22 FileGocHOMOMoi

Contents Chapter 1.1 1.2 Chapter 2.1 Introduction Preface Glossary Examples for practice Arithmetic problems 2.1.1 Junior problems 2.1.2 Senior problems 2.1.3 Undergraduate problems 2.1.4 Olympiad problems 2.2 Algebraic problems 2.2.1 Junior problems 2.2.2 Senior problems 2.2.3 Undergraduate problems 2.2.4 Olympiad problems 2.3 Geometric problems 2.3.1 Junior problems 2.3.2 Senior problems 2.3.3 Olympiad problems 2.4 Analysic problems 2.4.1 Junior problems 2.4.2 Senior problems 2.4.3 Undergraduate problems 2.4.4 Olympiad problems 2.5 Problems of Other Topics 2.5.1 Junior problems 2.5.2 Senior problems 2.5.3 Undergraduate problems 2.5.4 Olympiad problems Chapter Exercises for training 3.1 Exercises 3.1.1 Exercises from Mathematic Reflextion 3.1.2 Exercises from OP from Around the World 3.2 Problems of Hanoi Open Mathematical Olympiad 3.2.1 Hanoi Open Mathematical Olympiad 2006 3.2.2 Hanoi Open Mathematical Olympiad 2007 3.2.3 Hanoi Open Mathematical Olympiad 2008 3.2.4 Hanoi Open Mathematical Olympiad 2009 3.2.5 Hanoi Open Mathematical Olympiad 2010 Senior Section 3.3 Singapore Open Mathematical Olympiad 2009 3.3.1 Junior Section 3.3.2 Senior Section Bibliography i 1 14 15 15 36 45 47 60 60 77 86 88 101 101 119 138 153 153 156 159 170 174 174 180 184 185 196 196 196 210 215 215 216 219 221 224 226 226 229 234 Chapter Introduction 1.1 Preface Although mathematical olympiad competitions are carried out by solving problems, the system of Mathematical Olympiads and the related training courses cannot involve only the techniques of solving mathematical problems Strictly speaking, it is a system of mathematical advancing education To guide students who are interested in mathematics and have the potential to enter the world of Olympiad mathematics, so that their mathematical ability can be promoted efficiently and comprehensively, it is important to improve their mathematical thinking and technical ability in solving mathematical problems Technical ability in solving mathematical problems does not only involve producing accurate and skilled computations and proofs, the standard methods available, but also the more unconventional, creative techniques It is clear that the usual syllabus in mathematical educations cannot satisfy the above requirements, hence the mathematical olympiad training books must be self-contained basically The book is based on the lecture notes used by the editor in the last 25 years for Olympiad training courses in BAC GIANG SPECIALIZING UPPER SECONDARY SCHOOL Its scope and depth significantly exceeds that of the usual syllabus, and introduces many concepts and methods of modern mathematics The core of each lecture are the concepts, theories and methods of solving mathematical problems Examples are then used to explain and enrich the lectures, and indicate their applications And from that, a number of questions are included for the reader to try Detailed solutions are provided in the book The examples given are not very complicated so that the readers can understand them more easily However, the practice questions include many from actual competitions which students can use to test themselves These are taken from a range of countries, e.g China, Russia, the USA and Singapore The questions are for students to practise, and test students’ ability to apply their knowledge in solving real competition questions Each section can be used for training courses with a few hours per week The test questions are not considered part of the lectures, since students can complete them on their own Acknowledgments My great thanks to Doctor of Science, Professor Nguyen Van Mau, and Doctor Associate Professor Nguyen Vu Luong for their strong support I would also like to thank my colleagues, MA Bach Dang Khoa, MA Tran Thi Ha Phuong, MA Nguyen Danh Hao and MA Tran Anh Duc for their careful reading of my manuscript, and their helpful suggestions This book would be not written today without their efficient assistance Nguyen Van Tien Chapter Introduction 1.2 Glossary Abel summation For an integer n > and reals a1 , a2 , , an and b1 , b2 , , bn, ∑ bi = bn ∑ + ∑ ((bi − bi+1 ) ∑ aj ) n n n−1 i i=1 i=1 i=1 j=1 Angle bisector theorem If D is the intersection of either angle bisector of angle ABC with line AC, then BA DA = ⋅ BC DC Arithmetic mean-geometric mean (AM-GM) inequality If a1 , a2 , , an are n n nonnegative numbers, then their arithmetic mean is defined as ∑ ak and their n k=1 geometric mean is defined as (a1 a2 ⋯an ) n The arithmetic mean - geometric mean inequality states that 1 n ∑ ak ⩾ (a1 a2 ⋯an ) n n k=1 with equality if and only if a1 = a2 = ⋯ = an The inequality is a special case of the power mean inequality Arithmetic mean-harmonic mean (AM-HM) inequality If a1 , a2 , , an are n n positive numbers, then their arithmetic mean is defined as ∑ ak and their harmonic n k=1 mean is defined as ⋅ The arithmetic mean - harmonic mean inequality states that n ∑ n k=1 ak n ∑ ak ⩾ n 1 n k=1 ∑ n k=1 ak ( ∑ ak ) ( ∑ n or k=1 ) ⩾ n2 k=1 ak n with equality if and only if a1 = a2 = ⋯ = an Like the arithmetic mean-geometric mean inequality, this inequality is a special case of the power mean inequality Bernoulli’s inequality For x > −1 and a > 1, with equality when x = (1 + x)a ⩾ + ax, 1.2 Glossary Binomial coefficient Cnk = n! , k!