All Russian Mathematical Olympiad

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   17-th All-Russian Mathematical Olympiad 1991

   www.imomath.com/othercomp/Rus/RusMO491.pdf

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   17-th All-Russian Mathematical Olympiad 1991. Final (Fourth) Round – March 22–29. Grade 9. First Day. 1. Find the locus of the foci of the parabolas given by y ...
   
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   28-th All-Russian Mathematical Olympiad 2002

   www.imomath.com/othercomp/Rus/RusMO402.pdf

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   28-th All-Russian Mathematical Olympiad 2002. Fourth Round. Grade 8. First Day. 1. Can a natural number be written in each square of a 9× 2002 rectangular ...
   
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   21-th All-Russian Mathematical Olympiad 1995

   www.imomath.com/othercomp/Rus/RusMO495.pdf

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   21-th All-Russian Mathematical Olympiad 1995. Fourth Round. Grade 9. First Day. 1. If x and y are positive numbers, prove the inequality x x4 + y2 + y y4 + x2 ≤ ...
   
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   25-th All-Russian Mathematical Olympiad 1999

   www.imomath.com/othercomp/Rus/RusMO99.pdf

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   25-th All-Russian Mathematical Olympiad 1999. Final Round. Grade 9. First Day. 1. The decimal digits of a natural number A form an increasing sequence ...
   
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   18-th All-Russian Mathematical Olympiad 1992

   www.imomath.com/othercomp/Rus/RusMO492.pdf

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   18-th All-Russian Mathematical Olympiad 1992. Final (Fourth) Round – March 22–29. Grade 9. First Day. 1. Solve the system of equations x3 − 5 y2 x. = 6 ...
   
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   31-st All-Russian Mathematical Olympiad 2005

   www.imomath.com/othercomp/Rus/RusMO05.pdf

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   31-st All-Russian Mathematical Olympiad 2005. Final Round – Nizhniy Novgorod, April 24–29. Grade 9. First Day. 1. A parallelogram ABCD with AB < BC is ...
   
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   15-th All-Russian Mathematical Olympiad 1989

   www.imomath.com/othercomp/Rus/RusMO489.pdf

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   15-th All-Russian Mathematical Olympiad 1989. Final (Fourth) Round. Grade 8. First Day. 1. Find three different natural numbers in an arithmetic progression ...
   
8. [PDF] 

   19-th All-Russian Mathematical Olympiad 1993

   www.imomath.com/othercomp/Rus/RusMO93.pdf

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   19-th All-Russian Mathematical Olympiad 1993. Final Round – Anapa, April. Grade 9. First Day. 1. A natural number n is such that 2n + 1 and 3n + 1 are perfect ...
   
9. [PDF] 

   22-nd All-Russian Mathematical Olympiad 1996

   www.imomath.com/othercomp/Rus/RusMO96.pdf

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   22-nd All-Russian Mathematical Olympiad 1996. Final Round – Ryazan', April 19–20. Grade 9. First Day. 1. What numbers are more numbered among the ...
   
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    20-th All-Russian Mathematical Olympiad 1994

    www.imomath.com/othercomp/Rus/RusMO94.pdf

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    20-th All-Russian Mathematical Olympiad 1994. Final Round – Tver, April 19–25. Grade 9. First Day. 1. Prove that if (x + √x2 + 1)(y + √y2 + 1) = 1, then x + y = 0 ...
    
11. [PDF] 

    34-th All-Russian Mathematical Olympiad 2008

    www.imomath.com/othercomp/Rus/RusMO08.pdf

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    34-th All-Russian Mathematical Olympiad 2008. Final Round – Kislovodsk, April 19–24. Grade 9. First Day. 1. Do there exist 14 positive integers such that, upon ...
    

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12. [PDF] 

    24-th All-Russian Mathematical Olympiad 1998

    www.imomath.com/othercomp/Rus/RusMO498.pdf

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    24-th All-Russian Mathematical Olympiad 1998. Fourth Round. Grade 8. First Day. 1. Do there exist n-digit numbers M and N such that all digits of M are even, all ...
    
13. [PDF] 

    23-rd All-Russian Mathematical Olympiad 1997

    www.imomath.com/othercomp/Rus/RusMO97.pdf

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    23-rd All-Russian Mathematical Olympiad 1997. Final Round. Grade 9. First Day. 1. Let P(x) be a quadratic polynomail with nonnegative coefficients. Prove ...
    
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    26-th All-Russian Mathematical Olympiad 2000

    www.imomath.com/othercomp/Rus/RusMO00.pdf

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    26-th All-Russian Mathematical Olympiad 2000. Final Round – Kazan, April 14–15. Grade 9. First Day. 1. Let a, b, c be distinct numbers such that the equations ...
    
