Jul 13, 2011

International Mathematical Olympiad problems


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    37th International Mathematical Olympiad Solutions
    imo.math.ca/IMO96/solutions96.pdf
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    37th International Mathematical Olympiad. Solutions. Problem 1. We shall work on the array A of lattice points defined by. A = f i; j 2 Z2 ...
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    ACB = p=2. p or q. xj.
    imo.math.ca/IMO96/imo96q.pdf
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    International Mathematical Olympiad | 1996. Day 1. July 10, 1997. 1. Let ABCD be a rectangular board, with AB = 20 and BC = 12. ...
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    THE 36 th IMO, CANADA, 1995
    imo.math.ca/Sol/95/95_prob.pdf
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    1. First Day, July 19, 1995. 1. Let A, B, C and D be four distinct points on a line, in that order. The circles with diameters AC and BD intersect at the ...
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    IMO2000 Problems
    imo.math.ca/Exams/exam00/imo2000-problems.pdf
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    20 Jul 2000 – 41st IMO, Taejon, South Korea. First Day, 19 July 2000. Duration : 4 hr 30 min. 7 points each problem. Problem 1 ...
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    37th International Mathematical Olympiad Mumbai, India Day I 9 ...
    imo.math.ca/Exams/1996imo.pdf
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    37th International Mathematical Olympiad. Mumbai, India. Day I. 9 a.m. - 1:30 p.m.. July 10, 1996. 1. We are given a positive integer r and a rectangular ...
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    Twenty-third International Olympiad, 1982
    imo.math.ca/Exams/1982imo.pdf
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    Twenty-third International Olympiad, 1982. 1982/1. The function f(n) is defined for all positive integers n and takes on non-negative integer values. ...
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    30thInternational Mathematical Olympiad Braunschweig, Germany Day...
    imo.math.ca/Exams/1989imo.pdf
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    30thInternational Mathematical Olympiad. Braunschweig, Germany. Day I. 1. Prove that the set {1, 2,...,1989} can be expressed as the disjoint union of ...
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    Twenty-first International Olympiad, 1979
    imo.math.ca/Exams/1979imo.pdf
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    2. +. 1. 3. −. 1. 4. + ···−. 1. 1318. +. 1. 1319 . Prove that p is divisible by 1979. 1979/2. A prism with pentagons A1A2A3A4A5 and B1B2B3B4B5 as top and ...
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    33rd International Mathematical Olympiad First Day - Moscow - July ...
    imo.math.ca/Exams/1992imo.pdf
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    33rd International Mathematical Olympiad. First Day - Moscow - July 15, 1992. Time Limit: 41. 2 hours. 1. Find all integers a, b, c with 1 <a<b<c such that ...
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    Eighth International Olympiad, 1966
    imo.math.ca/Exams/1966imo.pdf
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    Eighth International Olympiad, 1966. 1966/1. In a mathematical contest, three problems, A,B,C were posed. Among the participants there were 25 students who ...
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    Fifteenth International Olympiad, 1973
    imo.math.ca/Exams/1973imo.pdf
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    Fifteenth International Olympiad, 1973. 1973/1. Point O lies on line g;. −−→. OP1,. −−→. OP2, ...,. −−→. OPn are unit vectors such that points ...
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    The 35th International Mathematical Olympiad (July 13-14, 1994 ...
    imo.math.ca/Exams/1994imo.pdf
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    The 35th International Mathematical Olympiad (July 13-14,. 1994, Hong Kong). 1. Let m and n be positive integers. Let a1,a2,...,am be distinct elements ...
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    Twenty-sixth International Olympiad, 1985
    imo.math.ca/Exams/1985imo.pdf
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    Twenty-sixth International Olympiad, 1985. 1985/1. A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to ...
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    Ninth International Olympiad, 1967
    imo.math.ca/Exams/1967imo.pdf
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    Ninth International Olympiad, 1967. 1967/1. Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with. BAD = α. If ∆ABD is acute, ...
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    32nd International Mathematical Olympiad First Day July 17, 1991 ...
    imo.math.ca/Exams/1991imo.pdf
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    32nd International Mathematical Olympiad. First Day. July 17, 1991. Time Limit: 41. 2 hours. 1. Given a triangle ABC, let I be the center of its inscribed ...