(n − k)! the coefficient of xk in the expansion of (x + 1)n Binomial theorem (x + y)n = ∑ Cnk xn−k y k = ∑ Cnk xk y n−k n n k=0 k=0 Brianchon’s theorem If hexagon ABCDEF is circumscribed about a conic in the projective plane such that A ≠ D, B ≠ E, and C ≠ F , then lines AD, BE, and CF concur (If they lie on a conic in the afine plane, then these lines either concur or are parallel.) This theorem the dual to Pascal’s theorem Brocard angle ( See Brocard points.) Brocard points Given a triangle ABC, there exists a unique point P such that ∠ABP = ∠BCP = ∠CAP and a unique point Q such that ∠BAQ = ∠CBQ = ∠ACQ The points P and Q are the Brocard points of triangle ABC Moreover, ∠ABP and ∠BAQ are equal; their value φ is the Brocard angle of triangle ABC Cauchy-Schwarz inequality For any real numbers a1 , a2 , , an , and b1 , b2 , , bn n (∑ a2i ) (∑ b2i ) i=1 i=1 n ⩾ (∑ bi ) n i=1 with equality if and only if and bi are proportional, i = 1, 2, , n Centrally symmetric A geometric figure is centrally symmetric (centrosymmetric) about a point O if, whenever P is in the figure and O is the midpoint of a segment P Q, then Q is also in the figure Centroid of a triangle Point of intersection of the medians Centroid of a tetrahedron Point of the intersection of the segments connecting the midpoints of the opposite edges, which is the same as the point of intersection of the segments connecting each vertex with the centroid of the opposite face Ceva’s theorem and its trigonometric form Let AD, BE, CF be three cevians of triangle ABC The following are equivalent: (i) AD, BE, CF are concurrent; Chapter Introduction AF BD CE ⋅ ⋅ = 1; F B DC EA sin ∠ABE sin ∠BCF sin ∠CAD ⋅ ⋅ = (iii) sin ∠EBC sin ∠F CA sin ∠DAB (ii) Cevian A cevian of a triangle is any segment joining a vertex to a point on the opposite side Chinese remainder theorem Let k be a positive integer Given integers a1 , a2 , , ak and pairwise relatively prime positive integers n1 , n2 , , nk , there exists a unique integer a such that ⩽ a < ∏ ni and a ≡ (mod ni ) for i = 1, 2, , k k i=1 Circumcenter Center of the circumscribed circle or sphere Circumcircle Circumscribed circle Complex numbers in planar geometry If we introduce a Cartesian coordinate system in the Euclidean plane, we can assign a complex number to each point in the plane by assigning α + βi to the point (α, β) for all reals α and β Suppose that A, B, , F are points and a, b, , f are the corresponding complex numbers Then: → a + (c − b) corresponds to the translation of A under the vector BC; given an angle θ, b+eiθ (a−b) corresponds to the image of A under a rotation through θ about B; given a real scalar λ, b + λ(a − b) corresponds to thee image of A under a homothety of ratio λ centered at B; the absolute value of a − b equals AB; (c − b) equals ∠ABC (directed and modulo 2π) the argument of (a − c) Using these facts, one can translate much of the language of geometry in the Euclidean plane into language about complex numbers Congruence For integers a, b, and n with n ≠ 1, a ≡ b (mod n) (or ”a is congruent to b modulo n”) means that a − b is divisible by n Concave up (down) function A function f (x) is concave up (down) on [a, b] ⊆ R if f (x) lies under (above) the line connecting (a1 , f (a1 )) and (b1 , f (b1 )) for all a ⩽ a1 < x < b1 ⩽ b A function g(x) is concave up (down) on the Euclidean plane if it is concave up (down) on each line in the plane, where we identify the line naturally with R Concave up and down functions are also called convex and concave, respectively Convex hull Given a nonempty set of points S in Euclidean space, there exists a convex set T such that every convex set containing S also contains T We call T the 1.2 Glossary convex hull of S Cyclic polygon Polygon that can be inscribed in a circle De Moivre’s formula For any angle a and for any integer n, (cos a + i sin a)n = cos na + i sin na Derangement A derangement of n items a1 , , an is a permutation (b1 , b2 , , bn ) of these items such that bi ≠ for all i According to a formula of Euler’s, there are exactly n! − derangements of n items n! n! n! n! + − + ⋯ + (−1)n 1! 2! 3! n! Desargues’ theorem Two triangles have corresponding vertices joined by lines which are concurrent or parallel if and only if the intersections of corresponding sides are collinear Directed angles A directed angle contains information about both the angle’s measure and the angle’s orientation (clockwise or counterclockwise) If two directed angles sum to zero, then they have the same angle measure but opposite orientations One often takes directed angles modulo π or 2π Some important features of directed angles modulo p follow: If A, B, C, D are points such that ∠ABC and ∠ABD are welldefined, then ∠ABC = ∠ABD if and only if B, C, D are collinear If A, B, C, D are points such that ∠ABC and ∠ADC are welldefined, then ∠ABC = ∠ADC if and only if A, B, C, D are concyclic π π Because 2(θ) = 2( + θ), but θ ≠ + θ, one cannot divide directed angles by For 2 example, if ∠ABC = 2∠ADC, D lies either on the internal angle bisector of angle ABC, or on the external angle bisector of angle ABC we cannot write ∠ADC = ∠ABC to determine which line D lies on These features show that using directed angles modulo π allows one to deal with multiple possible configurations of a geometry problem at once, but at the expense of possibly losing important information about a configuration Euler’s formula (for planar graphs) If F, V, and E are the number of faces, vertices, and edges of a planar graph, then F + V − E = This is a special case of an invariant of topological surfaces called