HD De HOMC 2013 du

<span class='text_page_counter'>(1)</span>Hanoi Mathematical Society Hanoi Opens Mathematics Competition 2013 Junior Section. Sunday, March 24, 2013 Important: Answer all 15 questions. Enter yor answers on the answer sheet provided. For the multiple choice questions, enter only the letters ( A,B,C,D or E) corresponding to the correct answers in the answer sheet. No calculators are allowed. Multiple Choice Questions : Q1 : Write 2013 as a sum of m prime numbers. The smallest value of m is: (A) : 2 (B) : 3 (C) : 4 (D) : 1 (E) : None of the above. 2 Q2 : How many natural numbers n are there so that n + 2014 is a perfect square. (A) : 1 (B) : 2 (C) : 3 (D) : 4 (E) : None of the above. Q3 : The largest integer not exceeding [(n + 1)  ] - [n  ], where n is a natural number,  =. 2013 2014 , is :. (A) : 1 (B) : 2 (C) : 3 (D) : 4 (E) : None of the above. 20 Q4 : Let A be an even number but not divisible by 10. The last two digits of A are :. (A) : 46 (B) : 56 (C) : 66 (D) : 76 (E) : None of the above. Q5 : The number of integer solutions x of the equation below: (12 x  1)(6 x  1)(4 x  1)(3 x  1) 330 is :. (A) : 0 (B) : 1 (C) : 2 (D) : 3 (E) : None of the above. Short Questions 2. Q6 : Let ABC be a triangle with area 1 ( cm ). Points D,E and F lie on the sides AB,BC and CA, respectively. Prove that : 2 Min{Area of ADF, area of BED, area of CEF}  1/4 ( cm )..

<span class='text_page_counter'>(2)</span> Q7 : Let ABC be a triangle with  A = 90,  B = 60 and BC = 1cm. Draw outside of ABC three regular triangles ABD, ACE and BCF. Determine the area of DEF. Q8 : Let ABCDE be a convex pentagon. Gives that SABC = SBCD = SCDE = 2 SDEA = SEAB = 2 ( cm ). Find the area of the pentagon. Q9 : Solve the following system in positive numbers.  x  y 1  1 2   xy x 2  y 2 10  Q10 : Consider the set of all rectangles with a given perimeter p. Find the largest value of S M = 2S  p  2. Where S is denoted the area of the rectangle. Q11 : The positive numbers a,b,c,d,e are such that the following identify hold for all number x ( x  a )( x  b)( x  c)  x 3  3dx 2  3x  e3 .. Find the smallest value of d. 2 Q12 : If f ( x) ax  bx  c satisfies the condition. | f ( x )|  1,  x ä   1,1. Prove that the equation f ( x)=2 x 2 − 1 has two real roots. Q13 : Solve the system of equations 1 1 1  x  y 6    3  2 5  x y 6. Q14 : Solve the system of equations  x3  y x 2  1  3 2 2 y  z 2 y  1 3z 3  x 3z 2  1 .

<span class='text_page_counter'>(3)</span> * Q15 : Denote by Q and N the set of all rational and positive integer numbers,. ax  b Q * respectively. Suppose that x for every x  N . ax  b Ax  B  Cx for all x  N* . Prove that there exist integers A, B , C such that x. Hướng dẫn Multiple Choice Questions : Q1. Q2. Q3. Q4. Q5. A(2). E. E. D. B. Short Questions 2. Q6 : Let ABC be a triangle with area 1 ( cm ). Points D,E and F lie on the sides AB,BC and CA, respectively. Prove that : 2 Min{Area of ADF, area of BED, area of CEF}  1/4 ( cm ).. A. F D. B. C. E S ADF AD . AF. S BED. BD . BE. S CEF. CE .CF. Ta có S = AB . AC (1) ; S = AB . BC (2); S = BC. AC (3) ABC ABC ABC.

