Đề olympic toán quốc tế international tại trung quốc national

A collection of National Mathematics Olympiads July 19, 2014 Contents Preface Due to problems with the pdf generator at AoPS I decided to make my own collection of math Olympiad problems in pdf format with links to problems at AoPS The idea is also that the list may be printed, and therefore a secondary aim is to not waste as much space as the AoPS pdf’s, but rather to start a new page only for the next country’s Olympiads Of course, when the document it printed, one will not be able to follow links Therefore I include the link location in as minimalist a form I could think of The links are all of the form http://www.artofproblemsolving.com/Forum/viewtopic.php?p=***, where *** is a number (the post number on AoPS), mostly consisting of digits, but earlier posts may have less Thus after each problem that appears on AoPS (to my knowledge) I include the post number, which also links to the problem there Of course, this is a work in progress, since there is quite a lot to and there are constantly new contests being written Therefore I start with the most popular contest, and the ones I have most complete collections of Also, in stead of defining common terms over and over in problems that refer to them, I include a glossary at the end, where undefined terms can be looked up I also changed the margins in which the text is written This is not something I normally do, since the appearance turns out to be quite strange However, for printing purposes this saves a lot of pages This is only a private collection and not something professional That is why I feel this change is warranted There are also places where I have slightly altered the text This is mostly removing superfluous definitions like “where R is the set of real numbers” or “where x denotes the greatest integer ”, etc Austria Federal Competition Part Austria Federal Competition For Advanced Students, Part 2002 Determine all integers a and b such that (19a + b)18 + (a + b)18 + (a + 19b)18 is a perfect square AoPS:2297672 Find the greatest real number C such that, for all real numbers x and y = x with xy = it holds that ((x + y)2 − 6)((x − y)2 + 8) ≥ C (x − y)2 When does equality occur? AoPS:2297674 x Let f (x) = 9x9+3 Compute are coprime to 2002 k f k 2002 , where k goes over all integers k between and 2002 which AoPS:2297681 Let A, C, P be three distinct points in the plane Construct all parallelograms ABCD such that point P lies on the bisector of angle DAB and ∠AP D = 90◦ AoPS:2297685 Austria Federal Competition For Advanced Students, Part 2003 Find all triples of prime numbers (p, q, r) such that pq + pr is a perfect square AoPS:2317090 Find the greatest and smallest value of f (x, y) = y − 2x, if x, y are distinct non-negative real +y numbers with xx+y ≤ AoPS:2317091 Given a positive real number t, find the number of real solutions a, b, c, d of the system a(1 − b2 ) = b(1 − c2 ) = c(1 − d2 ) = d(1 − a2 ) = t AoPS:2317093 In a parallelogram ABCD, points E and F are the midpoints of AB and BC, respectively, and P is the intersection of EC and F D Prove that the segments AP, BP, CP and DP divide the parallelogram into four triangles whose areas are in the ratio : : : AoPS:2317094 Austria Federal Competition For Advanced Students, Part 2004 Find all quadruples (a, b, c, d) of real numbers such that a + bcd = b + cda = c + dab = d + abc AoPS:2317197 A convex hexagon ABCDEF with AB = BC = a, CD = DE = b, EF = F A = c is inscribed in a circle Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals AoPS:2317198 For natural numbers a, b, define Z(a, b) = (3a)!