the Euler characteristic Euler’s formula (in planar geometry) Let O and I be the circumcenter and incenter, respectively, of a triangle with circumradius R and inradius r Then OI = R2 − 2rR Chapter Introduction Euler line The orthocenter, centroid and circumcenter of any triangle are collinear The centroid divides the distance from the orthocenter to the circumcenter in the ratio of : The line on which these three points lie is called the Euler line of the triangle Euler’s theorem Given relatively prime integers a and m with m ⩾ aφ(m) ≡ a(modm), where φ(m) is the number of positive integers less than or equal to m and relatively prime to m Euler’s theorem is a generalization of Fermat’s little theorem Excircles or escribed circles Given a triangle ABC, there are four circles tangent to the lines AB, BC, CA One is the inscribed circle, which lies in the interior of the triangle One lies on the opposite side of line BC from A, and is called the excircle (escribed circle) opposite A, and similarly for the other two sides The excenter opposite A is the center of the excircle opposite A; it lies on the internal angle bisector of A and the external angle bisectors of B and C Excenters See excircles Exradius The radius of the three excircles of a triangle n Fermat number A number of the form 22 for some positive integer n Fermat’s little theorem If p is prime, then ap ≡ a(modp) for all integers a Feuerbach circle The feet of the three altitudes of any triangle, the midpoints of the three sides, and the midpoints of segments from the three vertices to the orthocenter, all lie on the same circle, the Feuerbach circle or the nine-point circle of the triangle Let R be the circumradius of the triangle The nine-point circle of the triangle has radius R/2 and is centered at the midpoint of the segment joining the orthocenter and the circumcenter of the triangle Feuerbach’s theorem The nine-point circle of a triangle is tangent to the incircle and to the three excircles of the triangle Fibonacci sequence The sequence F0 , F1 , defined recursively by F0 = 0, F1 = 1, and Fn+2 = Fn+1 + Fn for all n ⩾ Generating function If a0 , a1 , a2 , is a sequence of numbers, then the generating function for the sequence is the infinite series a0 + a1 x + a2 x2 + ⋯ If f is a function such that f (x) = a0 + a1 x + a2 x2 + ⋯, 1.2 Glossary then we also refer to f as the generating function for the sequence Graph A graph is a collection of vertices and edges, where the edges are distinct unordered pairs of distinct vertices We say that the two vertices in one of these unordered pairs are adjacent and connected by that edge The degree of a vertex is the number of edges which ontain it A path is a sequence of vertices v1 , v2 , , such that vi is adjacent to vi+1 for each i A graph is called connected if for any two vertices v and w, there exists a path from v to w A cycle of the graph is an ordered collection of vertices v1 , v2 , , such that v1 ≡ and such that the (vi , vi+1 ) are distinct edges A connected graph which contains no cycles is called a tree, and every tree contains at least two leaves, vertices with degree Harmonic conjugates Let A, C, B, D be four points on a line in that order If the points C and D divide AB internally and externally in the same ratio, (i.e., AC ∶ CB = AD ∶ DB), then the points C and D are said to be harmonic conjugates of each other with respect to the points A and B, and AB is said to be harmonically divided by the points C and D If C and D are harmonic with respect to A and B, then A and B are harmonic with respect to C and D Harmonic range The four points A, B, C, D are referred to as a harmonic range, denoted by (ABCD), if C and D are harmonic conjugates with respect to A and B Helly’s theorem If n > d and C1 , , Cn are convex subsets of Rd , each d+1 of which have nonempty intersection, then there is a point in common to all the sets Heron’s formula The area of a triangle with sides a, b, c is equal to √ a+b+c s(s − a)(s − b)(s − c), where s = Hă olders inequality Let w1 , , wn be positive real numbers whose sum is For any positive real numbers aij , ∏ (∑ aij ) n m i=1 j=1 wi i ⩾ ∑ ∏ aw ij m n j=1 i=1 Homothety A homothety (central similarity) is a transformation that fixes one point O (its center) and maps each point P to a point P ′ for which O, P, P ′ are collinear and the ratio OP ∶ OP ′ = k is constant (k can be either positive or negative), where k is called the magnitude of the homothety Homothetic triangles Two triangles ABC and DEF are homothetic if they have parallel sides Suppose that AB ∥ DE, BC ∥ EF, and CA ∥ F D Then lines AD, BE, and CF concur at a point X, as given by a special case of Desargues’ theorem Furthermore, Chapter Introduction some homothety centered at X maps triangle ABC onto triangle DEF Incenter Center of inscribed circle Incircle Inscribed circle Inversion of center O and ratio r Given a point O in the plane and a real number r > 0, the inversion through O with radius r maps every point P ≠ O to the point P ′ on → the ray OP such that OP ⋅ OP ′ = r We also refer to this map as inversion through ω, the circle with center O and radius r Key properties of inversion are: Lines through O invert to themselves (though the individual points on the line are not all fixed) Lines not through O invert to circles through O and vice versa Circles not through O invert to other circles not through O A circle other than ω inverts to itself (as a whole, not