203 x, y, z (x, y, z) x 6 + y 6 + z 6 = 3xyz ( 203 5) a 0, b 0, c 0 a3 + b 3 + c 3 abc 3 a = b = c 3xyz = x 6 + y 6 + z 6 = (x 2 ) 3 + (y 2 ) 3

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2 203 x, y, z (x, y, z) x 6 + y 6 + z 6 = 3xyz ( 203 5) a 0, b 0, c 0 a3 + b 3 + c 3 abc 3 a = b = c 3xyz = x 6 + y 6 + z 6 = (x 2 ) 3 + (y 2 ) 3 + (z 2 ) 3 3x 2 y 2 z 2 ( ) 3xyz 3(xyz) 2. 0 xyz xyz ( ) xyz = 0 xyz =. x 2 = y 2 = z (x, y, z, ) = (0, 0, 0), (,, ), (,, ), (,, ), (,, ) ( ) 2 3 IV 2

3 2 x 2 + y 2 + z 2 = 2xyz x, y, z x = y = z = 0 ( ) x, y, z (i) x, y, z x 2 + y 2 + z 2 4 (ii) x, y, z 2 x 2 + y 2 + z (iii) x, y, z x 2 + y 2 + z [ ] k, l, m (i) x, y, z x 2 + y 2 + z 2 = (2k + ) 2 + (2l) 2 + (2m) 2 = 4(k 2 + k + l 2 + m 2 ) + x 2 + y 2 + z 2 4 (ii) x, y, z 2 x 2 + y 2 + z 2 = (2k + ) 2 + (2l + ) 2 + (2m) 2 = 4(k 2 + k + l 2 + l + m 2 ) + 2 x 2 + y 2 + z (iii) x, y, z x 2 + y 2 + z 2 = (2k + ) 2 + (2l + ) 2 + (2m + ) 2 = 4(k 2 + k + l 2 + l + m 2 + m) + 3. k 2 +k = k(k +), l 2 +l = l(l +), m 2 +m = m(m+) x 2 +y 2 +z x 2 + y 2 + z 2 = 2xyz (x, y, z) = (0, 0, 0) (x 0, y 0, z 0 ) x 0, y 0, z 0 0 x 0 \= 0 3

4 x y0 2 + z0 2 = 2x 0 y 0 z 0 x y0 2 + z0 2 (i) (iii) (ii) x y0 2 + z x 0y 0 z 0 4 x y0 2 + z0 2 = 2x 0 y 0 z 0 x 0, y 0, z 0 x 0 = 2x, y 0 = 2y, z 0 = 2z (x, y, z ) x 2 + y 2 + z 2 = 4x y z 2 x 2 + y 2 + z 2 4 x, y, z x = 2x 2, y = 2y 2, z = 2z 2 (x 2, y 2, z 2 ) 2 x y z 2 2 = 8x 2 y 2 z 2 3 x y z x 2, y 2, z 2 x 2 = 2x 3, y 2 = 2y 3, z 2 = 2z 3 (x 3, y 3, z 3 ) 3 x y z 2 3 = 6x 3 y 3 z 3 4 x 0, x, x 2,, y 0, y, y 2,, z 0, z, z 2, x 0 \= 0 x 0 > x > x 2 > > x n >. x 0 (0, 0, 0) 3 x 2 + y 2 + z 2 + w 2 = 2xyzw x, y, z, w ( 9 0 ( 3 ) ) x 2 + y 2 + z 2 + w 2 = 2xyzw x, y, z, w (0, 0, 0, 0) 2 2 x, y, z, w x 2 + y 2 + z 2 + w 2 x, y, z, w (i) x, y, z, w 2 x 2 + y 2 + z 2 + w (ii) x, y, z, w x 2 + y 2 + z 2 + w

5 n 3 x n + 2y n = 4z n x, y, z x = y = z = 0 (2000 ) 2 n ( ) x n + 2y n = 4z n () n = 2 ( ) x, y, z (2) n 3 ( ) x, y, z (2005 ) 2 x 2 + y 2 + z 2 = 2xyz x 2 + y 2 + z 2 = xyz 2006 (x, y, z) (A) : x, y, z x 2 + y 2 + z 2 = xyz x y z () (A) (x, y, z) y 3 (2) (a, b, c) (A) (b, c, z) (A) z (3) (A) (x, y, z) (2006 ) 5

6 2 AIME n n + 0 n (AIME 986 5) American Invitational Mathematics Examination Find the largest integer n such that n + 0 divides n x x + 0 x 2 0x + 00 ) x + 0 x x 3 + 0x 2 0x 2 0x 2 00x 00x x n = (n + 0)(n 2 0n + 0) 900. n n + 0 = n 2 0n n + 0. n n+0 n n+0 = 900 n = 890 6

7 5 m 4 + 4m 2 2m + m (203 ) x 4 + 4x 2 2x + ) 2x m 4 + 4m 2 2m + 2 x3 4 x x 57 6 x 4 + 4x 2 x x3 2 x3 + 4x 2 2 x3 4 x x x x m 4 + 4m 2 = (2m + ) m 4 + 4m 2 2m x 57 8 x ( 2 m3 4 m m 57 ) = 2 m3 4 m m (2m + ). = M (M ) 6M = 6(m4 + 4m 2 ) 2m + = 8m 3 4m 2 + 4m m + 6 2m + 6M = 6(m4 + 4m 2 ) 2m + m 4 + 4m 2 = M M 6M 2m + 6M ( ) 2m m + = ±, ±3, ±9, ±57 ( ) m = 29, 0, 2,, 0,, 9, 28 7 ( )

