inmo-2004 previous year question papers of indian national mathematical olympiad (inmo) with...

Upload: akshay-pandey

Post on 04-Apr-2018

220 views

Category:

Documents


0 download

TRANSCRIPT

  • 7/29/2019 inmo-2004 Previous year Question Papers of Indian National Mathematical Olympiad (INMO) with solutions

    1/5

    " $ ' )

    1 2 4 6 8 9 @ B D F G I 6 8 Q D T U W G B F @ Y G a D F G Y d e g i q @ 8 r t @ I t x q q q G F D a t D @ B 6 @ 8 a 9 6 a t D 9 @ B D 9 d e q

    e g q g i q i d F D 9 D I a @ Q D Y 2 W 6 9 D

    G e i @ 9 D I a 9 x G a k m

    k d o k e o k g o k i m G 8 B

    I x o g i g e 2

    F 6 Q D a t G a d e g i @ 9 G

    ) z { |2

    " $ ' }

    A K B

    L

    CMD

    N

    Q

    Fig. 1.

    ~

    9 D F Q D a t G a x @ 9 G G F G Y D Y Y 6 F G q k @ 9 a t D @ B 6 @ 8 a 6 x G 8 B t D 8 I D G Y 9 6 G 9 9 D 9

    a t F 6 W t k 2 D a D a t D 6 @ 8 a 6 @ 8 a D F 9 D I a @ 6 8 6 d g G 8 B e i m G 8 B Y D a D a t D 6 @ 8 a 6 @ 8 a D F 9 D I a @ 6 8

    6 e i G 8 B 2

    4 6 8 9 @ B D F a t D a F @ G 8 Y D x 2 6 a D a t G a k @ 9 G F G Y Y D Y a 6 x G 8 B k @ 9 a t D @ B 6 @ 8 a 6 x 2

    D 8 I D @ 9 a t D @ B 6 @ 8 a 6 2 @ 8 I D @ 9 G F G Y Y D Y a 6 d g q @ a 6 Y Y 6 r 9 a t G a @ 9 a t D @ B 6 @ 8 a

    6 d g 2 6 r k @ 9 a t D I @ F I W I D 8 a F D 6 d e g G 8 B a t D D B @ G 8 e G 9 9 D 9 a t F 6 W t k 2

    D F D a t D F D

    G F D a r 6 6 9 9 @ @ Y @ a @ D 9

    @ d e g @ 9 G F @ t a a F @ G 8 Y D r @ a t d e g o G 8 B o k m G 8 B

    @ @ o k @ 8 r t @ I t I G 9 D e @ 9 D F D 8 B @ I W Y G F a 6 d g 2

    W 6 9 D d e g o G 8 B o k 2

    ~

    9 D F Q D a t G a k @ 9 a t D I @ F I W I D 8 a F D 6 a t D a F @ G 8 Y D i g e G 8 B

    t D 8 I D i g e o 2 @ @ Y G F Y i d e o 2 a 6 Y Y 6 r 9 a t G a d i g o b 2 G 8 B d e g i @ 9 G

    F D I a G 8 Y D 2 t @ 9 @ Y @ D 9 a t G a x @ 9 G F t 6 W 9 2

    D 8 I D x o 1 G 8 B a t @ 9 @ Q D 9 g i o g e 2

    t W 9 d e g i @ 9 G 9 U W G F D 2

    8 a t D 9 D I 6 8 B I G 9 D q 6 9 D F Q D a t G a e i @ 9 D F D 8 B @ I W Y G F a 6 d g q x @ 9 G F G Y Y D Y a 6 d g G 8 B @ 9

    G F G Y Y D Y a 6 e i 2

    D 8 I D @ a 6 Y Y 6 r 9 a t G a @ 9 D F D 8 B @ I W Y G F a 6 x 2 @ @ Y G F F D G 9 6 8 @ 8 9 t 6 r 9 a t G a

    x @ 9 G F D I a G 8 Y D 2

    e JEE Mains/JEE Advanced Online Prep and Guidance on www.100Marks.in

  • 7/29/2019 inmo-2004 Previous year Question Papers of Indian National Mathematical Olympiad (INMO) with solutions

    2/5

    9 @ 8 x o g i g e q r D D a a t G a g e i @ 9 9 @ @ Y G F a 6 x 2 8 G F a @ I W Y G F q x o

    g e i o 9 G 2 @ 8 I D @ 9 G F G Y Y D Y a 6 i e q r D G Y 9 6 D a e k x o 2 @ 8 I D x @ 9 G I I Y @ I

