30 đề thi học sinh giỏi toán lớp 9

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 x y z x y z + + − = − + − + − AB+&,() * -.@\.3           x x P x x + − = + + + − − DF # 4 x = AB,&-() * %.2-C>'*.GC$XP-C1Q1∈> 23H    AD AB AC = + \EC1T(9P-A%(.IE.L7.M.2W.E(%{ AB-&+() * 6 §Ò 8      %2S\+:02%%2d\≤ -.@p^.J2\.3   Q a b a b = + + + ./0/12/3/1C0'556D'55 V*.9:1 290 phút (không kể thời gian phát đề) (Học sinh không phải chép đề vào giấy thi) AB!Q#3H'F0?.ZM.  4 #  !  # 4n n n n+ + + − 2E.%4$ AB'Q#]-@-80?2'(\2.3C` 4 #   #x x x ax b− + + + T(\9: J2=.2.3$ AB+Q# 2 3HDF0?.Z2S\SS+.2A ( ) ( ) ( )      ab cd a c b d+ ≤ + + \ DF2 ≥ B\ ≥ Bf$3H ( ) ( ) c a c c b c ab− + − ≤ AB,Q4 #* = + + + − +Ruùt goïn 4 10 2 5 4 10 2 5B E* 5\.302•-.@p^.S.-.@p^.• C x x= − − AB-Q# %.2-C>QC>gCSVT(.MG>'u>t⊥CVS)⊥CV$ ]-@'@.J2V.MG>.W>td)TF^.$ ABFQ4%'*C>1G\H2$~.<L2Oy.-GC>'( C1Ir%+(.Gz'([$ 2€3H1[$>zI*W$ \€3H   DE AE BF AF = )•6-%'Mp'q.& ~U.V*%- $%! (2 điểm)Đồng chí hãy hướng dẫn học sinh giải bi tập sau 0?.ZM02%%.W0?A'F-m0?J2A\H& 7 .G1H §Ò 9 §Ò 10      $%' (3 điểm) 5SSe.%-._902 2€5``#e'(5  d  de  `44 \€5  d  de  d4#4& ≤ 4Q45d"d4e $%+& 2 điểm) %2S\S.%7/2 # d\ # d # `#2\'(2d\d`!4$ -.@J2\.3X`Q2  & dQ\&  dQ & $%,(3 điểm)%.2-C>A+8.I*W$~VBBX .=C>S>SC02%%  k PA CP NC BN MB AM === QIf 23 SinAACABS ABC $$   = 2 3H  Q + = k k S S AMP \I VX p^.  Câu 1Q2C` #  #  −− + ++ \%0- &   & + '( & + Câu 2Q2679:.5  d5d +x `#! 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(Đề 19) đề khảo sát chất lợng HSG lớp 9 Môn: Toán Bài 1: (2 điểm): a) Chứng minh rằng: 2( x 2 + 6 x + 1) = ( ) ( 4 x +1 + ) 4 x 1 vi mi x khụng õm; x 2 + 5x = 1 x b) Gii h phng trỡnh: 4 4 x +1 + x 1 = 0 ( ) ( ) c) Gii phng trỡnh: 10 x 2 5 x 2 3 x 10 + 3 x = 0 Bi 2 (2 im): a) Tỡm ba s nguyờn t liờn tip a, b, c sao cho a 2 + b 2 + c 2 cng l s nguyờn t? 2 2 b) Cho hai s x, y tho món: x + 9. .. xỳc trong vi ng trũn (O) ng trũn (O4) tip xỳc vi 2 tia CA v CD v tip xỳc ngoi vi ng trũn (O1) Tớnh bỏn kớnh ca cỏc ng trũn (O1), (O2), (O3), (O4) theo R Đề 18 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Câu 1: a) Tính giá trị của biểu thức: A= 11 + 3 59 4+ 5+ 3 3+ 5+ 5+ 3 3 5+ 5+ 3 b) Cho s N=k4+2k3 16k2 2k+15 vi k l s nguyờn Tỡm iu kin ca k s N chia ht cho 16 Cõu 2: Cho biu thc: M = iu kin: x2 +... EF (H EF) Chng minh rng ABH = ACH Cõu 3: 1) Chng minh rng F(n) = 4n + 15n 1 luụn chia ht cho 9 vi mi n nguyờn dng 2) Tỡm tt c cỏc cp s nguyờn (x; y) tho món x3 = y3 +2y2+1 Cõu 4: Cho a, b, c 1 v a3+b3+c3 = 6 Tỡm giỏ tr ln nht ca biu thc A = a2 + b2 + c2 Đề 24 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Câu 1: a) Cho a, b > 0, c 0 Chng minh rng: 1 1 1 + + = 0 a+b = a+c + b+c a b c 1 1 x2 + y... Bi 4 (2 im): Tỡm giỏ tr ln nht ca biu thc A = 13 x 2 x 4 + 9 x 2 + x 4 vi 0 x 1 Bi 5 (2 im): Cho hỡnh vuụng ABCD cú cnh 1 n v Trờn cnh BC v CD lõn lt ly M v N sao cho MC+CN+MN=2 Cỏc on thng AM, AN lõn lt ct ng cheo DB ti I, K Chng minh cỏc on thng BI, LK, KD lõp thnh ba cnh ca mt tam giỏc vuụng Đề 26 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Câu 1: a) Chứng minh rằng với mọi số tự nhiên n 1 thỡ... tõm O K ng thng AB tip xỳc vi (O) ti M v song song vi CD (A thuc ID, B thuc