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Lời giải HOMC 2013 (toán Hà Nội Mở rộng)UEE|ASP|JMS22 Mar 2014Chúng tôi giới thiệu lời giải các bài toán sử dụng trong kỳ thi HOMC 2013. Đề thi các năm khác cũng có thể tìm thấy trên trang Hexagon, chúng tôi sẽ tiếp tục cập nhật đưa lên lời giải các năm còn lại, theo đề nghị của một số bạn sử dụng website này.Question 1. 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 aboveAnswer is D. There is only one way2013=2011+2.Notice that 2013 is odd. The sum of two numbers is odd if and only if one of them is odd while the other is even. Let 2n+1 be the odd numbers and 2m be the even numbers. Then2n+1+2m=2(m+n)+1,where n,m are integers. Since the only even prime is 2, we have 2013=2+2011.Also notice that 2011 can be written as 11 consecutive primes:2011=157+163+167+173+179+181+191+193+197+199+211.Hence, the answer is E. Question 2. Let A be an even number but not divisible by 10. The last two digits of A20 areA) 46 B) 56 C) 66 D) 76 E) None of the Trước hết, ta có khẳng định: Nếu n là số chẵn và n không chia hết cho 5 thì n100 có ba chữ số tận cùng là 376. Từ đó, ta phải tìm hai chữ số tận cùng của A20 khi biết rằng A100 có tận cùng là 76. Trong các số đã cho chỉ có 76 thỏa mãn, nên 76 là hai chữ số tận cùng của A20.Question 3. The largest integer not exceeding (n+1)α−nα, where n is a natural number, α=20132014−−−−√ is A) 1 B) 2 C) 3 D) 4 E) None of the above. Bài này nên đặc biệt hóa và dự đoán kết quả từ các trường hợp đặc biệt. Đáp án là A. Question 4. How many natural numbers n are there so that n2+2014 is a perfect square.A) 1B) 2C) 3D) 4E) None of the aboveLời giải. Let b2 be the number, where b is some integer. Then n2+2014=b2. We have (b−n)(b+n)=2014.By prime factorisation, we have 2014=2×1007=2×19×53. Two numbers in each of the pairs (2,1007), (19,106), (38,53) do not have the same parity.Hence, there are no such b. Answer: E) None of the above. Question 5. The number of integer solutions x of the equation below(12x−1)(6x−1)(4x−1)(3x−1)=330isA) 0B) 1C) 2D) 3E) None of the aboveLời giải. Expanding the lefthand side and rearranging gives a polynomial equation. The integer root of the equation is a factor of 329 which is factorised into 7×47. Hence, the possible factors are 1,7,47, and 329.If x=1, then11×5×3×2=330.If x=7 then 83×41×27×20≠330.If x=47, then 12x−1 exceeds 330. Hence, the number of integer roots is 1. Answer B.Question 6. Let ABC be a rightangled triangle with ∠CAB=90∘, ∠CBA=60∘, and CB=1 cm. Points D,E,F are chosen outside the triangle such that triangles ACD, AEB, and CBF are all equilateral. If the area of triangle DEF is x cm2, what is the value of x?Lời giải. Notice that D,B,F are collinear, and AC=3√2. Construct a rectangle GHIF. Hence, ∠GCE=30∘, ∠GEC=60∘. Simple computations give GE=3√4, and GC=34. Since CF=1, then GF=1+34=74. The area of (GEF)=73√32. Notice that the length of GH is equal to that of AC plus the altitude h of triangle ABD. Computing h gives h=3√4. That is, GH=33√4. The area of the rectangle is 213√16. The area of triangle (DFI) is 93√32. The area of (EHD) is 3√4. The area of (DEF) is 213√16−93√32−3√4−73√32=93√16.