·(4b)! a!4 ·b!3 (a) Prove that Z(a, b) is an integer for a ≤ b (b) Prove that for each natural number b there are infinitely many natural numbers a such that Z(a, b) is not an integer AoPS:2317203 √ √ Each of the 2N = 2004 real numbers x1 , x2 , , x2004 equals either − or + Can the sum N k=1 x2k−1 x2 k take the value 2004? Which integral values can this sum take? AoPS:2317209 Austria Federal Competition For Advanced Students, Part 2005 Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2, ,9 an equal number of times AoPS:269328 For how many integers a with |a| ≤ 2005, does the system x2 = y+a = x+a y have integer solutions? AoPS:269331 For real numbers a, b, c let sn = an + bn + cn It is known that s1 = 2, s2 = and s3 = 14 Prove that for all natural numbers n > 1, we have |s2n − sn−1 sn+1 | = AoPS:269336 We are given two congruent, equilateral triangles ABC and P QR with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down One is placed above the other so that the area of intersection is a hexagon A1 A2 A3 A4 A5 A6 (labelled counterclockwise) Prove that A1 A4 , A2 A5 and A3 A6 are concurrent AoPS:269346 Austria Federal Competition For Advanced Students, Part 2006 Let n be a non-negative integer, which ends written in decimal notation on exactly k zeros, but which is bigger than 10k For a n is only k = k(n) ≥ known In how many different ways (as a function of k = k(n) ≥ 2) can n be written as difference of two squares of non-negative integers at least? AoPS:1402125 Show that the sequence an = n 2−n (n+1) n 7n2 +1 is strictly increasing, where n = 0, 1, 2, AoPS:1402086 In the triangle ABC let D and E be the boundary points of the incircle with the sides BC and AC Show that if AD = BE holds, then the triangle is isosceles AoPS:1402097 Given is the function f = x2 + {x} for all x ∈ R+ Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator 3, if they are a reduced fraction, and don’t lie in the range of the function f AoPS:1402126 Austria Federal Competition For Advanced Students, Part 2007 In a quadratic table with 2007 rows and 2007 columns is an odd number written in each field For ≤ i ≤ 2007 is Zi the sum of the numbers in the i-th row and for ≤ j ≤ 2007 is Sj the sum of the numbers in the j-th column A is the product of all Zi and B the product of all Sj Show that A + B = AoPS:1399124 For every positive integer n determine the highest value C(n), such that for every n-tuple (a1 , a2 , , an ) of pairwise distinct integers  n (n + 1) j=1 a2j −  n j=1 2 aj  ≥ C(n) AoPS:1399131 Let M (n) = {−1, −2, , −n} For every non-empty subset of M (n) we consider the product of its elements How big is the sum over all these products? AoPS:1399135 Let n > be a non-negative integer Given is the in a circle inscribed convex n-gon A0 A1 A2 An−1 An (An = A0 ) where the side Ai−1 Ai = i (for ≤ i ≤ n) Moreover, let φi be the angle between the line Ai Ai+1 and the tangent to the circle in the point Ai (where the angle φi is less than or equal 90o , i.e φi is always the smaller angle of the two angles between the two lines) Determine the sum n−1 Φ= φi i=0 of these n angles AoPS:1399145 Austria Federal Competition For Advanced Students, Part 2008 What is the remainder of the number 1· 2008 2008 2008 +2· + · · · + 2009 · 2008 is divided by 2008? Given a ∈ R+ and an integer n > determine all n-tuples (x1 , , xn ) of positive real numbers that satisfy the following system of equations: x1 x2 (3a − 2x3 ) = a3 x2 x3 (3a − 2x4 ) = a3 xn−2 xn−1 (3a − 2xn ) = a3 xn−1 xn (3a − 2x1 ) = a3 xn x1 (3a − 2x2 ) = a3 Let p > be a natural number Consider the set Fp of all non-constant sequences of non-negative integers that satisfy the recursive relation an+1 = (p + 1)an pan1 for all n > Show that there exists a sequence (an ) in Fp with the property that for every other sequence (bn ) in Fp , the inequality an ≤ bn holds for all n In a triangle ABC let E be the midpoint of the side AC and F the midpoint of the side BC Let G be the foot of the perpendicular from C to AB Show that EF G is isosceles if and only if ABC is isosceles Austria Federal Competition For Advanced Students, Part 2009 Show that for all positive integer n the following inequality holds: 3n > (n!)4 For a positive integers n, k we define k-multifactorial of n as Fk (n) = n · (n − k) · (n − 2k) · · · r, where r is the reminder when n is divided by k that satisfy ≤ r ≤ k Determine all non-negative integers n such that F20 (n) + 2009 is a perfect square There are n bus stops placed around the circular lake Each bus stop is connected by a road to the two adjacent stops (we call a segment the entire road between two stops) Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly n + segments Let D, E, and F be respectively the midpoints of the sides BC, CA, and AB of ABC Let Ha , Hb , Hc be the feet of perpendiculars from A, B, C to the