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    Twelfth International Olympiad, 1970
    imo.math.ca/Exams/1970imo.pdf
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    Twelfth International Olympiad, 1970. 1970/1. Let M be a point on the side AB of ∆ABC. Let r1,r2 and r be the radii of the inscribed circles of triangles ...
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    Seventh Internatioaal Olympiad, 1965
    imo.math.ca/Exams/1965imo.pdf
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    Seventh Internatioaal Olympiad, 1965. 1965/1. Determine all values x in the interval 0 ≤ x ≤ 2π which satisfy the inequality. 2 cosx ≤ ...
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    Third International Olympiad, 1961
    imo.math.ca/Exams/1961imo.pdf
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    Third International Olympiad, 1961. 1961/1. Solve the system of equations: x + y + z = a x2 + y2 + z2. = b2 xy = z2 where a and b are constants. ...
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    28thInternational Mathematical Olympiad Havana, Cuba Day I July 10...
    imo.math.ca/Exams/1987imo.pdf
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    28thInternational Mathematical Olympiad. Havana, Cuba. Day I. July 10, 1987. 1. Let pn(k) be the number of permutations of the set {1,...,n}, n ≥ 1, which ...
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    Thirteenth International Olympiad, 1971
    imo.math.ca/Exams/1971imo.pdf
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    Thirteenth International Olympiad, 1971. 1971/1. Prove that the following assertion is true for n = 3 and n = 5, and that it is ...
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    First International Olympiad, 1959
    imo.math.ca/Exams/1959imo.pdf
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    First International Olympiad, 1959. 1959/1. Prove that the fraction 21n+4. 14n+3 is irreducible for every natural number n. 1959/2. ...
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    39thInternational Mathematical Olympiad Taipei, Taiwan Day I July ...
    imo.math.ca/Exams/1998imo.pdf
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    39thInternational Mathematical Olympiad. Taipei, Taiwan. Day I. July 15, 1998. 1. In the convex quadrilateral ABCD, the diagonals AC and BD are ...
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    Fourteenth International Olympiad, 1972
    imo.math.ca/Exams/1972imo.pdf
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    Fourteenth International Olympiad, 1972. 1972/1. Prove that from a set of ten distinct two-digit numbers (in the decimal sys- ...
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    Twentieth International Olympiad, 1978
    imo.math.ca/Exams/1978imo.pdf
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    Twentieth International Olympiad, 1978. 1978/1. m and n are natural numbers with 1 ≤ m < n. In their decimal representations, the last three digits of ...
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    Seventeenth International Olympiad, 1975
    imo.math.ca/Exams/1975imo.pdf
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    Seventeenth International Olympiad, 1975. 1975/1. Let xi,yi (i = 1,2, ..., n) be real numbers such that x1 ≥ x2 ≥···≥ xn and y1 ≥ y2 ≥···≥ yn. ...
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    Twenty-second International Olympiad, 1981
    imo.math.ca/Exams/1981imo.pdf
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    Twenty-second International Olympiad, 1981. 1981/1. P is a point inside a given triangle ABC.D, E, F are the feet of the perpendiculars from P to the lines ...
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    Twenty-fifth International Olympiad, 1984
    imo.math.ca/Exams/1984imo.pdf
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    Twenty-fifth International Olympiad, 1984. 1984/1. Prove that 0 ≤ yz + zx + xy − 2xyz ≤ 7/27, where x, y and z are non-negative real numbers for which x ...
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    Twenty-fourth International Olympiad, 1983
    imo.math.ca/Exams/1983imo.pdf
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    Twenty-fourth International Olympiad, 1983. 1983/1. Find all functions f defined on the set of positive real numbers which ...
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    Fifth International Olympiad, 1963
    imo.math.ca/Exams/1963imo.pdf
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    Fifth International Olympiad, 1963. 1963/1. Find all real roots of the equation. √ x2 − p + 2. √ x2 − 1 = x, where p is a real parameter. 1963/2. ...
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    The Problem Selection Committee of the 36th International ...
    imo.math.ca/Exams/exam95/36solns.pdf
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    SHORTLISTED PROBLEMS for the 36 th. IMO. CANADA, 1995. 1. INTRODUCTION. The Problem Selection Committee of the 36th International Mathematical Olympiad ...
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    Second International Olympiad, 1960
    imo.math.ca/Exams/1960imo.pdf
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    Second International Olympiad, 1960. 1960/1. Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum ...