point-by-point) if and only if it is orthogonal to ω, that is, it intersects ω and the tangents to the circle and to ω at either intersection point are perpendicular Isogonal conjugate Let ABCbe a triangle and let P be a point in the plane which does not lie on any of the lines AB, BC, and CA There exists a unique point Q in the plane such that ∠ABP = ∠QBC, ∠BCP = ∠QCA, and ∠CAP = ∠QAB, where the angles in these equations are directed modulo π We call Q the isogonal conjugate of P With this definition, we see that P is also the isogonal conjugate of Q Jensen’s inequality If f is concave up on an interval [a, b] and λ1 , λ2 , , λn are nonnegative numbers with sum equal to 1, then λ1 f (x1 ) + λ2 f (x2 ) + ⋯ + λn f (xn ) ⩾ f (λ1 x1 + λ2 x2 + ⋯ + λn xn ) for any x1 , x2 , , xn in the interval [a, b] If the function is concave down, the inequality is reversed a Kummer’s Theorem Given nonnegative integers a and b and a prime p, pt ∣Ca+b if and only if t is less than or equal to the number of carries in the addition a + b in base p Lattice point In the Cartesian plane, the lattice points are the points (x, y) for which x and y are both integers Law of cosines In a triangle ABC, CA2 = AB + BC − 2AB ⋅ BC cos ∠ABC, and analogous equations hold for AB and BC Law of quadratic reciprocity If p, q are distinct odd primes, then (p−1)(q−1) p q ( ) ( ) = (−1) q p 1.2 Glossary q p where ( ) and ( ) are Legendre symbols q p Law of sines In a triangle ABC with circumradius equal to R one has sin A sin B sin C = = = 2R BC AC AB Legendre symbol If m is an integer and n is a positive prime, then Legendre symm bol ( ) is defined to equal if n ∣ m, if m is a quadratic residue modulo n, and −1 if n m is a quadratic nonresidue modulo n Lucas’s theorem Let p be a prime; let a and b be two positive integers such that a = ak pk + ak−1 pk−1 + ⋯ + a1 p + a0 , b = bk pk + bk−1 pk−1 + ⋯ + b1 p + b0 , where ⩽ , bi < p are integers for i = 0, 1, , k Then ⋯Cab11 Cab00 (mod p) Cabk−2 Cab ≡ Cabkk Cabk−1 k−2 k−1 Matrix A matrix is a rectangular array of objects A matrix A with m rows and n columns is an m × n matrix The object in the ith row and j th column of matrix A is denoted ai,j If a matrix has the same number of rows as it has columns, then the matrix is called a square matrix In a square n × n matrix A, the main diagonal consists of the elements a1,1 , a2,2 , , an,n Menelaus’ theorem Given a triangle ABC, let F, G, H be points on lines BC, CA, AB, respectively Then F, G, H are collinear if and only if, using directed lengths, AH BF CG ⋅ ⋅ = −1 HB F C GA Minkowski’s inequality Given a positive integer n, a real number r ⩾ 1, and positive reals a1 , a2 , , an and b1 , b2 , , bn , we have r r r (∑(ai + bi )r ) ⩽ (∑ ari ) + (∑ bri ) n i=1 n i=1 n i=1 Multiset Informally, a multiset is a set in which an element may appear more than once For instance, {1, 2, 3, 2} and {2, 2, 2, 3, 1} are distinct multisets Nine point circle (See Feuerbach circle.) Orbit Suppose that S is a collection of functions on a set T , such that S is closed under composition and each f ∈ S has an inverse T can be partitioned into its orbits under S, sets of elements such that a and b are in the same set if and only if f (a) = b for some f ∈ S 220 Chapter Exercises for training Question Suppose x, y, z, t are real numbers such that ⎧ ∣ − x + y + z + t∣ ⎪ ⎪ ⎪ ⎪ ⎪∣x − y + z + t∣ ⎪ ⎪ ⎨ ⎪ ∣x + y − z + t∣ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩∣x + y + z − t∣ Prove that x2 + y + z + t2 ⩽ ⩽1 ⩽1 ⩽1 ⩽ Question Let P (x) be a polynomial such that Find P (x2 + 1)? P (x2 − 1) = x4 − 3x2 + Question The figure ABCDE is a convex pentagon Find the sum ∠DAC + ∠EBD + ∠ACE + ∠BDA + ∠CEB? Question The sides of a rhombus have length a and the area is S What is the length of the shorter diagonal? Question Let be given a right-angled triangle ABC with ∠A = 900 , AB = c, AC = b Let E ∈ AC and F ∈ AB such that ∠AEF = ∠ABC and ∠AF E = ∠ACB Denote by P ∈ BC and Q ∈ BC such that EP ⊥ BC and F Q ⊥ BC Determine EP +EF +P Q? Question 10 Let a, b, c ∈ [1, 3] and satisfy the following conditions max{a, b, c} ⩾ 2, a + b + c = What is the smallest possible value of a2 + b2 + c2 ? Senior Section, Sunday, 30 March 2008 Question How many integers are there in (b, 2008b], where b (b > 0) is given Question Find all pairs (m, n) of positive integers such that m2 + 2n2 = 3(m + 2n) Question Show that the equation x2 + 8z = + 2y has no solutions of positive integers x, y and z Question Prove that there exists an infinite number of relatively prime pairs (m, n) of positive integers such that the equation x3 − nx + mn = 3.2 Problems of Hanoi Open Mathematical Olympiad 221 has three distint integer roots Question Find all polynomials P (x) of degree such that max P (x) − P (x) = b − a, ∀ a, b ∈ R a⩽x⩽b where a < b a⩽x⩽b Question Let a, b, c ∈ [1, 3] and satisfy the following conditions max{a, b, c} ⩾ 2, a + b + c = What is the smallest possible value of a2 + b2 + c2 ? Question Find all triples (a, b, c) of consecutive odd positive integers such that a < b < c and a2 + b2 + c2 is a four digit number with all digits equal Question Consider a convex quadrilateral ABCD Let O be the intersection of AC and BD; M, N be the centroid of AOB and COD and P, Q be orthocenter of BOC and DOA, respectively Prove that MN ⊥ P Q Question Consider a triangle ABC For every point M ∈ BC we define N ∈ CA and P ∈ AB such that AP MN is a parallelogram Let O be the intersection of BN and CP Find