opposite sides, respectively Let P , Q, R be the midpoints of the Hb Hc , Hc Ha , and Ha Hb respectively Prove that P D, QE, and RF are concurrent Austria Federal Competition For Advanced Students, Part 2010 2010 Let f (n) = k=0 nk Show that for any integer m satisfying ≤ m ≤ 2010, there exists no natural number n such that f (n) is divisible by m AoPS:1874739 n For a positive integer n, we define the function fn (x) = k=1 |x − k| for all real numbers x For any two-digit number n (in decimal representation), determine the set of solutions Ln of the inequality fn (x) < 41 AoPS:1874750 Given is the set Mn = {0, 1, 2, , n} of non-negative integers less than or equal to n A subset S of Mn is called outstanding if it is non-empty and for every natural number k ∈ S, there exists a k-element subset Tk of S Determine the number a(n) of outstanding subsets of Mn AoPS:1874757 The the parallel lines through an inner point P of triangle ABC split the triangle into three parallelograms and three triangles adjacent to the sides of ABC (a) Show that if P is the incentre, the perimeter of each of the three small triangles equals the length of the adjacent side (b) For a given triangle ABC, determine all inner points P such that the perimeter of each of the three small triangles equals the length of the adjacent side (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? AoPS:1874765 Austria Federal Competition For Advanced Students, Part 2011 Determine all integer triplets (x, y, z) such that x4 + x2 = 7z y AoPS:2270915 For a positive integer k and real numbers x and y, let fk (x, y) = (x + y) − x2k+1 + y 2k+1 If x2 + y = 1, then determine the maximal possible value ck of fk (x, y) AoPS:2270918 A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two How many three-element subsets of the set of integers {z ∈ Z | −2011 < z < 2011} are arithmetic and harmonic? AoPS:2270919 Inside or on the faces of a tetrahedron with five edges of length and one edge of length 1, there is a point P having distances a, b, c, d to the four faces of the tetrahedron Determine the locus of all points P such that a + b + c + d is minimal and the locus of all points P such that a + b + c + d is maximal AoPS:2270922 Austria Federal Competition For Advanced Students, Part 2012 Determine all functions f : Z → Z satisfying the following property: For each pair of integers m and n (not necessarily distinct), gcd(m, n) divides f (m) + f (n) AoPS:2693139 Determine all solutions (n, k) of the equation n! + An = nk with n, k ∈ N for A = and for A = 2012 AoPS:2693137 Consider a stripe of n fields, numbered from left to right with the integers to n in ascending order Each of the fields is coloured with one of the colours 1, or Even-numbered fields can be coloured with any colour Odd-numbered fields are only allowed to be coloured with the odd colours and How many such colourings are there such that any two neighbouring fields have different colours? AoPS:2693138 Let ABC be a scalene (i.e non-isosceles) triangle Let U be the centre of the circumcircle of this triangle and I the centre of the incircle Assume that the second point of intersection different from C of the angle bisector of γ = ∠ACB with the circumcircle of ABC lies on the perpendicular bisector of U I Show that γ is the second-largest angle in the triangle ABC AoPS:2693136 Austria Federal Competition For Advanced Students, Part 2013 Show that if for non-negative integers m, n, N , k the equation k (n2 + 1)2 · (44n3 + 11n2 + 10n + 2) = N m holds, then m = AoPS:3103094 Solve the following system of equations in rational numbers: (x2 + 1)3 = y + 1, (y + 1)3 = z + 1, (z + 1)3 = x + AoPS:3103104 Arrange the positive integers into two lines as follows: We start with writing in the upper line, in the lower line and again in the upper line Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line The number of consecutive integers in a block is determined by the first number in the previous block Let a1 , a2 , a3 , be the numbers in the upper line Give an explicit formula for an AoPS:3103138 Let A, B and C be three points on a line (in this order) For each circle k through the points B and C, let D be one point of intersection of the perpendicular bisector of BC