    U W G B F @ Y G a D F G Y r D G Y 9 6 D a x o x o 2

    9 @ 8 a t D G I a a t G a e i @ 9 G F G Y Y D Y a 6 x q r D

    D a k e o x o 2 @ 8 I D e i @ 9 D I a 9 g e d q r D G Y 9 6 t G Q D x e k o 2 t W 9

    k x o x e o e o k

    G 8 B e @ 9 G F G Y Y D Y a 6 k x 2 t @ 9 @ Q D 9 k @ 9 G F G Y Y D Y a 6 g G 8 B

    k o k o e o g

    a 6 Y Y 6 r 9 a t G a k g @ 9 G G F G Y Y D Y 6 F G 2 W a g o 2

    D 8 I D k o 2 t @ 9 @ Y @ D 9

    a t G a x @ 9 G 9 U W G F D 2 Y 9 6 6 9 D F Q D a t G a k x o G 8 B t D 8 I D g e d o k x o 2

    t @ 9 @ Q D 9 g i d o G 8 B t D 8 I D d e g i @ 9 G F D I a G 8 Y D3 2 @ 8 I D e d o e g q @ a 6 Y Y 6 r 9 a t G a d e g i

    @ 9 G 9 U W G F D 2

    2 W 6 9 D @ 9 G F @ D F D G a D F a t G 8 2 @ 8 B G Y Y G @ F 9 6 @ 8 a D D F 9

    9 G a @ 9 @ 8 a t D D U W G a @ 6 8

    o

    " $ ' }

    D r F @ a D a t D D U W G a @ 6 8 @ 8 a t D 6 F

    o

    D 8 I D

    o

    @ 9 G 8 @ 8 a D D F 2 t @ 9 9 t 6 r 9 a t G a

    B @ Q @ B D 9

    G 8 B t D 8 I D G Y 9 6 B @ Q @ B D 9

    2 W a q r D

    t G Q D

    o

    o

    a 6 Y Y 6 r 9 a t G a

    B @ Q @ B D 9

    2 @ 8 I D @ 9 G F @ D q a t D 6 8 Y B @ Q @ 9 6 F 9 6

    G F D 1

    G 8 B

    2

    @ 8 I D q r D G Y 9 6 t G Q D o

    1 6 F

    2

    z ) | }

    W 6 9 D o

    1 2

    ~

    Q @ 6 W 9 Y

    o 1 @ 9 8 6 a 6 9 9 @ Y D 2 8 G I a q r D D a 1 o

    o

    1

    o

    1 r t @ I t @ 9 @ 6 9 9 @ Y D 2

    ~

    8 a t D 6 a t D F t G 8 B

    o

    1 @ Q D 9

    o

    1 o

    o

    1 G 8 B

    o

    1

    1 o 2

    t W 9

    o

    o

    2 t W 9

    o

    1

    r t D 8 o

    1 2 @ @ Y G F Y q

    o

    o

    r t @ I t @ 9 8 6 a 6 9 9 @ Y D 2 4 6 8 9 @ B D F @ 8

    o

    q r D D a

    o

    6 F

    o

    o 2

    D 8 I D

    o

    1

    r t D F D o

    1 2

    D a W 9 I 6 8 9 @ B D F

    o

    2

    D 8 I D

    o

    o

    G 8 B 8 D @ a t D F 8 6 F

    @ 9 B @ Q @ 9 @ Y D

    2

    o

    q a t D 8

    o

    o

    1

    1 2

    D 8 I D

    o

    1

    1

    1 o

    1 2 s t @ 9 @ Q D 9

    o

    2 G @ 8

    o

    1

    1

    r t D 8 o

    1 2

    z ) | }

    W 6 9 D o

    1 2

    o 1 q a t D 8

    o

    6 F

    o

    1 2

    D G Y 9 6 D a

    o

    1

    o

    1

    1

    1

    o

    e JEE Mains/JEE Advanced Online Prep and Guidance on www.100Marks.in

  • 7/29/2019 inmo-2004 Previous year Question Papers of Indian National Mathematical Olympiad (INMO) with solutions