M ∈ BC such that ∠P MO = ∠OMN Question 10 Let be given a right-angled triangle ABC with ∠A = 900 , AB = c, AC = b Let E ∈ AC and F ∈ AB such that ∠AEF = ∠ABC and ∠AF E = ∠ACB Denote by P ∈ BC and Q ∈ BC such that EP ⊥ BC and F Q ⊥ BC Determine EP + EF + F Q? 3.2.4 Hanoi Open Mathematical Olympiad 2009 Junior Section, Sunday, 29 March 2009 Question What is the last two digits of the number 1000 ⋅ 1001 + 1001 ⋅ 1002 + 1002 ⋅ 1003 + ⋯ + 2008 ⋅ 2009? (A) 25; (B) 41; (C) 36; (D) 54; (E) None of the above Question Which is largest positive integer n satisfying the inequality 1 1 + + +⋯+ < ⋅ 1⋅2 2⋅3 3⋅4 n(n + 1) (A) 3; (B) 4; (C) 5; (D) 6; (E) None of the above 222 Chapter Exercises for training Question How many positive integer roots of the inequality −1 < there in (−10, 10) (A) 15; (B) 16; (C) 17; (D) 18; (E) None of the above x−1 < are x+1 Question How many triples (a, b, c) where a, b, c ∈ {1, 2, 3, 4, 5, 6} and a < b < c such that the number abc + (7 − a)(7 − b)(7 − c) is divisible by (A) 15; (B) 17; (C) 19; (D) 21; (E) None of the above Question Show that there is a natural number n such that the number a = n! ends exacly in 2009 zeros Question Let a, b, c be positive integers with no common factor and satisfy the conditions 1 + = a b c Prove that a + b is a square Question Suppose that a = 2b + 19, where b = 210n+1 Prove that a is divisible by 23 for any positive integer n Question Prove that m7 − m is divisible by 42 for any positive integer m Question Suppose that real numbers a, b, c, d satisfy the conditions a2 + b2 = c2 + d2 = and ac + bd = Find the set of all possible values the number M = ab + cd can take Question 10 Let a, b be positive integers such that a + b = 99 Find the smallest and the greatest values of the following product P = ab Question 11 Find all integers x, y such that x2 + y = (2xy + 1)2 Question 12 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 15 Question 13 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 Suppose that P S intersects QR at T Evaluate area (∆P QT ) Question 14 Let ABCbe an acute-angled triangle with AB = and CD be the altitude through C with CD = Find the distance between the midpoints of AD and BC 223 3.2 Problems of Hanoi Open Mathematical Olympiad Senior Section, Sunday, 29 March 2009 Question What is the last two digits of the number 1000 ⋅ 1001 + 1001 ⋅ 1002 + 1002 ⋅ 1003 + ⋯ + 2008 ⋅ 2009? (A) 25; (B) 41; (C) 36; (D) 54; (E) None of the above Question Which is largest positive integer n satisfying the inequality 1 + + +⋯+ < ⋅ 1⋅2 2⋅3 3⋅4 n(n + 1) (A) 3; (B) 4; (C) 5; (D) 6; (E) None of the above Question How many integral roots of the inequality −1 < (−10, 10) (A) 15; (B) 16; (C) 17; (D) 18; (E) None of the above x−1 < are there in x+1 Question How many triples (a, b, c) where a, b, c ∈ {1, 2, 3, 4, 5, 6} and a < b < c such that the number abc + (7 − a)(7 − b)(7 − c) is divisible by (A) 15; (B) 17; (C) 19; (D) 21; (E) None of the above Question Suppose that a = 2b + 19, where b = 210n+1 Prove that a is divisible by 23 for any positive integer n Question Determine all positive integral pairs (u, v) for which 5u2 + 6uv + 7v = 2009 Question Prove that for every positive integer n there exists a positive integer m such that the last n digists in decimal representation of m3 are equal to Question Give an example of a triangle whose all sides and altitudes are positive integers Question Given a triangle ABC with BC = 5, CA = 4, AB = and the points E, F, G lie on the sides BC, CA, AB, respectively, so that EF is parallel to AB and area (∆EF G) = Find the minimum value of the perimeter of triangle EF G Question 10 Find all integers x, y, z satisfying the system ⎧ ⎪ ⎪x + y + z ⎨ 3 ⎪ ⎪ ⎩x + y + z =8 = Question 11 Let be given three positive numbers p, q and r Suppose that real numbers a, b, c, d satisfy the conditions 224 Chapter Exercises for training ⎧ a2 + b2 = p ⎪ ⎪ ⎪ ⎪ ⎨c + d2 = q ⎪ ⎪ ⎪ ⎪ ⎩ac + bd = r Find the set of all possible values the number M = ab + cd can take Question 12 Let a, b, c, d be positive integers such that a + b + c + d = 99 Find the smallest and the greatest values of the following product P = abcd Question 13.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 meets the outwards facing semicirle drawn on BC as diameter Points B, C are located similarly Evaluate the sum T = (area ∆BCA)2 + (area ∆CAB)2 + (area ∆ABC)2 Question 14 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 times less than 2009 3.2.5 Hanoi Open Mathematical Olympiad 2010 Senior Section Sunday, 28 March 2010 08h45-11h45 Important: Answer all 10 questions Enter your 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 Question The number of integers n ∈ [2000, 2010] such that 22n + 2n + is divisible by 7, is (A) : 0; ; (B): 1; (C) : 2; (D) : 3; (E) : None of the above Question The last digits of the number 52010 are (A) : 65625; (B) : 45625; (C) : 25625; (D) : 15625; (E) : None of the above Question How many real numbers a ∈ (1, 9) such that the corresponding number a − is an integer a (A) : 0; (B) : 1; (C) : 8; (D) : 9; (E) : None of the above 225 3.2 Problems of Hanoi Open Mathematical Olympiad Question Each box in a × table can be colored black or white How many different colorings of the table are there? Question Determine all positive integer a such that the equation 2x2 − 210x + a = has two prime roots, i.e both roots are prime numbers Question Let a, b be the roots of the equation x2 − px + q = and let c, d be the roots