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    35-th All-Russian Mathematical Olympiad, Kislovodsk, April, 21–27 ...

    www.imomath.com/othercomp/Rus/RusMO09.pdf

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    35-th All-Russian Mathematical Olympiad, Kislovodsk,. April, 21–27, 2009. Final Round. Grade 9. First Day. 1. The denominators of two irreducible fractions are ...
    

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16. [PDF] 

    21-st All-Russian Mathematical Olympiad 1995

    www.imomath.com/othercomp/Rus/RusMO95.pdf

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    21-st All-Russian Mathematical Olympiad 1995. Final Round – Saratov. Grade 9. First Day. 1. A freight train departed from Moscow at x hours and y minutes and ...
    

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17. [PDF] 

    33-rd All-Russian Mathematical Olympiad 2007

    www.imomath.com/othercomp/Rus/RusMO07.pdf

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    33-rd All-Russian Mathematical Olympiad 2007. Final Round – Maykop, April 23–28. Grade 8. First Day. 1. If a, b, c are real numbers, show that at least one of ...
    
18. [PDF] 

    26-th Inter-republic Mathematical Olympiad 1992

    www.imomath.com/othercomp/Rus/RusMO92.pdf

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    26-th Inter-republic Mathematical Olympiad 1992. Alma-Ata, April 16–23. Grade 9. First Day. 1. Prove that if x, y, z are positive real numbers, then x4 + y4 + z2 ≥ ...
    
19. [PDF] 

    29-th All-Russian Mathematical Olympiad 2003

    www.imomath.com/othercomp/Rus/RusMO03.pdf

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    29-th All-Russian Mathematical Olympiad 2003. Final Round – Gorod Oryol, April 14–20. Grade 9. First Day – April 15. 1. Suppose that M is a set of 2003 ...
    
20. [PDF] 

    19-th All-Russian Mathematical Olympiad 1993

    www.imomath.com/othercomp/Rus/RusMO493.pdf

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    19-th All-Russian Mathematical Olympiad 1993. Fourth Round. Grade 9. First Day. 1. If a and b are positive numbers, prove the inequality a2 + ab + b2 ≥ 3(a + b ...
    
21. [PDF] 

    20-th All-Russian Mathematical Olympiad 1994

    www.imomath.com/othercomp/Rus/RusMO494.pdf

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    20-th All-Russian Mathematical Olympiad 1994. Fourth Round. Grade 9. First Day. 1. One day, Rabbit was about to go for a meeting with Donkey, but Winnie ...
    
22. [PDF] 

    24-th All-Russian Mathematical Olympiad 1998

    www.imomath.com/othercomp/Rus/RusMO98.pdf

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    24-th All-Russian Mathematical Olympiad 1998. Final Round. Grade 9. First Day. 1. Each of the rays y = x and y = 2x (x ≥ 0) cuts off an arc from a given parabola ...
    
23. [PDF] 

    28-th All-Russian Mathematical Olympiad 2002

    www.imomath.com/othercomp/Rus/RusMO02.pdf

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    28-th All-Russian Mathematical Olympiad 2002. Final Round – Maykop, April 21–29. Grade 9. First Day – April 23. 1. Can the numbers from 1 to 20022 be ...
    
24. [PDF] 

    29-th All-Russian Mathematical Olympiad 2003

    www.imomath.com/othercomp/Rus/RusMO403.pdf

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    29-th All-Russian Mathematical Olympiad 2003. Fourth Round. Grade 8. First Day. 1. The numbers 1 through 10 are to be divided into two groups so that the ...
    
25. [PDF] 

    25-th All-Soviet Mathematical Olympiad 1991

    www.imomath.com/othercomp/Rus/RusMO91.pdf

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    25-th All-Soviet Mathematical Olympiad 1991. Smolensk, April 17–24. Grade 9. First Day. 1. Find all integer solutions of the system. { xz − 2yt = 3, xt + yz = 1. (Yu.
    
26. [PDF] 

    32-nd All-Russian Mathematical Olympiad 2006

    www.imomath.com/othercomp/Rus/RusMO06.pdf

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    32-nd All-Russian Mathematical Olympiad 2006. Final Round – Pskov, April 21–26. Grade 9. First Day. 1. Some pairs of centers of neighboring squares (by side) ...
    
27. [PDF] 

    16-th All-Russian Mathematical Olympiad 1990

    www.imomath.com/othercomp/Rus/RusMO490.pdf

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    16-th All-Russian Mathematical Olympiad 1990. Final (Fourth) Round. Grade 9. First Day. 1. Among 25 apparently equal coins 3 are fakes. All true coins have ...
    
28. [PDF] 

    27-th All-Russian Mathematical Olympiad 2001

    www.imomath.com/othercomp/Rus/RusMO01.pdf

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    27-th All-Russian Mathematical Olympiad 2001. Final Round – Tver, April 21–22. Grade 9. First Day. 1. The integers from 1 to 999999 are partioned into two ...