Đáp án: 93√16. Question 7. Let ABCDE be a convex pentagon. If the areas of triangles ABC,BCD,CDF,DEA, and EAB are all 2 cm2, what is the area of ABCDE?Lời giải. Vì diện tích hai tam giác EAB,CCAB bằng nhau nên AB||EC. Tương tự, BC,CD,DE,EA theo thứ tự song song với AD,BE,CA,DB. Gọi M là giao điểm của AC với BE. Vì area(EAB)=area (CAB), nên area(AEM)=area(BCM)=x. Lại có tứ giác EMCD là hình bình hành nên Area(CEM)=Area(CED)=2. Từ đó suy rax2−x=area(AEM)area(ABM)=EMBM=area(CEM)area(CBM)=2x.Do đó, x2+2x−4=0. Giải phương trình này, ta được x=5√−1. Từ đây, ta thu đượcarea(ABCDE)=area(ABE)+area(CED)+area(CEM)+area(BCM)=6+5√−1=5+5√. Question 7. Solve the simultaneous equationsx+y≤1,2xy+1x2+y2=10.Lời giải. Let s=x+y, p=xy. Then s2=x2+y2+2p or x2+y2≤1−2p. Hence, we have 2p+11−2p≥10.Simple manipulations give 2−4p+p≥10p(1−2p).20p2−13p+2≥0.The discriminant of the quadratic equation is 169−160=9. Hence, p=(13±3)40, or p=25, p=14. That is,20(p−25)(p−14)≥0.We have p≤14 from the fact that 4xy≤(x+y)2. This implies that the last inequality is true. Equality occurs when p=14, and s=1. Hence,x=y=12. Question 8. If a,b,c,d,e are positive real numbers such that (x+a)(x+b)(x+c)=x3+3dx2+3x+e3,for all x∈R,what is the minimum value of d?Lời giải. Expanding the lefthand side of the equation gives x3+(a+b+c)x2+(ab+bc+ca)x+abc=x3+3dx2+3x+e3.Comparing the coefficients gives a+b+c=3d,ab+bc+ca=3,abc=e3.Notice that (a+b+c)2≥3(ab+bc+ca),for all a,b,c∈R.This implies that 9d2≥3×3, or d≥1. Hence, the minimum value of d is 1. Question 9. Solve the system {1x+1y3x+2y=16,=56.Lời giải. Let 1x=a, 1y=b. Then a+b=16 and 3a+2b=56. Solving these simultaneous equations gives a=12, b=−13. Hence, x=2, y=−3.Question 10. Consider the set of all rectangles with a givenperimeter p. Find the largest value ofM=S2S+p+2.where S is denoted the area of the rectangle.Lời giải. Let a,b be the sidelength of a rectangle in the set. Then S=ab and p=2(a+b). Notice that (a+b)2≥4ab for all a,b. We have S≤p216.That is p≥4S√. Hence 2S+p+2≥2S+4S√+2=2(S√+1)2. Hence, M≤12(S√S√+1)2=12(1−1S√+1)2≤12.Answer: 12.Question 11. If f(x)=ax2+bx+c satisfies the condition|f(x)| and a + b Determine the minimum value of M= 1.6.2 1 1 + + + ab a + ab ab + b2 a2 + b2 Senior Section Question An integer is called ”octal” if it is divisible by or if at least one of its digits is How many integers between and 100 are octal? (A): 22; (B): 24; (C): 27; (D): 30; (E): 33 Question What is the smallest number √ √1 (A) 3; (B) 2 ; (C) 21+ ; (D) 2 + ; (E) Question What is the largest integer less than to (2011)3 + × (2011)2 + × 2011 + 5? 21 (A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of the above Question Prove that + x + x2 + x3 + · · · + x2011 for every x −1 Question Let a, b, c be positive integers such that a + 2b + 3c = 100 Find the greatest value of M = abc Question Find all pairs (x, y) of real numbers satisfying the system x+y =2 x4 − y = 5x − 3y Question How many positive integers a less than 100 such that 4a2 + 3a + is divisible by Question Find the minimum value of S = |x + 1| + |x + 5| + |x + 14| + |x + 97| + |x + 1920| Question For every pair of positive integers (x; y) we define f (x; y) as follows: f (x, 1) = x f (x, y) = if y > x f (x + 1, y) = y[f (x, y) + f (x, y − 1)] Evaluate f (5; 5) 22 Question 10 Two bisectors BD and CE of the triangle ABC intersect at O Suppose that BD.CE = 2BO.OC Denote by H the point in BC such that OH ⊥ BC Prove that AB.AC = 2HB.HC Question 11 Consider a right-angle triangle ABC with A = 90o , AB = c and AC = b Let P ∈ AC and Q ∈ AB such that ∠AP Q = ∠ABC