IC) K ng kớnh MN ca ng trũn (O) Gi giao im ca IN vi AB l M/ Chng minh rng: a) AM/.CN=BM/.DN; b) AM/=BM đề khảo sát chất lợng HSG lớp 9 Môn: Toán Đề 29 Câu 1: ( ) ( ) 3 3 ab b a b + 2a a + b b a) Rút gọn biểu thức: A = + a b a a +b b vi a>0, b>0 v a b ( x 3) 3 b) Tớnh giỏ tr ca biu thc: y = 2 vi x = 2 + 3 x x +1 Cõu 2:... thng d qua im M ct AC v AB theo th t ti N v P Chng minh rng: BM CM khụng i khi M v d thay i BP CN Cõu 5: Chng minh rng vi mi x, y khỏc nhau v khỏc 0 thỡ: Đề 30 x2 y2 + xy ( x y) 2 x y x2 x y2 = 1 y đề khảo sát chất lợng HSG lớp 9 Môn: Toán Câu 1: (2,0 điểm) 1 1 a 2 + 1 1 P= + a) Rút gọn biểu thức: 2 + 2 a 2 2 a 1 a 2 1 + a vi a>0 v a 1 2+ x 2 x + = 2 b) Gii phng trỡnh: 2 + 2+... cnh a ngoi tip ng trũn (0) Mt tip tuyn ca ng trũn ct cnh AB, AC th t D, E t AD=x; AE=y; DE=z Chng minh rng : a, x 2 + y 2 xy = z 2 b, AD AE + khụng i khi tip tuyn DE thay i DB EC Đề 23 đề khảo sát chất lợng HSG lớp 9 Môn: Toán 3 x + y = a (a, b l tham s) ax y = b Câu1: 1) Cho hệ phơng trình Xỏc nh b h luụn cú nghim (x; y) vi mi a 2) Vit phng trỡnh ng thng i qua im M(2; 3), ct trc honh v trc tung... DAF = NAD + MAB = BAD MAN = 90 0 450 = 450 T suy ra: MAN = FAN(cnh, gc, cnh) MN = FN =BM + DN Xet tam giỏc vung CMN, ta c: MN2 = CM2 + CN2 (BM + DN)2 = (a BM)2 + (a DN)2 (1) Khai trin (1) ri rỳt gn, ta c: a(BM + DN) + BM.DN =a2 PCM b)Ta c: EAF = MAN + NAF = 450 + 450 = 90 0 EAF l tam giỏc vung (H thc lng trong tam giỏc vung) Hay l: Đề 13 PCM Mt s luyn thi vo chuyờn Toỏn 9 11 Trng THCS Nguyn ỡnh Chiu... vuụng ABCD tõm O Gi K, N lõn lt l trung im ca AB, BC v F l trung im ca NC T A k ng thng song song vi KF ct CD ti G Chng minh FG l tip tuyn ca ng trũn tõm O ni tip trong hỡnh vuụng đề khảo sát chất lợng HSG lớp 9 Môn: Toán Đề 25 Bài 1 (2 điểm): a) Chứng minh rằng nếu b là số nguyên tố lớn hơn 3 và 10b+1 cũng là số nguyên tố thì 5b+1 chia hết cho 6 b) Tìm nghiệm nguyên của phơng trình x2+x+13=y2 Bài 2... tớch S ca tam giỏc AHK theo R v x Tỡm x sao cho S t giỏ tr ln nht b) Chng minh rng khi A thay i, tng AH2+HK2 luụn l mt i lng khụng i Tớnh s o gúc B ca tam giỏc ABC bit Câu1: Đề 20 AH 3 = HK 5 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Chứng minh rằng A = 2 + 2 12n 2 + 1 l s chớnh phng khi A N v n N 15 Trng THCS Nguyn ỡnh Chiu Nm hc 2010-2011 Cõu 2: Cho a thc P(x) nguyờn v P(x) chia ht cho 3 khi x . 1^H.%^.<.3.M572I.2-C>A;{ AB,Q#$%.qQNBS+P>`2I*W$CT(= M TFC>02%%.2-C>A#A$-2%C1S>[S)y.2.G$ %. _9 ã ã BHC BOC= S.C.Y%2 '@.J2C.1$1Cn-.@TF^.$ AB-Q%2-UQA4O$3H.%!O^.IJ2Q T*A4OT(-OJ2.2$ Đề thi học sinh giỏi vòng 2 lớp 9 năm học 2008-20 09 Đề thi môn: Toán (Thời gian. $%'2 %9 "T(0?M.?02%%9dT(0?M.?$3H 9d2E.%!' (9 dI *97 T(0?M.?$ 6 79: .M5 # ` # d5d& $%+2.^.7-0?M^.<.3 Q i5g# # i4 ddc'F0?M+:5$ \%5SSeT(-0?.ZI*P'(5dde`$-.@TF^.J2X` 5dede5$ $%,%'*C>1.PN$6)STxT_.T(.J2C>S>'(zT( .J2$wCI}.<0%0%'F)zy.1.G6$3z6T( .E9.EJ2.q.PN=.E9.%'*$ Đề 25 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Bài 1 (2 điểm): a) Chứng minh rằng nếu b là số nguyên tố. ) = + y y x x yx yx xy yx $ Đề 30 đề khảo sát chất lợng HSG lớp 9 Môn: Toán Câu 1: (2,0 điểm) 2 Rút gọn biểu thức: + + + + = aa a aa P 'F2f'(2 $ 6 79: . = + ++ + x x x x $ AB'(3,0