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    34nd International Mathematical Olympiad First Day July 18, 1993 ...
    imo.math.ca/Exams/1993imo.pdf
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    34nd International Mathematical Olympiad. First Day. July 18, 1993. Time Limit: 41. 2 hours. 1. Let f(x) = xn + 5xn−1 + 3, where n > 1 is an integer. ...
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    m : mn, 1
    imo.math.ca/Exams/exam94/imo1994.pdf
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    The 35th International Mathematical Olympiad July 13-14, 1994, Hong. Kong. 1. Let m and n be positive integers. Let a1; a2;:::;am be distinct elements ...
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    30thInternational Mathematical Olympiad Braunschweig, Germany Day I
    imo.math.ca/Exams/exam89/imo1989.pdf
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    30thInternational Mathematical Olympiad. Braunschweig, Germany. Day I. 1. Prove that the set {1; 2;:::; 1989} can be expressed as the disjoint union of ...
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    29thInternational Mathematical Olympiad Canberra, Australia Day I
    imo.math.ca/Exams/exam88/imo1988.pdf
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    29thInternational Mathematical Olympiad. Canberra, Australia. Day I. 1. Consider two coplanar circles of radii R and r R r with the same center. ...
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    38thInternational Mathematical Olympiad Mar del Plata, Argentina ...
    imo.math.ca/Exams/exam97/imo1997.pdf
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    38thInternational Mathematical Olympiad. Mar del Plata, Argentina. Day I. July 24, 1997. 1. In the plane the points with integer coordinates are the ...
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    28thInternational Mathematical Olympiad Havana, Cuba Day I July 10...
    imo.math.ca/Exams/exam87/imo1987.pdf
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    28thInternational Mathematical Olympiad. Havana, Cuba. Day I. July 10, 1987. 1. Let pn k be the number of permutations of the set {1;:::;n}, n > 1, which ...
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    31stInternational Mathematical Olympiad Beijing, China Day I July ...
    imo.math.ca/Exams/exam90/imo1990.pdf
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    31stInternational Mathematical Olympiad. Beijing, China. Day I. July 12, 1990. 1. Chords AB and CD of a circle intersect at a point E inside the circle. ...
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    40thInternational Mathematical Olympiad Bucharest Day I July 16 ...
    imo.math.ca/Exams/exam99/imo1999.pdf
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    40thInternational Mathematical Olympiad. Bucharest. Day I. July 16, 1999. 1. Determine all finite sets S of at least three points in the plane which satisfy ...
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    Premier Jour

     - [ Translate this page ]
    imo.math.ca/Exams/exam97/oim1997.pdf
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    Version: French. Premier Jour. Mar del Plata, Argentine - 24 Juillet 1997. 1. Dans le plan, les points a coordonn ees enti eres sont les sommets de carr es ...
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    39thInternational Mathematical Olympiad Taipei, Taiwan Day I July ...
    imo.math.ca/Exams/exam98/imo1998.pdf
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    39thInternational Mathematical Olympiad. Taipei, Taiwan. Day I. July 15, 1998. 1. In the convex quadrilateralABCD, the diagonalsAC andBD are perpendicular ...
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    36thInternational Mathematical Olympiad First Day - Toronto - July ...
    imo.math.ca/Exams/exam95/imo1995.pdf
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    36thInternational Mathematical Olympiad. First Day - Toronto - July 19, 1995. Time Limit: 41. 2 hours. 1. LetA; B; C; D be four distinct points on a line, ...
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    33rdInternational Mathematical Olympiad First Day - Moscow - July ...
    imo.math.ca/Exams/exam92/imo1992.pdf
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    15 Jul 1992 – 33rdInternational Mathematical Olympiad. First Day - Moscow - July 15, 1992. Time Limit: 41. 2 hours. 1. Find all integers a; b; ...
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    34nd International Mathematical Olympiad First Day July 18, 1993 ...
    imo.math.ca/Exams/exam93/imo1993.pdf
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    19 Jul 1993 – 34nd International Mathematical Olympiad. First Day July 18, 1993. Time Limit: 41. 2 hours. 1. Let f x = xn + 5xn-1 + 3, where n 1 is an ...
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    32nd International Mathematical Olympiad First Day July 17, 1991 ...
    imo.math.ca/Exams/exam91/imo1991.pdf
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    18 Jul 1991 – 32nd International Mathematical Olympiad. First Day July 17, 1991. Time Limit: 41. 2 hours. 1. Given a triangle ABC; let I be the center of ...

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