and ∠AQP = ∠ACB Calculate P Q + P E + QF, where E and F are the projections of P and Q onto BC, respectively Question 12 Suppose that |ax2 + bx + c| numbers x Prove that |b2 − 4ac| 1.7 1.7.1 |x2 − 1| for all real Hanoi Open Mathematics Competition 2012 Junior Section Question Assume that a − b = −(a − b) Then: (A) a = b; (B) a < b; (C) a > b; sible to compare those of a and b (D); It is impos- Question Let be given a parallelogram ABCD with the area of 12cm2 The line through A and the midpoint M of BC meets BD at N Compute the area of the quadrilateral M N DC (A): 4cm2 ; (B): 5cm2 ; (C): 6cm2 ; (D): 7cm2 ; (E) None of the above Question For any possitive integer a, let [a] denote the smallest prime factor of a Which of the following numbers is equal to [35]? (A) [10]; (B) [15]; (C) [45]; (D) [55]; (E) [75]; 23 Question A man travels from town A to town E through towns B, C and D with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively If the road between town A and town E can be classified as horizontal, up slope, down slope and horizontal and total length of each type of road is the same, what is the average speed of his journey? (A) 2km/h; (B) 2,5km/h; (C) 3km/h; (D) 3,5km/h; (E) 4km/h 24 Question How many different 4-digit even integers can be form from the elements of the set {1, 2, 3, 4, 5} (A): 4; (B): 5; (C): 8; (D): 9; (E) None of the above Question At 3:00 A.M the temperature was 13o below zero By noon it had risen to 32o What is the average hourly increase in teparature? Question Find all integers n such that 60 + 2n − n2 is a perfect square Question Given a triangle ABC and points K ∈ AB, N ∈ BC such that BK = 2AK, CN = 2BN and Q is the S∆ABC common point of AN and CK Compute · S∆BCQ Question Evaluate the integer part of the number H= 1+ 20112 20112 2011 + + · 20122 2012 Question 10 Solve the following equation 1 13 + = · (x + 29)2 (x + 30)2 36 Question 11 Let be given a sequence a1 = 5, a2 = and an+1 = an +3an−1 , n = 2, 3, Calculate the greatest common divisor of a2011 and a2012 Question 12 Find all positive integers P such that the sum and product of all its divisors are 2P and P , respectively Question 13 Determine the greatest value of the sum M = 11xy + 3xz + 2012yz, where x, y, z are non negative integers satisfying the following condition x + y + z = 1000 25 Question 14 Let be given a triangle ABC with ∠A = 900 and the bisectrices of angles B and C meet at I Suppose that IH is perpendicular to BC (H belongs to BC) If HB = 5cm, HC = 8cm, compute the area of ABC Question 15 Determine the greatest value of the sum M = xy + yz + zx, where x, y, z are real numbers satisfying the following condition x2 + 2y + 5z = 22 1.7.2 Senior Section √ √ 6+2 5+ 6−2 √ · The value of 20 Question Let x = 11 H = (1 + x5 − x7 )2012 is (A): 1; (B): 11; (C): 21; (D): 101; (E) None of the above Question Compare the numbers: P = 2α , Q = 3, T = 2β , where α = √ 2, β = + √ (A): P < Q < T ; (B): T < P < Q; (C): P < T < Q; T < Q < P ; (E): Q < P < T (D): Question Let be given a trapezoidal ABCD with the based edges BC = 3cm, DA = 6cm (AD BC) Then the length of the line EF (E ∈ AB, F ∈ CD and EF AD) through the common point M of AC and BD is (A): 3,5cm; (B): 4cm; (C): 4,5cm; (D): 5cm; (E) None of the above Question What is the largest integer less than or equal to √ √ 3 4x3 − 3x, where x = 2+ 3+ 2− ? 