    3/5

    G 8 B r D D a a t D 9 6 Y W a @ 6 8

    o

    1

    2

    ~

    8 a t D 6 a t D F t G 8 B

    o

    1 B 6 D 9 8 6 a t G Q D

    G 8 9 6 Y W a @ 6 8 @ 8 a D F G Y 9 6 Y W a @ 6 8 6 F

    2 @ @ Y G F Y q a t D F D @ 9 8 6 9 6 Y W a @ 6 8 @ 8 a t D I G 9 D

    o 2 G @ 8

    o

    q r D D a

    o

    G 8 B t D 8 I D

    o 2

    D D a a t D 9 6 Y W a @ 6 8

    o

    1

    2

    o

    q a t D 8

    o

    o

    1

    @ Q @ 8

    o

    1

    1 G 8 B t D 8 I D

    o

    1

    1

    1 o

    1 2 t @ 9 @ Q D 9

    o

    G 8 B t D 8 I D

    o

    1

    2

    @ 8 G Y Y

    o

    B 6 D 9 8 6 a t G Q D G 8 9 6 Y W a @ 6 8 2

    2 @ 9 G F D G Y F 6 6 a 6 a t D D U W G a @ 6 8

    o q F 6 Q D a t G a o 2 6 F G 8 F D G Y 8 W D F

    q r D B D 8 6 a D

    a t D F D G a D 9 a @ 8 a D D F 8 6 a D T I D D B @ 8

    2

    " $ ' } W 6 9 D @ 9 G F D G Y F 6 6 a 6 a t D @ Q D 8 D U W G a @ 6 8 2 t D 8

    o

    t @ 9 @ Q D 9

    1 o 1 G 8 B t D 8 I D

    1

    1 o 1 2

    ~

    9 D F Q D a t G a

    1

    o

    2

    1 q a t D 8

    q @ Q @ 8

    G 8 B t D 8 I D

    1

    1 2

    1 q a t D 8

    o

    1 G 8 B t D 8 I D

    1 2

    t @ 9 G G @ 8 @ Q D 9

    1

    1 2

    t D G 6 Q D F D 9 6 8 @ 8 9 t 6 r 9 a t G a 6 F q r D t G Q D

    1 G 8 B t D 8 I D I G 8 8 6 a D D U W G Y

    a 6 1 2

    D I 6 8 I Y W B D a t G a G F D G Y F 6 6 a 6

    o @ 9 6 9 @ a @ Q D

    6 Q @ 6 W 9 Y o 2

    6 r W 9 @ 8

    o q r D D a

    o

    t D 9 a G a D D 8 a

    o @ 9 D U W @ Q G Y D 8 a a 6 2

    4 6 8 9 @ B D F

    2 @ 8 I D q a t @ 9 @ 9 D U W @ Q G Y D 8 a a 6

    2

    9 @ 8 a t D

    F D Y G a @ 6 8

    1 q r D I G 8 r F @ a D

    6 F

    2 F D G a @ 8 a t @ 9 G 9 G U W G B F G a @ I q r D

    D a a t @ 9 @ 9 D U W @ Q G Y D 8 a a 6

    1

    2 6 r 6 9 D F Q D a t G a @ a t D 8 1 o

    1

    1

    r t @ I t @ 9 @ 6 9 9 @ Y D 2

    1

    q a t D 8 1 o

    1

    1 r t @ I t G G @ 8 @ 9 @ 6 9 9 @ Y D 2

    D I 6 8 Y W B D a t G a

    1

    2 @ @ Y G F Y

    @ 9 D U W @ Q G Y D 8 a a 6

    r t @ I t @ 9 D U W @ Q G Y D 8 a a 6

    2 W a a t @ 9 @ 9

    1

    r t @ I t @ 9 Q G Y @ B 2

    D 8 I D

    G 8 B r D D a

    o 2

    2 D a B D 8 6 a D a t D I @ F I W F G B @ W 9 6 G a F @ G 8 Y D d e g m

    q

    q @ a 9 9 @ B D 9 e g q g d q d e m G 8 B q q @ a 9

    D T F G B @ @ 6 6 9 @ a D d q e q g 2 q F 6 Q D a t G a

    @

    G 8 B

    m

    @ @ G 8 B 2

    " $ ' } D 8 6 r a t G a o

    G 8 B o

    q r t D F D

    G F D a t D 9 @ B D 9 6 a t D a F @ G 8 Y D d e g q

    o

    G 8 B @ 9 a t D G F D G 6 d e gP 2 t W 9 a t D @ Q D 8 I 6 8 B @ a @ 6 8

    a F G 8 9 Y G a D 9 a 6

    e JEE Mains/JEE Advanced Online Prep and Guidance on www.100Marks.in

  • 7/29/2019 inmo-2004 Previous year Question Papers of Indian National Mathematical Olympiad (INMO) with solutions