<span class='text_page_counter'>(4)</span> Nhân (1); (2), (3) ta có S ABC ¿ 3 ¿ AD+DB 2 BE+EC 2 AF+ FC 2 . . 2 2 2 ( AD. . BD).(BE. EC).( AF . FC) 1 ¿ ≤ = 2 2 2 2 2 2 64 AB . AC BC AB . AC BC ¿ ¿ S ADF . S BDE . S CEF ¿ 1 Nên ít nhất phải có một tam giác cỏ diện tích không lớn hơn 4. (. )(. )(. ). Q7 : Let ABC be a triangle with  A = 90,  B = 60 and BC = 1cm. Draw outside of ABC three regular triangles ABD, ACE and BCF. Determine the area of DEF. F. B H. D. C. A. H. E 1 3 Do BC=1 góc B=600 nên AB= AB= ; AC= √ và D ,B <,F thẳng hàng 2. 2. Các tam giac đều biết cạnh thì tính được diện tích theo công thức với a là cạnh thì a2 √ 3 S= 4. Ta đi tính SADE kéo dài AD cắt CE tại H thì AH//BC vì góc BCE=900 nên AH ⊥ CE CE 3 AD . HE √ 3 = Suy ra EH=HC= = √ mà S ADE= 2. 4. 2. 16. Cách khác Gọi H là trung điểm BC ( áp dụng tính chất trung tuyến tam giác vuông và đường cao tam giác cân).

<span class='text_page_counter'>(5)</span> ta có. 1 3 3 S ABD = S ABC= √ ; SBCF =2 S ABC= √ 2 16 4. 3 3 3 3 9 3 Kết quả S DEF = √ + √ + √ + √ = √ (cm2 ) 16. 8. 4. 16. 16. Q8 : Let ABCDE be a convex pentagon. Gives that SABC = SBCD = SCDE = 2 SDEA = SEAB = 2 ( cm ). Find the area of the pentagon. A. O. E. D. B. C. Do SABC = SBCD = SCDE = SDEA = SEAB = 2 AB//EC; BC//AD;AC//DE;AE//BD gọi AC cắt BE tại O ta có EOCD là hình bình hành suy ra SEOC=SDCE=2 Vì ABCE là hình thang. S AOE=S BOC. Đặt S AOE=S BOC=x ; (0< x <2). Thì SAOB=2-x ta có. 2. S ABCDE =SDEOC + S ABC+ S AOE=4+2+ x=5+ √ 5 (cm ). OB S AOB S COB 2− x x = = ⇒ = OE S AOE SCOE x 2 2 x +1 ¿ =5 ⇔ ¿ x=√5 − 1 ¿ x= √5+1(loai ; vi: x <2) ¿ ¿ ¿ ¿ ¿ ¿ 2 2 ⇔ 4 −2 x=x ⇔ x +2 x+1=5 ⇔ ¿. Q9 : Solve the following system in positive numbers.

<span class='text_page_counter'>(6)</span>  x  y 1  1 2   xy x 2  y 2 10  2. HD Đặt. x+ y ¿ ¿ ¿ 2 1 1 1 3 4 P= + 2 2 = + 2 2+ ≥ xy x + y 2 xy x + y 2 xy ¿. 1. Dấu “=” xảy ra khi x=y= 2 Q10 : Consider the set of all rectangles with a given perimeter p. Find the largest value of S M = 2S  p  2. Where S is denoted the area of the rectangle. HD Gọi cạnh hình chữ nhật là a, b (0< a≤ b) 2 ab+2(a+b)+2. ab 1 Ta có S=ab; P=2(a+b) nên M = 2ab +2(a+ b)+2 ⇔ M = ab. Áp dụng BĐT. ab ≤. a+b 2 P2 = 2 16. ( ). nên. =2+. P 2 + ab ab. p+4 ¿2 ¿ 2¿ 2 16 32 2( p + 8 p+ 16) M ≥ 2+ + 2 = =¿ P p p2. 2. Vậy. P+4 ¿ ¿ P+4 ¿2 ¿ 2¿ 2¿ P2 M≤ ¿. Q11 : The positive numbers a,b,c,d,e are such that the following identify hold for all number x ( x  a )( x  b)( x  c)  x 3  3dx 2  3x  e3 .. Find the smallest value of d. HD GT suy ra ¿ a+b+c =3 d ab+ bc+ ca=3 abc=e 3 ¿{{ ¿.