26 (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question Let f (x) be a function such that f (x)+2f 4020 − x for all x = Then the value of f (2012) is x + 2010 = x−1 (A): 2010; (B): 2011; (C): 2012; (D): 2014; (E) None of the above Question For every n = 2, 3, , we put 1 × 1− ×· · ·× 1− 1+2 1+2+3 + + + ··· + n Determine all positive integer n (n ≥ 2) such that is an An integer An = 1− Question Prove that the number a = is a 2012 2011 perfect square Question Determine the greatest number m such that the system x2 + y = |x3 − y | + |x − y| = m3 has a solution Question Let P be the common point of internal bisectors of a given ABC The line passing through P and perpendicular to CP intersects AC and BC at M and N , respectively If AM AP = 3cm, BP = 4cm, compute the value of ? BN Question 10 Suppose that the equation x3 +px2 +qx+1 = 0, with p, q are rational numbers, has real roots x1 , x2 , x3 , where √ x3 = + 5, compute the values of p and q? 27 Question 11 Suppose that the equation x3 + px2 + qx + r = has real roots x1 , x2 , x3 , where p, q, r are integer numbers Put Sn = xn1 + xn2 + xn3 , n = 1, 2, Prove that S2012 is an integer Question 12 In an isosceles triangle ABC with the base AB given a point M ∈ BC Let O be the center of its circumscribed circle and S be the center of the inscribed circle in ∆ABC and SM AC Prove that OM ⊥ BS Question 13 A cube with sides of length 3cm is painted red and then cut into × × = 27 cubes with sides of length 1cm If a denotes the number of small cubes (of 1cm×1cm×1cm) that are not painted at all, b the number painted on one sides, c the number painted on two sides, and d the number painted on three sides, determine the value a − b − c + d? Question 14 Solve, in integers, the equation 16x + = (x2 − y )2 Question 15 Determine the smallest value of the sum M = xy − yz − zx, where x, y, z are real numbers satisfying the following condition x2 + 2y + 5z = 22 1.8 1.8.1 Hanoi Open Mathematics Competition 2013 Junior Section Question 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 28 Question How many natural numbers n are there so that n2 + 2014 is a perfect square (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question The largest integer not √ exceeding [(n+1)α]−[nα], 2013 where n is a natural number, α = √ , is: 2014 (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question Let A be an even number but not divisible by 10 The last two digits of A20 are: (A): 46; (B): 56; (C): 66; above (D): 76; (E): None of the Question The number of integer solutions x of the equation below (12x − 1)(6x − 1)(4x − 1)(3x − 1) = 330 is: (A): 0; (B): 1; (C): 2; (D): 3; (E): None of the above Short Questions Question Let ABC be a triangle with area (cm2 ) Points D, E and F lie on the sides AB, BC and CA, respectively Prove that min{Area of ∆ADF, Area of ∆BED, Area of ∆CEF } ≤ 29 (cm2 ) Question Let ABC be a triangle with A = 900 , B = 600 and BC = 1cm Draw outside of ∆ABC three equilateral triangles ABD, ACE and BCF Determine the area of ∆DEF Question Let ABCDE be a convex pentagon Given that area of ∆ABC = area of ∆BCD = area of ∆CDE = area of ∆DEA = area of ∆EAB = 2cm2 , Find the area of the pentagon Question Solve the following system in positive numbers x + y ≤ + = 10 xy x + y Question 10 Consider the set of all rectangles with a given perimeter p Find the largest value of M= S , 