    4/5

    W a a @ 8

    o

    o

    o q r D D a

    o

    o

    o

    G 8 B a t D I 6 8 B @ a @ 6 8 8 6 r

    @ 9

    W a

    D F 6 8 9 6 F W Y G @ Q D 9 q

    o

    o

    2

    D 6 a G @ 8

    2 T G 8 B @ 8 G 8 B D D I a @ 8 9 6 D I G 8 I D Y Y G a @ 6 8 9 q r D D a

    W 6 9 D

    2 t @ 9 @ Y @ D 9 a t G a

    G 8 B t D 8 I D 2 t @ 9 @ Y @ D 9 a t G a

    G 8 B

    @ Q @ 8

    o

    r t @ I t

    I 6 8 a F G B @ I a 9

    2 @ @ Y G F Y q

    @ 9 G Y 9 6 8 6 a 6 9 9 @ Y DS 2 t @ 9 F 6 Q D 9

    @ 2

    W 6 9 D

    2 9 G 6 Q D a t @ 9 a G D 9 a t D 6 F

    @ 8 I D

    G 8 B

    q r D t G Q D

    2 t W 9

    G 8 B

    2

    D 8 I D

    o

    r t @ I t I 6 8 a F G B @ I a 9

    2

    D 8 I D

    2 @ @ Y G F Y q r D I G 8 F 6 Q D a t G a

    2 t @ 9 F 6 Q D 9

    @ @

    2 D a B D 8 6 a D a t D 9 D a 6 G Y Y

    a W Y D 9

    6 6 9 @ a @ Q D @ 8 a D D F 9 9 W I t a t G a

    o

    2 4 6 8 9 @ B D F a t D 9 D a

    o

    @ 8 B a t D F D G a D 9 a I 6 6 8 B @ Q @ 9 6 F 6 G Y Y a t D D D F 9 6 2

    " $ ' }

    D 9 t 6 r a t G a a t D F D U W @ F D B I B @ 9 2 4 6 8 9 @ B D F G 8 D Y D D 8 a

