and nbe arbitrary non-negative integers. Prove that

(2m)!(2n)!

mn!(m + n)!is an integer. (0! = 1.)

14.4 1972/4.

Find all solutions (x1, x2, x3, x4, x5) of the system of inequalities

(x21 x3x5)(x22 x3x5) 0(x22 x4x1)(x23 x4x1) 0(x23 x5x2)(x24 x5x2) 0(x24 x1x3)(x25 x1x3) 0(x25 x2x4)(x21 x2x4) 0

where x1, x2, x3, x4, x5 are positive real numbers.

14.5 1972/5.

Let f and g be real-valued functions defined for all real values of x and y, andsatisfying the equation

f(x + y) + f(x y) = 2f(x)g(y)for all x, y.Prove that iff(x) is not identically zero, and if|f(x)| 1 for all x,

then|g(y)| 1 for all y.

14.6 1972/6.

Given four distinct parallel planes, prove that there exists a regular tetrahedron witha vertex on each plane.

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15 Fifteenth International Olympiad, 1973

15.1 1973/1.

Point Olies on line g; OP1, OP2, ...,OPnare unit vectors such that points P1, P2,...,Pnall lie in a plane containing g and on one side ofg. Prove that ifn is odd,

OP1+ OP2+ + OPn1Here

OM denotes the length of vectorOM.15.2 1973/2.

Determine whether or not there exists a finite set Mof points in space not lying in

the same plane such that, for any two points A and B ofM,one can select two otherpoints Cand D ofMso that lines AB and CD are parallel and not coincident.

15.3 1973/3.

Let a and bbe real numbers for which the equation

x4 + ax3 + bx2 + ax + 1 = 0

has at least one real solution. For all such pairs (a, b), find the minimum valueofa2 + b2.

15.4 1973/4.

A soldier needs to check on the presence of mines in a region having the shape of anequilateral triangle. The radius of action of his detector is equal to half the altitudeof the triangle. The soldier leaves from one vertex of the triangle. What pathshouid he follow in order to travel the least possible distance and still accomplishhis mission?

15.5 1973/5.

Gis a set of non-constant functions of the real variable xof the form

f(x) =ax + b, aand bare real numbers,

and G has the following properties:(a) Iff and g are in G, then g f is in G; here (g f)(x) =g[f(x)].(b) Iff is inG,then its inverse f1 is in G; here the inverse off(x) =ax + bis

f1(x) = (x b)/a.(c) For everyf inG,there exists a real number xfsuch that f(xf) =xf.Prove that there exists a real number k such that f(k) =k for all f inG.

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15.6 1973/6.

Let a1, a2,...,an be n positive numbers, and let qbe a given real number such that0< q

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16 Sixteenth International Olympiad, 1974

16.1 1974/1.

Three playersA, BandCplay the following game: On each of three cards an integeris written. These three numbers p, q,r satisfy 0 < p < q < r. The three cards areshuffled and one is dealt to each player. Each then receives the number of countersindicated by the card he holds. Then the cards are shuffled again; the countersremain with the players.

This process (shuffling, dealing, giving out counters) takes place for at least tworounds. After the last round,A has 20 counters in all, B has 10 and Chas 9. Atthe last round B received r counters. Who received qcounters on the first round?

16.2 1974/2.

In the triangleABC,prove that there is a point D on sideAB such that C Dis thegeometric mean ofAD and DB if and only if

sin A sin Bsin2C2

.

16.3 1974/3.

Prove that the numbern

k=0

2n+12k+1

23k is not divisible by 5 for any integer n0.

16.4 1974/4.Consider decompositions of an 88 chessboard into p non-overlapping rectanglessubject to the following conditions:

(i) Each rectangle has as many white squares as black squares.(ii) Ifai is the number of white squares in the i-th rectangle, then a1 < a2 p.

17.3 1975/3.

On the sides of an arbitrary triangle ABC, triangles ABR,BCP,CAQ are con-structed externally with CBP = CAQ = 45, BC P = ACQ = 30, ABR =BAR = 15. Prove that QRP = 90 and QR = RP.

17.4 1975/4.