2S + p + where S is denoted the area of the rectangle Question 11 The positive numbers a, b, c, d, e are such that the following identity hold for all real number x (x + a)(x + b)(x + c) = x3 + 3dx2 + 3x + e3 Find the smallest value of d Question 12 If f (x) = ax2 + bx + c safisfies the condition |f (x)| < 1, ∀x ∈ [−1, 1], 30 prove that the equation f (x) = 2x2 − has two real roots Question 13 Solve the system of equations 1 1 + = x y + = x y Question 14 Solve the system of equations = x2 + x + y 2y + z = 2y + 3z + x = 3z + Question 15 Denote by Q and N∗ the set of all rational and ax + b ∈Q positive integer numbers, respectively Suppose that x for every x ∈ N∗ Prove that there exist integers A, B, C such that ax + b Ax + B = for all x ∈ N∗ x Cx 1.8.2 Senior Section Question How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square? (A): 1; (B): 2; (C): 4; (D): 8; (E) None of the above Question The smallest value of the function f (x) = |x| + where x ∈ [−1, 1] is 31 − 2013x 2013 − x 1 (A): ; (B): ; (C): ; 2012 2013 2014 of the above (D): ; (E): None 2015 Question What is the largest integer not exceeding 8x3 + √ √ 3 2+ 5+ 2− ? 6x − 1, where x = (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question Let x0 = [α], x1 = [2α] − [α], x2 = √ [3α] − [2α], 2013 x4 = [5α] − [4α], x5 = [6α] − [5α], , where α = √ The 2014 value of x9 is (A): 2; (B): 3; (C): 4; (D): 5; (E): None of the above Question The number n is called a composite number if it can be written in the form n = a × b, where a, b are positive integers greater than Write number 2013 in a sum of m composite numbers What is the largest value of m? (A): 500; (B): 501; (C): 502; above (D): 503; (E): None of the Question Let be given a ∈ {0, 1, 2, 3, , 1006} Find all n a k n ∈ {0, 1, 2, 3, , 2013} such that C2013 > C2013 , where Cm = m! k!(m − k)! Question Let ABC be an equilateral triangle and a point M inside the triangle such that M A2 = M B + M C Draw an equilateral triangle ACD where D = B Let the point N inside 32 ∆ACD such that AM N is an equilateral triangle Determine BM C Question Let ABCDE be a convex pentagon and area of ∆ABC = area of ∆BCD = area of ∆CDE = area of ∆DEA = area of ∆EAB Given that area of ∆ABCDE = Evaluate the area of area of ∆ABC Question A given polynomial P (t) = t3 + at2 + bt + c has distinct real roots If the equation (x2 + x + 2013)3 + a(x2 + x + 2013)2 + b(x2 + x + 2013) + c = has no real roots, prove that P (2013) > 64 Question 10 Consider the set of all rectangles with a given area S Find the largest value of M= 16S − p , p2 + 2p where p is the perimeter of the rectangle Question 11 The positive numbers a, b, c, d, p, q are such that (x+a)(x+b)(x+c)(x+d) = x4 +4px3 +6x2 +4qx+1 holds for all real numbers x Find the smallest value of p or the largest value of q Question 12 The function f (x) = ax2 + bx + c safisfies the √ following conditions: f ( 2) = 3, and |f (x)| ≤ 1, for all x ∈ [−1, 1] 33 √ Evaluate the value of f ( 2013) Question 13 Solve the system of equations xy = x y + =1 x4 + y x2 + y Question 14 Solve the system of equations: x3 + y = x2 + x − 3 y3 + z = y2 + y − 4 z + x = z + z − 5 Question 15 Denote by Q and N∗ the set of all rational and ax + b positive integral numbers, respectively Suppose that ∈ cx + d Q for every x ∈ N∗ Prove that there exist integers A, B, C, D such that ax + b Ax + B = for all x ∈ N∗ cx + d Cx + D ——————————————————- 34 [...]