Toán Hà Nội Mở Rộng 2006-2013

HANOI MATHEMATICAL SOCIETY NGUYEN VAN MAU HANOI OPEN MATHEMATICS COMPETITON PROBLEMS 2006 - 2013 HANOI - 2013 www.VNMATH.com Contents 1 Hanoi Open Mathematics Competition 3 1.1 Hanoi Open Mathematics Competition 2006 . . . 3 1.1.1 Junior Section . . . . . . . . . . . . . . . . 3 1.1.2 Senior Section . . . . . . . . . . . . . . . . 4 1.2 Hanoi Open Mathematics Competition 2007 . . . 6 1.2.1 Junior Section . . . . . . . . . . . . . . . . 6 1.2.2 Senior Section . . . . . . . . . . . . . . . . 8 1.3 Hanoi Open Mathematics Competition 2008 . . . 10 1.3.1 Junior Section . . . . . . . . . . . . . . . . 10 1.3.2 Senior Section . . . . . . . . . . . . . . . . 11 1.4 Hanoi Open Mathematics Competition 2009 . . . 13 1.4.1 Junior Section . . . . . . . . . . . . . . . . 13 1.4.2 Senior Section . . . . . . . . . . . . . . . . 15 1.5 Hanoi Open Mathematics Competition 2010 . . . 16 1.5.1 Junior Section . . . . . . . . . . . . . . . . 16 1.5.2 Senior Section . . . . . . . . . . . . . . . . 18 1.6 Hanoi Open Mathematics Competition 2011 . . . 19 1.6.1 Junior Section . . . . . . . . . . . . . . . . 19 1.6.2 Senior Section . . . . . . . . . . . . . . . . 21 1.7 Hanoi Open Mathematics Competition 2012 . . . 23 1 www.VNMATH.com 1.7.1 Junior Section . . . . . . . . . . . . . . . . 23 1.7.2 Senior Section . . . . . . . . . . . . . . . . 26 1.8 Hanoi Open Mathematics Competition 2013 . . . 28 1.8.1 Junior Section . . . . . . . . . . . . . . . . 28 1.8.2 Senior Section . . . . . . . . . . . . . . . . 31 2 www.VNMATH.com Chapter 1 Hanoi Open Mathematics Competition 1.1 Hanoi Open Mathematics Competition 2006 1.1.1 Junior Section Question 1. What is the last two digits of the number (11 + 12 + 13 + ··· + 2006) 2 ? Question 2. Find the last two digits of the sum 2005 11 + 2005 12 + ··· + 2005 2006 . Question 3. Find the number of different positive integer triples (x, y, z) satisfying the equations x 2 + y −z = 100 and x + y 2 − z = 124. Question 4. Suppose x and y are two real numbers such that x + y −xy = 155 and x 2 + y 2 = 325. Find the value of |x 3 − y 3 |. 3 www.VNMATH.com Question 5. Suppose n is a positive integer and 3 arbitrary numbers are choosen from the set {1, 2, 3, . . . , 3n + 1} with their sum equal to 3n + 1. What is the largest possible product of those 3 numbers? Question 6. The figure ABCDEF is a regular hexagon. Find all points M belonging to the hexagon such that Area of triangle MAC = Area of triangle MCD. Question 7. On the circle (O) of radius 15cm are given 2 points A, B. The altitude OH of the triangle OAB intersect (O) at C. What is AC if AB = 16cm? Question 8. In ∆ABC, PQ  BC where P and Q are points on AB and AC respectively. The lines PC and QB intersect at G. It is also given EF//BC, where G ∈ EF , E ∈ AB and F ∈ AC with P Q = a and EF = b. Find value of BC. Question 9. What is the smallest possible value of x 2 + y 2 − x − y −xy? 1.1.2 Senior Section Question 1. What is the last three digits of the sum 11! + 12! + 13! + ··· + 2006! Question 2. Find the last three digits of the sum 2005 11 + 2005 12 + ··· + 2005 2006 . Question 3. Suppose that a log b c + b log c a = m. 4 www.VNMATH.com Find the value of c log b a + a log c b ? Question 4. Which is larger 2 √ 2 , 2 1+ 1 √ 2 and 3. Question 5. The figure ABCDEF is a regular hexagon. Find all points M belonging to the hexagon such that Area of triangle MAC = Area of triangle MCD. Question 6. On the circle of radius 30cm are given 2 points A, B with AB = 16cm and C is a midpoint of AB. What is the perpendicular distance from C to the circle? Question 7. In ∆ABC, PQ  BC where P and Q are points on AB and AC respectively. The lines P C and QB intersect at G. It is also given EF//BC, where G ∈ EF , E ∈ AB and F ∈ AC with P Q = a and EF = b. Find value of BC. Question 8. Find all polynomials P (x) such that P (x) +  1 x  = x + 1 x , ∀x = 0. Question 9. Let x, y, z be real numbers such that x 2 +y 2 +z 2 = 1. Find the largest possible value of |x 3 + y 3 + z 3 − xyz|? 