of the equation x2 −rx+s = 0, where p, q, r, s are some positive real numbers Suppose that 2(abc + bcd + cda + dab) M= p2 + q + r + s2 is an integer Determine a, b, c, d 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, AM respectively If AP = 3cm, BP = 4cm, compute the value of ? BN Question If n and n3 + 2n2 + 2n + are both perfect squares, find n? Question Let x, y be the positive integers such that 3x2 + x = 4y + y Prove that x − y is a perfect integer Question 10 Find the maximum value of T= y z x + + , 2x + y 2y + z 2z + x x, y, z > 226 3.3 3.3.1 Chapter Exercises for training Singapore Open Mathematical Olympiad 2009 Junior Section Tuesday, June 2009 0930-1200 hrs Important: Answer ALL 35 questions Enter your answer sheet provided For the multiple choice questions, enter your answer on the answer sheet by shading the bubble containing the letter (A, B, C, D or E) corresponding to the correct answer For the other short questions, write your answer in the answer sheet No steps are needed to justify your answer Each question carries mark No calculators are allowed Part A Multiple Choice Questions Question Let C1 and C2 be distinct circles of radius cm that are in the same plane and tangenr to each other Find the number of circles of radius 26 cm in this plane that are tangent to both C1 and C2 (A) ; (B) ; (C) ; (D) ; (E) non of the above Question In the diagram below, the radius of quadrant ODA is and the radius of quadrant OBC is Given that ∠COD = 30o , find the area of the shades region ABCD (A) 12p ; (B) 13p ; (C) 15p ; (D) 16p ; (E) non of the above Question Let k be√a real number Find the maximum value of k such that the √ following inequality holds: x − + − x ⩾ k (A) √ √ √ √ √ ; (B) ; (C) + ; (D) 10 ; (E) Question Three circles of radius 20 are arranged with their respective centres A, B and C in a row If the line W Z is tangent to the third circle, find the langth of XY (A) 30 ; (B) 32 ; (C) 34 ; (D) 36 ; (E) 38 Question Given that x and y are both negative integers satisfying the equation 10x , find the maximum value of y y= 10 − x (A) -10 ; (B) -9 ; (C) -6 ; (D) -5 ; (E) non of the above Question The sequence an satisfy an = an−1 + n2 and a0 = 2009 Find a50 (A) 42434 ; (B) 42925 ; (C) 44934 ; (D) 45029 ; (E) 45359 3.3 Singapore Open Mathematical Olympiad 2009 227 Question Coins of the same size are arranged on a very large table (the infinite plane) such that each coin touches six other coins Find the percetage of the plane that is covered by the coins √ √ √ 50 20 (A) √ π% ; (B) √ π% ; (C) 16 3π% ; (D) 17 3π% ; (E) 18 3π% 3 Question Given that x and y are real numbers satisfying the following equations: √ x + xy + y = + and x2 + y = 6, find the value of ∣x + y + 1∣ (A) + √ √ √ √ √ ; (B) − ; (C) + ; (D) − ; (E) + Question Given that y = (x − 16)(x − 14)(x + 14)(x + 16), find the minimum value of y (A) − 896 ; (B) − 897 ; (C) − 898 ; (D) − 899 ; (E) − 900 Question 10 The number of positive integral solutions (a, b, c, d) satisfying 1 1 + + + =1 a b c d with the condition that a < b < c < d is (A) ; (B) ; (C) ; (D) ; (E) 10 Part B Short Questions Question 11 There are two models of LCD television on sale One is a ”20 inch” standard model while the other is a ”20 inch” widescreen model The ratio of length to the height of the standard mode is : 3, while that of the widescreen model is 16 : Television screens are measured by the length of their diagonals, so both models have the same diagonal length of 20 inches If the ratio of the area of the standard model to that of the widescreen model is A ∶ 300, find the value of A Question 12 The diagram below shows a pentagon (made up of region A and region B) and a rectangle (made up of region B and region C ) that overlap The overlapped region B is of the pentagon and of the rectangle If the ratio of region A of the 16 m pentagon to region C of the rectangle is in its lowest term, find the value of m + n n Question 13 2009 students are taking a test which comprises ten true or fales questions Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers Question 14 The number of ways to arrange boys and giels in a row such that girls can be adjacent to other girls but boys cannot be adjacent to other boys is 6! × k 228 Chapter Exercises for training Find the value of k Question 15 ABC is a right-angled triangle with ∠BAC = 900 A square is constructed on the side AB and BC as shown The area of the square ABDE is 8cm2 and the area of the square BCF G is 26cm2 Find the area of triangle DBG in cm2 Question 16 The sum of 1 1 + + +⋯+ + 2×3×4 3×4×5 4×5×6 13 × 14 × 15 14 × 15 × 16 m in its lowest terms Find the value of m + n n 1 = b+ − and a − b + ≠ 0, find the value of Question 17 Given that a + a+1 b−1 ab − a + b is Question 18 If ∣x∣ + x + 5y = and ∣y∣ − y + x = 7, find the value of x + y + 2009 Question 19 Let p and q represent two consecutive prime number For some fixed integer n, the set {n−1, 3n−19, 38−5n, 7n−45} represents {p, 2p, q, 2q}, but not necessarily in that order Find the value of n Question 20 Find the number of ordered pairs of positive intergers (x, y) that satisfy the equation √ √ √ √ √ x y + y x + 2009xy − 2009x − 2009y − 2009 = Question 21 Find the intergers part of ⋅ 1 1 1 + + + + + + 2003 2004 2005 2006 2007 2008 2009 Question 22 Given the rectangle ABLJ, where the area of ACD, BCEF, DEIJ and F GH are 22cm2 , 500cm2 and 22cm2 respectively Find the area of HIK in cm2 √ √ √ √ 3 Question 23 Evaluate 77 − 20 