When 44444444 is written in decimal notation, the sum of its digits is A. Let B bethe sum of the digits ofA. Find the sum of the digits ofB.(Aand B are written indecimal notation.)

17.5 1975/5.

Determine, with proof, whether or not one can find 1975 points on the circumferenceof a circle with unit radius such that the distance between any two of them is arational number.

17.6 1975/6.

Find all polynomials P, in two variables, with the following properties:(i) for a positive integer n and all realt, x, y

P(tx, ty) =tnP(x, y)

(that is,P is homogeneous of degree n),(ii) for all real a, b, c,

P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,

(iii)P(1, 0) = 1.

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18 Eighteenth International Olympiad, 1976

18.1 1976/1.

In a plane convex quadrilateral of area 32, the sum of the lengths of two oppositesides and one diagonal is 16.Determine all possible lengths of the other diagonal.

18.2 1976/2.

Let P1(x) = x2 2 and Pj(x) = P1(Pj1(x)) for j = 2, 3, . Show that, for any

positive integer n,the roots of the equation Pn(x) =x are real and distinct.

18.3 1976/3.

A rectangular box can be filled completely with unit cubes. If one places as manycubes as possible, each with volume 2, in the box, so that their edges are parallelto the edges of the box, one can fill exactly 40% of the box. Determine the possibledimensions of all such boxes.

18.4 1976/4.

Determine, with proof, the largest number which is the product of positive integerswhose sum is 1976.

18.5 1976/5.

Consider the system ofpequations in q= 2punknowns x1, x2, , xq :a11x1+ a12x2+ + a1qxq = 0a21x1+ a22x2+ + a2qxq = 0

ap1x1+ ap2x2+ + apqxq = 0

with every coefficient aij member of the set{1, 0, 1}. Prove that the systemhas a solution (x1, x2, , xq) such that

(a) all xj (j= 1, 2,...,q) are integers,

(b) there is at least one value ofj for which xj= 0,(c)|xj| q(j = 1, 2,...,q).

18.6 1976/6.

A sequence{un} is defined byu0 = 2, u1= 5/2, un+1 = un(u

2n1 2) u1for n= 1, 2,

Prove that for positive integers n,

[un] = 2[2n(1)n]/3

where [x] denotes the greatest integerx.

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19 Nineteenth International Olympiad, 1977

19.1 1977/1.

Equilateral triangles ABK,BCL,CDM,DAN are constructed inside the squareABCD.Prove that the midpoints of the four segments KL, LM,MN, NKand themidpoints of the eight segmentsAKBK,BL,CL,CM,DM,DN,ANare the twelvevertices of a regular dodecagon.

19.2 1977/2.

In a finite sequence of real numbers the sum of any seven successive terms is negative,and the sum of any eleven successive terms is positive. Determine the maximumnumber of terms in the sequence.

19.3 1977/3.

Let n be a given integer > 2, and let Vn be the set of integers 1 +kn, wherek = 1, 2,.... A numberm Vn is called indecomposable in Vn if there do not existnumbersp, qVn such that pq=m. Prove that there exists a number rVn thatcan be expressed as the product of elements indecomposable in Vn in more than oneway. (Products which differ only in the order of their factors will be considered thesame.)

19.4 1977/4.Four real constants a, b, A, B are given, and

f() = 1 a cos b sin A cos2 B sin2.Prove that iff()0 for all real, then

a2 + b2 2 and A2 + B2 1.

19.5 1977/5.

Let aand b be positive integers. When a

2

+ b

2

is divided bya + b,the quotient is qand the remainder is r.Find all pairs (a, b) such that q2 + r= 1977.

19.6 1977/6.

Let f(n) be a function defined on the set of all positive integers and having all itsvalues in the same set. Prove that if

f(n + 1) > f(f(n))

for each positive integer n,then

f(n) =n for each n.

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20 Twentieth International Olympiad, 1978

20.1 1978/1.

mand n are natural numbers with 1m < n.In their decimal representations, thelast three digits of 1978m are equal, respectively, to the last three digits of 1978n.Findmand n such that m + n has its least value.

20.2 1978/2.