    2

    D t G Q D

    o

    D

    F 9 a 6 9 D F Q D a t G a 8 6 a G Y Y

    I G 8 D 6 B B 2

    ~

    a t D F r @ 9 D q r D t G Q D

    1

    6 B G 8 B t D 8 I D

    6 B q r t @ I t @ 9 @ 6 9 9 @ Y D D I G W 9 D 8 6 9 U W G F D I G 8 D I 6 8 F W D 8 a a 6

    6 B W Y 6 2 t W 9 G a Y D G 9 a 6 8 D 6

    @ 9 D Q D 8 2

    @ @ Y G F Y @ 8 6 8 D 6

    @ 9 B @ Q @ 9 @ Y D q a t D 8

    1

    6 B G 8 B t D 8 I D

    6 B r t @ I t G G @ 8 @ 9 @ 6 9 9 @ Y D D I G W 9 D 8 6 9 U W G F D @ 9 I 6 8 F W D 8 a a 6 6 B W Y 6 2 t W 9

    B @ Q @ B D 9

    2

    t D F D G F D 9 D Q D F G Y 6 9 9 @ @ Y @ a @ D 9 6 F

    2

    z ) | }

    W 6 9 D 6 8 D 6 a t D @ 9 D Q D 8 G 8 B a t D 6 a t D F 6 W F G F D 6 B B m 9 G

    @ 9 D Q D 8 q

    G F D 6 B B 2

    t D 8

    6 B 2

    6 B q a t D 8

    6 B G 8 B t D 8 I D

    @ Q @ 8

    2

    6 B q a t D 8

    6 B r t @ I t G G @ 8 @ Q D 9 a t G a

    G 8 B

    9 6 a t G a

    2 a 6 Y Y 6 r 9 a t G a

    G 8 B t D 8 I D

    2

    z ) | }

    W 6 9 D

    G F D D Q D 8 G 8 B

    G F D 6 B B 2 t D 8

    6 B 2 @ 8 I D

    6 F 6 B W Y 6 q @ a 6 Y Y 6 r 9 a t G a

    6 F !

    6 B r t @ I t @ 9 @ 6 9 9 @ Y D 2

    D 8 I D a t @ 9 I G 9 D B 6 D 9 8 6 a

    G F @ 9 D 2

    z ) | " }

    a t F D D 6

    G F D D Q D 8 G 8 B a r 6 6 B B q a t D 8

    G 8 B t D 8 I D

    2

    z ) | }

    6 W F 6

    G F D D Q D 8 q a t D 8 G G @ 8

    G 8 B

    2

    D F D G G @ 8 6 F G 8 9 @ T

    a W Y D

    @ 8 q r D 6 9 D F Q D a t G a

    2 @ 8 I D

    1

    1

    1

    o

    D 9 D D a t G a

    1 q 1 q 1 q q q G 8 B t D 8 I D 2 t W 9 @ 9 a t D I B 6 2

    e JEE Mains/JEE Advanced Online Prep and Guidance on www.100Marks.in

  • 7/29/2019 inmo-2004 Previous year Question Papers of Indian National Mathematical Olympiad (INMO) with solutions

    5/5

    2

    F 6 Q D a t G a a t D 8 W D F 6

    a W Y D 9 6 6 9 @ a @ Q D @ 8 a D D F 9

    9 G a @ 9 @ 8 a t D D U W G a @ 6 8

    o

    @ 9 G 8 $ $

    @ 8 a D D F 2

    " $ ' } D r F @ a D a t D D U W G a @ 6 8 @ 8 a t D 6 F

    1

    1

    1

    1

    1

    o

    1

    t D 8 W D F 6 Q D a W Y D

    r t @ I t 9 G a @ 9 a t D @ Q D 8 F D Y G a @ 6 8 G 8 B 6 F r t @ I t

    o

    @ 9 D Q D 8 q

    D I G W 9 D 6 F @

    @ 9 G 9 6 Y W a @ 6 8 q a t D 8 9 6 @ 9

    r t @ I t @ 9 B @ 9 a @ 8 I a F 6

    2

    @ @ Y G F Y a t D 8 W D F 6 Q D a W Y D 9 r t @ I t 9 G a @ 9 a t D D U W G a @ 6 8 G 8 B 6 F r t @ I t o

    @ 9 G Y 9 6 D Q D 8 2

    D 8 I D @ a 9 W ( I D 9 a 6 I 6 W 8 a 6 8 Y a t 6 9 D Q D a W Y D 9

    6 F r t @ I t

    o

    o

    2 t W 9 a t D

    D U W G a @ 6 8 F D B W I D 9 a 6

    1

    o

    1

    D F D G G @ 8 a t D a W Y D

    6 F r t @ I t

    o @ 9 D Q D 8 D I G W 9 D r D I G 8 G 9 9 6 I @ G a D B @ D F D 8 a 9 6 Y W a @ 6 8

    a 6 a t @ 9 Q D a W Y DS 2 t W 9 @ a 9 W ( I D 9 a 6 I 6 8 9 @ B D F a t D D U W G a @ 6 8

    1

    o

    1

    G 8 B 9 t 6 r a t G a a t D 8 W D F 6 G @ F 9

    9 G a @ 9 @ 8 a t @ 9 D U W G a @ 6 8 @ 9 6 B B 2

    t @ 9 F D B W I D 9 a 6

    o

    6 F

    o 1

    W a 6 9 D F Q D a t G a

    1 o 1 2 1 o 2

    o 2

    o

    2

    o 1 2 1 o 2

    o

    2 o

    2 o 1 2 1

    6 a D a t G a 8 6 G I a 6 F @ 9 G a @ 6 8 6 1 G 9 F 6 B W I a 6 a r 6 8 D G a @ Q D 8 W D F 9 @ D Y B G 6 9 @ a @ Q D a W Y D

    2

    D 8 I D r D D a a t D 9 D 9 6 Y W a @ 6 8 9 2 t @ 9 F 6 Q D 9 a t G a a t D a 6 a G Y 8 W D F 6 Q D a W Y D 9

    9 G a @ 9 @ 8 a t D @ Q D 8 D U W G a @ 6 8 @ 9 6 B B 2

    e JEE Mains/JEE Advanced Online Prep and Guidance on www.100Marks.in