5 www.VNMATH.com 1.2 Hanoi Open Mathematics Competition 2007 1.2.1 Junior Section Question 1. What is the last two digits of the number (3 + 7 + 11 + ··· + 2007) 2 ? (A) 01; (B) 11; (C) 23; (D) 37; (E) None of the above. Question 2. What is largest positive integer n satisfying the following inequality: n 2006 < 7 2007 ? (A) 7; (B) 8; (C) 9; (D) 10; (E) 11. Question 3. Which of the following is a possible number of diagonals of a convex polygon? (A) 02; (B) 21; (C) 32; (D) 54; (E) 63. Question 4. Let m and n denote the number of digits in 2 2007 and 5 2007 when expressed in base 10. What is the sum m + n? (A) 2004; (B) 2005; (C) 2006; (D) 2007; (E) 2008. Question 5. Let be given an open interval (α; eta) with eta − α = 1 2007 . Determine the maximum number of irreducible frac- tions a b in (α; eta) with 1 ≤ b ≤ 2007? (A) 1002; (B) 1003; (C) 1004; (D) 1005; (E) 1006. Question 6. In triangle ABC, ∠BAC = 60 0 , ∠ACB = 90 0 and D is on BC. If AD bisects ∠BAC and CD = 3cm. Then DB is (A) 3; (B) 4; (C) 5; (D) 6; (E) 7. 6 www.VNMATH.com Question 7. Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side 4. Prove that some three of these points are vertices of a triangle whose area is not greater than √ 3. Question 8. Let a, b, c be positive integers. Prove that (b + c − a) 2 (b + c) 2 + a 2 + (c + a − b) 2 (c + a) 2 + b 2 + (a + b − c) 2 (a + b) 2 + c 2 ≥ 3 5 . Question 9. A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides a, b, c satisfy the equation b = a(a − c). Prove that the triangle is isosceles. Question 10. Let a, b, c be positive real numbers such that 1 bc + 1 ca + 1 ab ≥ 1. Prove that a bc + b ca + c ab ≥ 1. Question 11. How many possible values are there for the sum a + b + c + d if a, b, c, d are positive integers and abcd = 2007. Question 12. Calculate the sum 5 2.7 + 5 7.12 + ··· + 5 2002.2007 . Question 13. Let be given triangle ABC. Find all points M such that area of ∆MAB= area of ∆MAC. Question 14. How many ordered pairs of integers (x, y) satisfy the equation 2x 2 + y 2 + xy = 2(x + y)? Question 15. Let p = abc be the 3-digit prime number. Prove that the equation ax 2 + bx + c = 0 has no rational roots. 7 www.VNMATH.com 1.2.2 Senior Section Question 1. What is the last two digits of the number  11 2 + 15 2 + 19 2 + ··· + 2007 2  2 ? (A) 01; (B) 21; (C) 31; (D) 41; (E) None of the above. Question 2. Which is largest positive integer n satisfying the following inequality: n 2007 > (2007) n . (A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above. Question 3. Find the number of different positive integer triples (x, y, z) satsfying the equations x + y −z = 1 and x 2 + y 2 − z 2 = 1. (A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above. Question 4. List the numbers √ 2, 3 √ 3, , 4 √ 4, 5 √ 5 and 6 √ 6 in order from greatest to least. Question 5. Suppose that A, B, C, D are points on a circle, AB is the diameter, CD is perpendicular to AB and meets AB at E, AB and CD are integers and AE −EB = √ 3. Find AE? Question 6. Let P (x) = x 3 + ax 2 + bx + 1 and |P(x)| ≤ 1 for all x such that |x| ≤ 1. Prove that |a|+ |b| ≤ 5. Question 7. Find all sequences of integers x 1 , x 2 , . . . , x n , . . . such that ij divides x i + x j for any two distinct positive integers i and j. 8 www.VNMATH.com Question 8. Let ABC be an equilateral triangle. For a point M inside ∆ABC, let D, E, F be the feet of the perpendiculars from M onto BC, CA, AB, respectively. Find the locus of all such points M for which ∠F DE is a right angle. Question 9. Let a 1 , a 2 , . . . , a 2007 be real numbers such that a 1 +a 2 +···+a 2007 ≥ (2007) 2 and a 2 1 +a 2 2 +···+a 2 2007 ≤ (2007) 3 −1. Prove that a k ∈ [2006; 2008] for all k ∈ {1, 2, . . . , 2007}. Question 10. What is the smallest possible value of x 2 + 2y 2 − x − 2y −xy? Question 11. Find all polynomials P (x) satisfying the equation (2x − 1)P (x) = (x − 1)P (2x), ∀x. Question 12. Calculate the sum 1 2.7.12 + 1 7.12.17 + ··· + 1 1997.2002.2007 . Question 13. Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove ∠F HP = ∠BAC. Question 14. How many ordered pairs of integers (x, y) satisfy the equation x 2 + y 2 + xy = 4(x + y)? 9 www.VNMATH.com