<span class='text_page_counter'>(7)</span> Ta có. a+b +c ¿ 2 ≥ 3(ab +bc +ca)=9 ⇔9 d 2 ≥ 9 ⇔d ≥1 a2 +b 2+ c2 ≥ ab+ bc+ca ⇔ ¿. Nên Min( d)=1⇔ a=b=c=e=d=1 2 Q12 : If f ( x) ax  bx  c satisfies the condition. | f ( x )|  1,  x ä   1,1. Prove that the equation f ( x)=2 x 2 − 1 has two real roots. ¿. |f (−1)|<1 |f (1)|<1 |f (0)|<1. Từ GT. ⇔ ¿ − 1< a −b+ c< 1 − 1< a+b+ c< 1 −1<c <1 ⇔ ¿ −1 −c <a − b<1 −c −1 −c <a+ b<1 −c −1<c <1 ⇔ ¿ − 2< a+b< 2 − 2< a −b< 2 ⇔− 2< a<2 ¿{{ ¿. Ta có ax 2+ bx +c=2 x 2 −1 ⇔ (a − 2) x2 + bx+ c+ 1=0 (*) Ta thấy a-2<0 và c+1>0 nên Δ=b2 − 4 ac> 0 phân biệt. suy ra PT (*) luôn có 2 nghiệm thực. Q13 : Solve the system of equations 1 1 1  x  y 6    3  2 5  x y 6 1 1 1 −1 Đặt x =a ; y =b giải ra ta được a= a= 2 ; b= 3 ⇒ x=2 ; y=−3. Q14 : Solve the system of equations  x3  y x 2  1  3 2 2 y  z 2 y  1 3z 3  x 3z 2  1 . HD.

<span class='text_page_counter'>(8)</span>  x3  y x 2  1 ⇔  3 2 2 y  z 2 y  1 x 2( x −1)=1 − y 3z 3  x 3z 2  1 2 y 2( y −1)=1 − z  2 3 z ( z − 1)=1− x ¿{{. Ta thấy x=y=z =1 là một nghiệm x=0 suy ra y=1 suy ra z=1 suy ra x=1 vô lí tương tự x,y, z khác 0 Với x,y,z khác 0 và khác nhân cả 3 PT ta được 6 x 2 y 2 z 2 ( x −1)( y − 1)( z −1)=( 1− x )(1 − y )(1− z) ⇔ 6 x 2 y 2 z2 =−1 ( vô lí). Vậy x=y=z=1 * Q15 : Denote by Q and N the set of all rational and positive integer numbers,. ax  b Q * respectively. Suppose that x for every x  N . ax  b Ax  B  Cx for all x  N* . Prove that there exist integers A, B , C such that x ax  b Q * HD vì x for every x  N . Suy ra a, b. Q. ax  b Ax  B  ⇒ aCx+ bC=Ax+ B ⇔( A − aC) x=( B − bC) ;(∗) Cx Ta có C khác 0 từ x * Đẳng thức (*) Đúng với mọi x  N khi. ¿ A −aC=0 B − bC=0 ⇔ ¿ A=aC B=bC (**) ¿{ ¿. Vì a, b Q đặt. a=. m p ; b= n q. trong đó n; q khác 0, m,n,p,q là số nguyên. (m;n)=1; (p;q)=1 Khi đó chọn C=BCNN(n;q) thay ào (**) thì A,B, C là số nguyên khác 0 thỏa mãn ax  b Ax  B  x Cx. Ví dụ. 1 2 x+ ax+ b 2 3 = x x. ta chọn C=6 khi đó A=3; B=4 thì. 1 2 x+ 2 3 3 x+ 4 = x 6x.

<span class='text_page_counter'>(9)</span> GV Nguyễn Minh Sang THCS Lâm Thao –H Lâm Thao- Phú Thọ ( HD vội và chưa hiểu hết đề có thể chưa chinh xác mong các bạn kiểm tra lại có thể còn cách khác hay hơn tôi sẽ bổ sung gửi sau).

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