13 + 77 + 20 13 Question 24 Find the number of integers in the set {1, 2, 3, , 2009} whose sum of the digits is 11 Question 25 Given that x + (1 + x)2 + (1 + x)3 + ⋯ + (1 + x)n = a0 + a1 x + a2 x2 + ⋯ + an xn , where each ar is an integer, r = 0, 1, 2, , n Find the value of n such that a0 + a2 + a3 + a4 + ⋯ + an−2 + an−1 = 60 − n(n + 1) ⋅ 229 3.3 Singapore Open Mathematical Olympiad 2009 Question 26 In the diagram, OAB is a triangle with ∠AOB = 900 and OB = 13cm P and Q are points on AB such that 26AP = 22P Q = 11QB If the vertical height of P Q = 4cm, find the area of the triangle OP Q in cm2 Question 27 Let x1 , x2 , x3 , x4 denote the four roots of the equation x4 + kx2 + 90x − 2009 = If x1 x2 = 49, Find the value of k Question 28 Three sides OAB, OAC and OBC of tetrahedron OABC are rightangled triangles, i.e ∠AOB = ∠AOC = ∠BOC = 900 Given that OA = 7, OB = and OC = 6, find the value of (Area of ∆OAB)2 +(Area of ∆OAC)2 +(Area of ∆OBC)2 +(Area of ∆ABC)2 n − 10 is a non-zero reQuestion 29 Find the least positive integer n for which 9n + 11 ducible fraction Question 30 Find the value of the smallest positive integer m such that the equation has only integer solutions x2 + 2(m + 5)x + (100m + 9) = Question 31 In a triangle ABC, the length of the altitudes AD and BE are and 12 respectively Find the largest possible integer calue for the length of third altitude CF Question 32 A four-digit number consists of two distinct pairs of repeated digit (for example 2211, 2626 and 7007) Find the total number of such possible number that are divisible by or 101 but not both Question 33 m and n are two positive integer satisfying ⩽ m ⩽ n ⩽ 40 Find the number of pairs of (m, n) such that their produc mn is divisible by 33 Question 34 Using the digits 0, 1, 2, 3, and 4, find the number of 13-digit sequences that can be written so that the diffenrence between any two consecutive digits is Question 35 m and n are two positive integers of reverse order (for example 123 and 321) such that mn = 1446921630 Find the value of m + n 3.3.2 Senior Section Tuesday, June 2009 0930-1200 hrs 230 Chapter Exercises for training Important: Answer ALL 35 questions Enter your answer sheet provided For the multiple choice questions, enter your answer on the answer sheet by shading the bubble containing the letter (A, B, C, D or E) corresponding to the correct answer For the other short questions, write your answer in the answer sheet No steps are needed to justify your answer Each question carries mark No calculators are allowed Part A Multiple Choice Questions Question Suppose that P is a plane and A and B are two points on the plane P If the distance between A and B is 33 cm, how many lines are there in the plane such that the distance between each line and A is cm and the distance between each line and B is 26 cm, respectively (A) ; (B) ; (C) ; (D) ; (E) infinitely many Question Let y = (17 − x)(19 − x)(19 + x)(17 + x), where x is a real number Find the smallest posible value of y (A) − 1296 ; (B) − 1295 ; (C) − 1294 ; (D) − 1293 ; (E) − 1292 Question If two real numbers a and b are randomly chosen from the interval (0, 1), find the probability that the equation x2 − ax + b = has real roots (A) 1 ; (B) ; (C) ; (D) ; (E) 16 16 Question If x and y are real numbers for which ∣x∣ + x + 5y = and ∣y∣ − y + x = 7, find the value of x + y (A) − ; (B) − ; (C) ; (D) ; (E) Question In a triangle ABC, sin A = (A) and cos B = ⋅ Find the value of cos C 13 16 56 16 56 56 16 56 or ; (B) ; (C) ; (D) − ; (E) or − 65 65 65 65 65 65 65 Question The area of a triangle ABC is 40cm2 Points D, E and F are on sides AB, BC and CA, respectively, as shown in the figure below If AD = 3cm, DB = 5cm, and the area of triangle ABE is equal to the area of quadrilateral DBEF, find the area of triangle AEC in cm2 (A) 11 ; (B) 12 ; (C) 13 ; (D) 14 ; (E) 15 Question Find the value of 22 + +⋯+ ⋅ 1! + 2! + 3! 2! + 3! + 4! 20! + 21! + 22! 3.3 Singapore Open Mathematical Olympiad 2009 231 1 1 1 1 ; (B) − ; (C) − ; (D) − ; (E) − 24! 23! 22! 22! 24! Question There are eight envolopes numbered to Find the number of ways in which identical red buttons and identical blue buttons can be put in the envolopes such that each envolope contains exactly one button, and the sum of the nimbers on the envolopes containing the blue buttons (A) − (A) 35 ; (B) 34 ; (C) 32 ; (D) 31 ; (E) 62 Question Determine the number of acute-angled triangles (i.e., all angles are less than 90o ) in which all angles (in degrees) are positive integers and the largest angle is three times the smallest angle (A) ; (B) ; (C) ; (D) ; (E) Question 10 Let ABCD be a quadrilateral inscribed in a circle with diameter AC, and let E be the foot of perpendicular from D onto AB, as shown in the figure below If AD = DC and the area of quadrilateral ABCD is 24cm2 , find the length of DE in cm √ √ √ √ (A) ; (B) ; (C) ; (D) ; (E) Part B Short Questions Question 11 Find the number of positive divisors of (20083 +(3 ×2008 ×2009)+1))2 Question 12 Suppose that a, b and c are real numbers greater than 1 1 + + ⋅ Find the value of c a + loga2 b ( a ) + logb2 c ( b ) + logc2 a ( cb ) Question 13 Find the remainder when is divided by 2009 (1! × 1) + (2! × 2) + (3! × 3) + ⋯ + (286! × 286) √ √ Question 14 Find the value of (25 + 10 5) + (25 − 10 5) √ + 2009 ⋅ Find the value of (a3 − 503a − 500)10 Question 15 Let a = Question 16 ABC is a triangle and D is a point on side BC Point E is on side AB such that DE