... function f (x) = |x| + where x ∈ [−1, 1] is 31 1 − 2013x 2013 − x 1 1 1 (A): ; (B): ; (C): ; 2012 2013 2014 of the above (D): 1 ; (E): None 2015 Question 3 What is the largest integer not exceeding 8x3 + √ √ 1 3 3 2+ 5+ 2− 5 ? 6x − 1, where x = 2 (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question 4 Let x0 = [α], x1 = [2α] − [α], x2 = √ [3α] − [2α], 2013 x4 = [5α] − [4α], x5 = [6α] − [5α], ,... form n = a × b, where a, b are positive integers greater than 1 Write number 2013 in a sum of m composite numbers What is the largest value of m? (A): 500; (B): 501; (C): 502; above (D): 503; (E): None of the Question 6 Let be given a ∈ {0, 1, 2, 3, , 1006} Find all n a k n ∈ {0, 1, 2, 3, , 2013} such that C2013 > C2013 , where Cm = m! k!(m − k)! Question 7 Let ABC be an equilateral triangle... of ∆ABCDE = 2 Evaluate the area of area of ∆ABC Question 9 A given polynomial P (t) = t3 + at2 + bt + c has 3 distinct real roots If the equation (x2 + x + 2013) 3 + a(x2 + x + 2013) 2 + b(x2 + x + 2013) + c = 0 has no real roots, prove that 1 P (2013) > 64 Question 10 Consider the set of all rectangles with a given area S Find the largest value of M= 16S − p , p2 + 2p where p is the perimeter of the... Mathematics Competition 2013 Junior Section Question 1 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 28 Question 2 How many natural numbers n are there so that n2 + 2014 is a perfect square (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Question 3 The largest integer not √ exceeding [(n+1)α]−[nα], 2013 where n is a natural... the greatest values of the following product P = abcd Question 8 Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than 1004 Question 9 Let be given ∆ABC with area (∆ABC) = 60cm2 Let R, S lie in BC such that BR = RS = SC and P, Q be midpoints of AB and AC, respectively... greatest values of the following product P = abcd 15 Question 8 Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than 1004 Question 9.Given an acute-angled triangle ABC with area S, let points A , B , C be located as follows: A is the point where altitude from A on BC... A = 72011 is (A) 1; (B) 3; (C) 7; (D) 9; (E) None of the above Question 3 What is the largest integer less than or equal to 19 3 (2011)3 + 3 × (2011)2 + 4 × 2011 + 5? (A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of the above Question 4 Among the four statements on real numbers below, how many of them are correct? “If “If “If “If “If a < b < 0 then a < b2 ”; 0 < a < b then a < b2 ”; a3 < b3 then... number √ √1 1 2 5 (A) 3; (B) 2 2 ; (C) 21+ 2 ; (D) 2 2 + 2 3 ; (E) 2 3 Question 3 What is the largest integer less than to 3 (2011)3 + 3 × (2011)2 + 4 × 2011 + 5? 21 (A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of the above Question 4 Prove that 1 + x + x2 + x3 + · · · + x2011 for every x 0 −1 Question 5 Let a, b, c be positive integers such that a + 2b + 3c = 100 Find the greatest value of M =... value of p or the largest value of q Question 12 The function f (x) = ax2 + bx + c safisfies the √ following conditions: f ( 2) = 3, and |f (x)| ≤ 1, for all x ∈ [−1, 1] 33 √ Evaluate the value of f ( 2013) Question 13 Solve the system of equations xy = 1 x y + =1 x4 + y 2 x2 + y 4 Question 14 Solve the system of equations: 1 4 x3 + y = x2 + x − 3 3 5 1 y3 + z = y2 + y − 4 4 z 3