8 [ ] m 4 + 4m 2 = m 2 (m 2 + 4) m 2 2m + (2m + ) 2 m = m 2m + m 2 2m + m 4 + 4m 2 = m 2 (m 2 + 4) 2m + m m + m m + 8n n 2n + n (999 ) 8n n = 2n(4n ) 2n 2n + 4n n + n =, 3, 0 p 5 p 3 p 4 4 p (203 ) n 2 < n + + n n < 3 4 ( ) n + > n + 2 > > 2n > 2n n + + n n > n 2n = 2 8

9 y y = x n n + O A C B D n n + n + 2 2n 2n x y y = x n n + O A C B D n n + n + 2 2n 2n x 2 n + + n n ABDC 2 9

10 n + + n n < 2 (AB+CD)BD = ( 2 n + 2n = 3 4 ) (2n n) [ ] III y = x = n, x = 2n, y = 0 x 0

11 g 80g g 200g g 550 = 400g g 300 = 200g g 400 = 300g g 600

12 2760g 200g 0 300g g 5 500g 2 00g 300g 6 500g g 400g 200g 200g 2 400g 3 200g g x 300g x 2 200g 2 (400g ) x 3 500g x 4 300g 3 (900g ) x 5 x, x 2 0 x 3, x 4, x 5 200x + 300x x x x x + 3x 2 + 4x 3 + 5x 4 + 9x 5 28 L = 300x + 400x x x x 5 = 00(3x + 4x 2 + 5x 3 + 6x 4 + 0x 5 ) x, x 2, x 3, x 4, x 5 a = 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 L = 00(a + x + x 2 + x 3 + x 4 + x 5 ) a 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = a x + x 2 + x 3 + x 4 + x 5 2

13 3 a 6 x, x 2 0 x 3, x 4, x 5 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = a x + x 2 + x 3 + x 4 + x 5 m(a) l () a = 9l 3 a = (l ), m(9l 3) = l +. (2) a = 9l 2 a = (l ) = (l ), m(9l 2) = l +. (3) a = 9l a = (l ) = 4 2+9(l ), m(9l ) = l +. (4) a = 9l a = 9 l, m(9l) = l. (5) a = 9l + a = (l ), m(9l + ) = l +. (6) a = 9l + 2 a = 2 + 9l, m(9l + 2) = l +. (7) a = 9l + 3 a = 3 + 9l, m(9l + 3) = l +. (8) a = 9l + 4 a = 4 + 9l, m(9l + 4) = l +. (9) a = 9l + 5 a = 5 + 9l, m(9l + 5) = l +. [ ] l (i) l = a = 6 a = 6 = m(6) 2. x, x 2, x 3, x 4, x 5 0 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = 6 m(6) =\. m(6) = 2. a = 7, 8,..., 4 (ii) l = k a = 9(k + ) 3 = 9k + 6 a = k m(9k + 6) k + 2. m(9k + 6) k + 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = 9k + 6 x, x 2, x 3, x 4 x i ( i 4) 9k + 6 (i + ) = 2x + + (i + )(x i ) + + 9x 5 m(9k + 6 i ) x + + (x i ) + + x 5 = x + x 2 + x 3 + x 4 + x 5 k + = k. 3

14 i =, 2, 3, 4 9k + 6 (i + ) 9k + 6 (i + ) = 9k + 4, 9k + 3, 9k + 2, 9k + m(9k+6 i ) = k+ m(9k+6 i ) k m(9k + 6) = k + 2. (iii) (i) (ii) l () (9) a 6 x, x 2 0 x 3, x 4, x 5 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = a L = 300x + 400x x x x 5 = 00(a + x + x 2 + x 3 + x 4 + x 5 ) L a {L a } L 9l 3 < L 9l 2 < L 9l = L 9l < L 9l+ < L 9l+2 < L 9l+3 < L 9l+4 < L 9l+5 < L 9l+6 (l =, 2,... ). [ ] a 6 x, x 2 0 x 3, x 4, x 5 2x + 3x 2 + 4x 3 + 5x 4 + 9x 5 = a x + x 2 + x 3 + x 4 + x 5 m(a) 3 a = 9l 3, 9l 3, 9l 2, 9l, 9l +, 9l + 2, 9l + 3, 9l + 4, 9l+ L a = 00(a + l + ). a = 9l L a = 00(a + l). L 9l 3 < L 9l 2 < L 9l = L 9l < L 9l+ < L 9l+2 < L 9l+3 < L 9l+4 < L 9l+5. a 28 L 28 < L 29 < L 30 < L 3 < L 32 <... < L 9l 3 < L 9l 2 < L 9l = L 9l < L 9l+ < L 9l+2 < L 9l+3 < L 9l+4 < L 9l+5 <... {L a } (a 28) a = 28 = x = x 2 = x 3 = 0, x 2 = x 5 = 2 L 28 = 00( }{{} +) = =9 3+ 4