is the angle bisector of ∠ADB, and point F is on side AC such that DF AE BD CF is the angle bisector of ∠ADC Find the value of ⋅ EB DC F A Question 17 Find the value of (cot 25o − 1)(cot 24o − 1)(cot 23o − 1)(cot 22o − 1)(cot 21o − 1)(cot 20o − 1) Question 18 Find the number of 2-element subset {a, b} of {1, 2, 3, , 99, 100} such that ab + a + b is a multiple of 232 Chapter Exercises for training Question 19 Let x be real number such that x2 −15x+1 = Find the value of x4 + ⋅ x4 Question 20 ABC is a triangle with AB = 10cm, BC = 40cm Points D and E lie on side AC and point F on side BC such that EF is parallel to AB and DF is parallel to EB Given that BE is an angle bisector of ∠ABC and that AD = 13.5cm, find the length of CD in cm Question 21 Let S = {1, 2, 3, , 64, 65} Determine the number of ordered triples (x, y, z) such that x, y, z ∈ S, x < z and y < z Question 22 Given that an+1 = find the value of an−1 , where n = 1, 2, 3, , and a0 = a1 = 1, + nan−1 an ⋅ a199 a200 Question 23 ABC is a triangle with AB = 5cm, BC = 13cm and AC = 10cm Points area of ∆AP Q P and Q lie on sides AB and AC respectively such that = ⋅ area of ∆ABC Given that the least posible length of P Q is kcm, find the value of k Question 24 If x, y are real numbers such that x + y + z = and xy + yz + zx = 24, find the largest possible value of z Question 25 Find the number of − binary sequences formed by six 0′ s and 1′ s such that no three 0′ s are together For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not Question 26 If cos 1000 = tanx, find x − sin 250 cos 250 cos 500 Question 27 Find the number of positive integers x, where x ≠ 9, such that log x 9 x2 < + log3 x Question 28 Let n be the positive integer such that 1 1 √ √ + √ √ +⋯+ √ √ = ⋅ 11 + 11 11 13 + 13 11 n n + + (n + 2) n Find the value of n Question 29 ABCD is a rectangle, E is the midpoint of AD and F is the midpoint of CE If the area of triangle BDF is 12cm2 , find the earea of rectangle ABCD in cm2 Question 30 In each of the following 6-digit positive integers: 555555, 555333, 818811, 300388, 3.3 Singapore Open Mathematical Olympiad 2009 233 every digit in the number appears at least twice Find the number of such 6-digit positive integers Question 31 Let x and y be positive integers such that 27x + 35y ⩽ 945 Find the largest posible value of xy Question 32 Determine the coefficient of x29 in the expansion (1 + x5 + x7 + x9 )16 Question 33 For n = 1, 2, 3, , let an = n2 + 100, and let dn denote the greatest common divisor of an and an+1 Find the maximun value of dn as n ranges over all positive integers Question 34 Using the digits 1, 2, 3, 4, 5, 6, 7, 8, we can form 8! (= 40320) 8-digit numbers in which the eight digits are all distinct For ⩽ k ⩽ 40320, let ak denote the k th number if these numbers are arranged in increasing order: 12345678, 12345687, 12345768, , 87654321; that is, a1 = 12345678, a2 = 12345687, , a40320 = 87654321 Find a2009 − a2008 100 ] x Here [c] denotes the greatest integer less than or equal to c Find the largest possible value of 2a2 − 3b2 Question 35 Let x be a positive integer, and write a = [log10 x] and b = [log10 Bibliography [1] Nguyen Van Mau Hanoi Open Mathematical Olympiad, Problems and Solutions HMS Hanoi, 2009 [2] Dusan Djukic- Vladimir Jankovic- Ivan Matic-Nikola Petrovic, The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads, 1959-2004 Springer, 2006 [3] Titu Andreescu and Zuming Feng, 102 Combinatorial Problems from the Training of the USA IMO Team Bikhauser, 2002 [4] Titu Andreescu and Zuming Feng, A path to combinatorics for undergraduates counting strategies Bikhauser, 2004 [5] Titu Andreescu, Razvan Gelca, Mathematical Olympiad Challenges Bikhauser, 2000 [6] Titu Andreescu and Zuming Feng, Mathematical Olympiads 1998 - 2000 Problems and Solutions From Around the World MMA 2002 [7] Arthur Engel, Problem - Solving Strategies Springer, 1998 [8] Loren C Larson, Problem - Solving Through Problems Springer, 1983 [9] Walter Mientka et al., Mathematical Olympiads 1996 - 1998, Problems and Solutions From Around the World, MMA 1997, 1998 [10] Acz’el, J.: 1966, Lectures on Functional Equations and Their Applications.New York, Birkhauser [11] Gleason, A M., Greenwood, R E and Kelly, L M.: 1980, The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965- 1984 MMA [12] Kedlaya, K S., Poonen, B., and Vakil, R.: 2002, The William Lowell Putnam Mathematical Competition 1985-2000 Problems, Solutions and Commentary MMA [13] A Engel, Problem-Solving Strategies, Springer Verlag, 1998 [14] Websites of mathemathics: http://diendantoanhoc.net http://mathnfriend.net www.kalva.co.uk http://www.ams.org/ http://www.math.ac.vn/ http://math.ca/crux/ http://kvant.mccme.ru/ http://mathforum.org http://www.math.com/ http://www.bymath.com/ http://lib.mexmat.ru/ http://hms.org.vn/hms/ http://sms.math.nus.edu.sg/ http://thesaurus.maths.org/ 234 ... + 16)(16n + 9) = (12n)2 + (92 + 162 )n + 122 22 Chapter Examples for practice is also a perfect square Since (12n + 12)2 ⩽ (12n)2 + (92 + 162 )n + 122 < (12n + 15)2 it follows that if n > then... a { 12 x1 + x22 + ⋯ + x2n = a has no integer solutions Solution First, we notice that if xi ≠ 0, for an integer component xi then x2i > xi and we have a contradiction a = x21 + x22 + ⋯ + x2n... then 2a ≡ ( mod 3) Proof ≠ ( mod 3) ⇒ 2k ≠ ( mod 3) ⇒ (2k − 1) (2k + 1) ≡ ( mod 3) Thus 22k ≡ ( mod 3) ⇒ 22k + = ( mod 3) Lemma If p = is a prime then p2 = ( mod 3) Proof If p is such a prime