Pis a given point inside a given sphere. Three mutually perpendicular rays from Pintersect the sphere at pointsU, V,and W; Qdenotes the vertex diagonally oppositeto Pin the parallelepiped determined by PU, PV ,and P W.Find the locus ofQforall such triads of rays from P

20.3 1978/3.

The set of all positive integers is the union of two disjoint subsets {f(1), f(2),...,f(n),...}, {g(1), g(2), .where

f(1)< f(2)

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21 Twenty-first International Olympiad, 1979

21.1 1979/1.

Let p and qbe natural numbers such thatp

q = 1 1

2+

1

31

4+ 1

1318+

1

1319.

Prove that p is divisible by 1979.

21.2 1979/2.

A prism with pentagons A1A2A3A4A5 and B1B2B3B4B5 as top and bottom facesis given. Each side of the two pentagons and each of the line-segments AiBj for all

i, j = 1, ..., 5,is colored either red or green. Every triangle whose vertices are verticesof the prism and whose sides have all been colored has two sides of a different color.Show that all 10 sides of the top and bottom faces are the same color.

21.3 1979/3.

Two circles in a plane intersect. LetA be one of the points of intersection. Startingsimultaneously fromAtwo points move with constant speeds, each point travellingalong its own circle in the same sense. The two points return to A simultaneouslyafter one revolution. Prove that there is a fixed point P in the plane such that, atany time, the distances fromPto the moving points are equal.

21.4 1979/4.

Given a plane , a point P in this plane and a point Q not in , find all points Rin such that the ratio (QP+ P A)/QRis a maximum.

21.5 1979/5.

Find all real numbers a for which there exist non-negative real numbers x1, x2, x3, x4, x5satisfying the relations

5k=1

kxk =a,5

k=1

k3xk=a2,

5k=1

k5xk=a3.

21.6 1979/6.

Let A and E be opposite vertices of a regular octagon. A frog starts jumping atvertex A. From any vertex of the octagon except E, it may jump to either of thetwo adjacent vertices. When it reaches vertex E, the frog stops and stays there..Let an be the number of distinct paths of exactly n jumps ending at E. Prove thata2n1 = 0,

a2n= 12

(xn1 yn1), n= 1, 2, 3, ,

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where x= 2 +

2 and y= 2 2.Note. A path ofn jumps is a sequence of vertices (P0,...,Pn) such that(i) P0=A, Pn = E;(ii) for every i, 0

i

n

1, Pi is distinct from E;(iii) for everyi, 0in 1, Pi and Pi+1 are adjacent.

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22 Twenty-second International Olympiad, 1980

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23 Twenty-second International Olympiad, 1981

23.1 1981/1.

P is a point inside a given triangle ABC.D,E,Fare the feet of the perpendicularsfrom P to the lines BC,CA, AB respectively. Find all P for which

BC

P D+

CA

P E+

AB

P F

is least.

23.2 1981/2.

Let 1 r n and consider all subsets ofr elements of the set{1, 2,...,n}. Eachof these subsets has a smallest member. Let F(n, r) denote the arithmetic mean ofthese smallest numbers; prove that

F(n, r) =n + 1

r+ 1.

23.3 1981/3.

Determine the maximum value of m3 +n3,where m and n are integers satisfyingm, n {1, 2, ..., 1981} and (n2 mn m2)2 = 1.

23.4 1981/4.(a) For which values ofn > 2 is there a set ofn consecutive positive integers suchthat the largest number in the set is a divisor of the least common multiple of theremaining n 1 numbers?

(b) For which values ofn >2 is there exactly one set having the stated property?

23.5 1981/5.

Three congruent circles have a common point O and lie inside a given triangle.Each circle touches a pair of sides of the triangle. Prove that the incenter and thecircumcenter of the triangle and the point O are collinear.

23.6 1981/6.

The function f(x, y) satisfies(1) f(0, y) =y + 1,(2)f(x + 1, 0) =f(x, 1),(3) f(x + 1, y+ 1) =f(x, f(x + 1, y)),for all non-negative integers x, y.Determine f(4, 1981).

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24 Twenty-third International Olympiad, 1982

24.1 1982/1.

The function f(n) is defined for all positive integers n and takes on non-negativeinteger values. Also, for all m, n

f(m + n) f(m) f(n) = 0 or 1f(2) = 0, f(3)> 0, and f(9999) = 3333.

