TRI Hẩ HNG VNG LN TH X THI MễN TON TRNG PT VNG CAO VIT BC KHI 10 THI XUT ny cú 01 trang, gm 05 cõu Cõu (4,0 im) Gii phng trỡnh sau trờn s thc 2(x + x - 1)2 + 2x + 2x = + + 4x Cõu (4,0 im) Cho t giỏc ABCD ngoi tip ng trũn (I) Cỏc tip im ca (I) vi AB, BC, CD, DA ln lt l E, F, G, H Chng minh AC, BD, EG, HF ng quy ti mt im ( ) Cõu (4,0 im) Gi s a thc P x = x + ax + b cú nm nghim x 1, x , x , x , x t f ( x ) = x - Chng minh rng f ( x ) ìf ( x ) ìf ( x ) ìf ( x ) ìf ( x ) - 243 Cõu (4,0 im) Cho 2014 hp m mi hp ny u cha ỳng 40 phn t Bit rng hai tu ý cỏc ny u cú ỳng mt phn t chung Hóy chng minh rng tn ti phn t thuc tt c 2014 ó cho Cõu (4,0 im) Chng t rng tng A= 2014 2014 2014 2014 + + + + + (2013 s hng) 20132 + 20132 + 20132 + n 2013 + 2013 khụng phi l s nguyờn dng - HT Ngi (Phm Th Lan in thoi 0982217044) TRI Hẩ HNG VNG LN TH X P N THI MễN TON TRNG PT VNG CAO VIT BC KHI 10 P N THI XUT ỏp ỏn ny cú 04 trang, gm 05 cõu Cõu ỏp ỏn Cõu Gii phng trỡnh sau trờn s thc 2(x + x - 1)2 + 2x + 2x = + + iu kin: x - im 4,0 + 4x 0,5 + t y = x + x - ta cú phng trỡnh: y + y - = - + + 4x + t z = - + + 4x ta cú phng trỡnh: z + z - 1=x ,z 2 ỡù x + x - = y ỡù x (x + 1) = y + ùù ùù ù + Ta cú h phng trỡnh: y + y - = z (1) ùớ y (y + 1) = z + (2) ùù ùù ùù z + z - = x ùù z (z + 1) = x + ợ ợ Do iu kin z suy y - 1, x - T (2) suy xyz = Mt khỏc, cng v vi v ca (1): x + y + z = ịÊ 3 x 2y 2z xyz ng thc xy v ch x = y = z = ị x = y = z = (vỡ x , y, z - ) Vy phng trỡnh ó cho cú nghim l x = 0,5 0,5 0,5 0,5 0,5 0,5 0,5 Cõu Cho t giỏc ABCD ngoi tip ng trũn (I) Cỏc tip im ca (I) vi AB, BC, CD, DA ln lt l E, F, G, H Chng minh AC, BD, EG, HF ng quy ti mt im 4,0 A E B H M D F G 1,5 C Gi M l giao ca AC v HF, M Â l giao ca AC v EG ã ã Vỡ sin Aã HM = sin BFM nờn ta cú = sin FMC S VA BM sin Aã MH A M A H A H HM sin Aã HM AM AH = = ị = ã S VMFC MC FC sin Aã MH MF MC MF FC sin BFM AM ' AE Tng t = (2) M 'C GC Do A H = A E , CF = CG nờn t (1), (2) ta cú (1) 0,5 1,0 AM AM ' = ị M M ' hay AC, EG, HF ng quy ti M MC M 'C Tng t ta cng cú BD, HF, EG ng quy ti M 1,0 Hay AC, BD, EG, HF ng quy Cõu Gi s a thc P x = x + ax + b cú nm nghim x , x , x , x , x ( ) ( ) t f x = x - Chng minh rng 4,0 f ( x ) ìf ( x ) ìf ( x ) ìf ( x ) ìf ( x ) - 243 ( ) Vỡ x 1, x , x , x , x l nghim ca P x nờn 1,0 P ( x ) = ( x - x1 ) ( x - x ) ( x - x ) ( x - x ) ( x - x ) f ( x1 ) f ( x ) f ( x ) f ( x ) f ( x ) ( = x1 - )( x1 + )( x2 - )( x2 + ) ( K x5 - )( x5 + ) 1,0 ( 3) ( x ) ( = P ( ) ìP ( - ) = ( - = x1 - K x5 - ) ( K x5 + + 1,0 ) )( + 3a + b + 3a + b ) 1,0 = ( 3a + b) - 243 - 243 Cõu Cho 2014 hp m mi hp ny u cha ỳng 40 phn t Bit rng hai tu ý cỏc ny u cú ỳng mt phn t chung Hóy chng 4,0 minh rng tn ti phn t thuc tt c 2014 ó cho Xột A tu ý 2014 ó cho Vỡ A cú phn t chung vi tng 2013 cũn li nờn A phi tn ti phn t a no ú thuc ớt nht 51 2013 cũn li (vỡ ngc 1,0 li, nu mt 40 phn t ca A ch thuc 50 cũn li thỡ s ó cho khỏc A s l 40.50 = 2000 < 2013 ) Nh vy, phn t a thuc 52 A , A1, A2, , A51 Chng minh a thuc bt k B 2014 ó cho Vỡ hai tu ý cỏc ó cho cú ỳng mt phn t chung nờn cỏc 1,0 A , A1, A2, , A 51 khụng th cú phn t chung no khỏc a Gi s phn t a khụng thuc B Khi ú vi mi Ai ( Ê i Ê 51) , B phi cú phn t chung a v cỏc phn t ny cng phi khỏc (Nu ( ) = a j thỡ cỏc Ai , A j Ê i, j Ê 52 s cú ớt nht hai phn t chung l 1,0 a v ) Bi vy B cha khụng ớt hn 52 phn t (trỏi gi thit) Do ú a thuc B ( ) Vỡ B l bt k 1962 cũn li B A, B Ai ( Ê i Ê 51) nờn a thuc tt c 2014 ó cho Cõu Chng t rng tng A= 2014 2014 2014 2014 + + + + + 20132 + 20132 + 20132 + n 2013 + 2013 (2013 s hng) 1,0 4,0 khụng phi l s nguyờn dng Trc ht ta gii bi toỏn tng quỏt: Chng minh rng tng (n s hng, n > ) A= n+1 n+1 n+1 + + + (n s hng) n +1 n +2 n +n khụng phi l s nguyờn dng n+1 n+1 n+1 Ta cú A < + + + (n s hng) 2 n n n2 n+1 = n = + < n n2 n+1 n+1 n+1 + + + Mt khỏc A > (n s hng) n +n n +n n +n n+1 = n = n +n Do ú < A < Vy A khụng phi l s nguyờn dng Vi n = 2013 thỡ ta cú bi toỏn ó cho 1,0 0,5 0,5 0,5 0,5 1,0