Determine f(1982).

24.2 1982/2.

A non-isosceles triangleA1A2A3 is given with sides a1, a2, a3 (ai is the side opposite

Ai). For all i= 1, 2, 3, Mi is the midpoint of side ai, and Ti. is the point where theincircle touches side ai. Denote by Si the reflection ofTi in the interior bisector ofangle Ai. Prove that the lines M1, S1, M2S2,and M3S3 are concurrent.

24.3 1982/3.

Consider the infinite sequences{xn} of positive real numbers with the followingproperties:

x0= 1, and for alli0, xi+1xi.(a) Prove that for every such sequence, there is an n

1 such that

x20x1

+x21x2

+ + x2n1

xn3.999.

(b) Find such a sequence for which

x20x1

+x21x2

+ +x2n1

xn

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24.6 1982/6.

LetSbe a square with sides of length 100,and let L be a path within Swhich doesnot meet itself and which is composed of line segments A0A1, A1A2, , An1AnwithA0=An. Suppose that for every pointPof the boundary ofSthere is a pointofL at a distance from Pnot greater than 1/2. Prove that there are two points Xand Y inL such that the distance between Xand Yis not greater than 1,and thelength of that part ofL which lies between Xand Y is not smaller than 198.

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25 Twenty-fourth International Olympiad, 1983

25.1 1983/1.

Find all functions fdefined on the set of positive real numbers which take positivereal values and satisfy the conditions:

(i) f(xf(y)) =yf(x) for all positive x, y;(ii)f(x)0 as x .

25.2 1983/2.

LetAbe one of the two distinct points of intersection of two unequal coplanar circlesC1 and C2 with centers O1 and O2, respectively. One of the common tangents tothe circles touches C1 at P1 and C2 at P2, while the other touches C1 at Q1 and C2

at Q2. Let M1 be the midpoint ofP1Q1,and M2 be the midpoint ofP2Q2. Provethat O1AO2 = M1AM2.

25.3 1983/3.

Let a, b and c be positive integers, no two of which have a common divisor greaterthan 1.Show that 2abcabbccais the largest integer which cannot be expressedin the form xbc + yca + zab,where x, y and z are non-negative integers.

25.4 1983/4.

LetABCbe an equilateral triangle andEthe set of all points contained in the threesegments AB, BC and CA (including A, B and C). Determine whether, for everypartition ofEinto two disjoint subsets, at least one of the two subsets contains thevertices of a right-angled triangle. Justify your answer.

25.5 1983/5.

Is it possible to choose 1983 distinct positive integers, all less than or equal to 105,no three of which are consecutive terms of an arithmetic progression? Justify youranswer.

25.6 1983/6.

Let a, band c be the lengths of the sides of a triangle. Prove that

a2b(a b) + b2c(b c) + c2a(c a)0.

Determine when equality occurs.

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26 Twenty-fifth International Olympiad, 1984

26.1 1984/1.

Prove that 0yz+ zx + xy 2xyz7/27,wherex, y and zare non-negative realnumbers for which x + y+ z= 1.

26.2 1984/2.

Find one pair of positive integers aand bsuch that:(i) ab(a + b) is not divisible by 7;(ii) (a + b)7 a7 b7 is divisible by 77 .Justify your answer.

26.3 1984/3.In the plane two different pointsO andA are given. For each point Xof the plane,other than O, denote by a(X) the measure of the angle between OA and OX inradians, counterclockwise from OA(0 a(X) < 2). Let C(X) be the circle withcenter O and radius of length OX+ a(X)/OX. Each point of the plane is coloredby one of a finite number of colors. Prove that there exists a point Y for whicha(Y)> 0 such that its color appears on the circumference of the circle C(Y).

26.4 1984/4.

Let ABCD be a convex quadrilateral such that the line CD is a tangent to thecircle on AB as diameter. Prove that the line AB is a tangent to the circle on CDas diameter if and only if the lines BCand AD are parallel.

26.5 1984/5.

Let d be the sum of the lengths of all the diagonals of a plane convex polygon withnvertices (n >3), and let p be its perimeter. Prove that

n

3

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