with the circle k Further, let E be the second point of intersection of the line AD with k Show that for each circle k, the ratio of lengths BE : CE is the same AoPS:3103150 Austria Federal Competition Part Austria Federal Competition For Advanced Students, Part 1997 Let a be a fixed integer Find all integer solutions x, y, z of the system 5x + (a + 2)y + (a + 2)z (2a + 4)x + (a + 3)y + (2a + 2)z (2a + 4)x + (2a + 2)y + (a + 3)z = a, = 3a − 1, = a + AoPS:2344288 A positive integer K is given Define the sequence (an ) by a1 = and an is the n-th positive integer greater than an−1 which is congruent to n modulo K (a) Find an explicit formula for an (b) What is the result if K = 2? AoPS:2344289 Let be given a triangle ABC Points P on side AC and Y on the production of CB beyond B are chosen so that Y subtends equal angles with AP and P C Similarly, Q on side BC and X on the production of AC beyond C are such that X subtends equal angles with BQ and QC Lines Y P and XB meet at R, XQ and Y A meet at S, and XB and Y A meet at D Prove that P QRS is a parallelogram if and only if ACBD is a cyclic quadrilateral AoPS:2344294 Determine all quadruples (a, b, c, d) of real numbers satisfying the equation 256a3 b3 c3 d3 = (a6 + b2 + c2 + d2 )(a2 + b6 + c2 + d2 )(a2 + b2 + c6 + d2 )(a2 + b2 + c2 + d6 ) AoPS:2344299 We define the following operation which will be applied to a row of bars being situated side-by-side on positions 1, 2, , N Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars After that, all bars will be put side-by- side in such a way that all bars form a new row and are situated on positions 1, , M From an initial number a0 > of bars there originates a sequence (an )n≥0 , where an is the number of bars after having applied the operation n times (a) Prove that for no n > can we have an = 1997 (b) Determine all natural numbers that can only occur as a0 or a1 AoPS:2344302 For every natural number n, find all polynomials x2 +ax+b, where a2 ≥ 4b, that divide x2n +axn +b AoPS:2344303 Find all poltnomials P (x) with real coefficients satisfying: For all a > 1995, the number of real roots of P (x) = a (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995 AoPS:1249019 Given an integer n ≥ and a reular 2n-gon Color all verices of the 2n-gon with n colors such that: (a) Each vertex is colored by exactly one color (b) Two vertices don’t have the same color Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon Find the total number of mutually non-equivalent ways of coloring AoPS:12385 Vietnam National Olympiad 1996 Solve the system of equations: √ ) x+y 7y(1 − ) x+y 3x(1 + = = √ AoPS:1247514 Given a trihedral angle Sxyz A plane (P ) not through S cuts Sx, Sy, Sz respectively at A, B, C On the plane (P ), outside triangle ABC, construct triangles DAB, EBC, F CA which are confruent to the triangles SAB, SBC, SCA respectively Let (T ) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC Prove that (T ) touches the plane (P ) at the circumcenter of triangle DEF AoPS:1247507 Let be given integers k and n such that ≤ k ≤ n Find the number of ordered k-tuples (a1 , a2 , , an ), where a1 , a2 , , ak are different and in the set {1, 2, , n}, satisfying (a) There exist s, t such that ≤ s < t ≤ k and as > at (b) There exists s such that ≤ s ≤ k and as is not congruent to s mod AoPS:926575 Find all f : N → N so that : f (n) + f (n + 1) = f (n + 2)f (n + 3) − 1996 AoPS:938887 The triangle ABC has BC = and ∠BAC = a Find the shortest distance between its incenter and its centroid Denote this shortest distance by f (a) When a varies in the interval ( π3 , π), find the maximum value of f (a) AoPS:1247512 Prove that a + b + c + d ≥ 32 (ab + bc + ca + ad + ac + bd) where a, b, c, d are positive real numbers satisfying 2(ab + bc + cd + da + ac + bd) + abc + bcd + cda + dab = 16 AoPS:759420 728 Vietnam National Olympiad 1997 Given a circle with centre O and radius R A point P lies inside the circle, OP = d, d < R We consider quadrilaterals ABCD, inscribed in the circle, such that AC is perp to BD at point P Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d AoPS:1247372 mn for all Let n be an integer which is greater than 1, not divisible by 1997 Let am = m + 1997 1997m m = 1, 2, , 1996, bm = m + n for all m = 1, 2, , n − We arrange the terms of two sequence (ai ), (bj ) in the ascending order to form a new sequence c1 ≤ c2 ≤ ≤ c1995+n Prove that ck+1 − ck < for all k = 1, 2, , 1994 + n AoPS:1247377 Find the number of functions f : N → N which satisfying: (a) f (1) = (b) f (n)f (n + 2) = f (n + 1) + 1997 for every natural numbers n AoPS:1247378 Let k = √ 3 (a) Find all polynomials p(x) with rationl coefficients whose degree are as least as possible such that p(k + k ) = + k (b) Does there exist a polynomial p(x) with integer coefficients satisfying p(k + k ) = + k AoPS:1247382 Prove that for evey positive integer n, there exits a positive integer k such that 2n |19k − 97 AoPS:1247384 In the unit cube, given 75 points, no three of which are collinear Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72 AoPS:1247374 Vietnam National Olympiad 1998 x2 Let a ≥ be a real number Put x1 = a, xn+1 = + ln ( 1+lnnxn ) (n = 1, 2, ) Prove that the sequence {xn } converges and find its limit AoPS:757778 Let be given a tetrahedron whose circumcenter is O Draw diameters AA1 , BB1 , CC1 , DD1 of the circumsphere of ABCD Let A0 , B0 , C0 , D0 be the centroids of triangle BCD, CDA, DAB, ABC Prove that A0 A1 , B0 B1 , C0 C1 , D0 D1 are concurrent at a point, say, F Prove that the line through F and a midpoint of a side of ABCD is perpendicular to the opposite side AoPS:758717 The sequence {an }n≥0 is defined by a0 = 20, a1 = 100, an+2 = 4an+1 + 5an + 20 (n = 0, 1, 2, ) Find the smallest positive integer h satisfying 1998|an+h − an ∀n = 0, 1, 2, AoPS:758726 Does there exist an infinite sequence {xn } of reals satisfying the following conditions (a) |xn | ≤ 0, 666 for all n = 1, 2, (b) |xm − xn | ≥ n(n+1) + m(m+1) for all m = n? AoPS:757785 729 Find minimum value of F (x, y) = where x, y ∈ R (x + 1)2 + (y − 1)2 + (x − 1)2 + (y + 1)2 + (x + 2)2 + (y + 2)2 , AoPS:758722 Find all positive integer n such that there exists a P ∈ R[x] satisfying P (x1998 − x−1998 ) = xn − x−n ∀x ∈ R − {0} AoPS:758728 Vietnam National Olympiad 1999 Solve the system of equations: (1 + 42x−y ).51−2x+y = + 22x−y+1 y + 4x + ln(y + 2x) + = AoPS:1248394 Let a triangle ABC and A , B , C be the midpoints of the arcs BC, CA, AB respectively of its circumcircle A B , A C meets BC at A1 , A2 respectively Pairs of point (B1 , B2 ), (C1 , C2 ) are similarly defined Prove that A1 A2 = B1 B2 = C1 C2 if and only if triangle ABC is equilateral AoPS:1248402 Let {xn }n≥0 and {yn }n≥0 be two sequences defined recursively as follows x0 = 1, x1 = 4, xn+2 = 3xn+1 − xn , y0 = 1, y1 = 2, yn+2 = 3yn+1 − yn (a) Prove that xn − 5yn + = for all non-negative integers (b) Suppose that a, b are two positive integers such that a2 − 5b2 + = Prove that there exists a non-negative integer k such that a = xk and b = yk AoPS:849861 Given are three positive real numbers a, b, c satisfying abc + a + c = b Find the max value of the expression: − + P = a + b2 + c2 + AoPS:413859 OA, OB, OC, OD are rays in space such that the angle between any two is the same Show that for a variable ray OX, the sum of the cosines of the angles XOA, XOB, XOC, XOD is constant and the sum of the squares of the cosines is also constant AoPS:1321 Let S = {0, 1, 2, , 1999} and T = {0, 1, 2, } Find all functions f : T → S such that (a) f (s) = s ∀s ∈ S (b) f (m + n) = f (f (m) + f (n)) ∀m, n ∈ T AoPS:10996 730 Vietnam National Olympiad 2000 √ Given a real number c > 0, a sequence (xn ) of real numbers is defined by xn+1 = c − c + xn for n ≥ Find all values of c such that for each initial value x0 in (0, c), the sequence (xn ) is defined for all n and has a finite limit lim xn when n → +∞ AoPS:1293817 Two circles (O1 ) and (O2 ) with respective centers O1 , O2 are given on a plane Let M1 , M2 be points on (O1 ), (O2 ) respectively, and let the lines O1 M1 and O2 M2 meet at Q Starting simultaneously from these positions, the points M1 and M2 move clockwise on their own circles with the same angular velocity (a) Determine the locus of the midpoint of M1 M2 (b) Prove that the circumcircle of M1 QM2 passes through a fixed point AoPS:1293816 Consider the polynomial P (x) = x3 + 153x2 − 111x + 38 (a) Prove that there are at least nine integers a in the interval [1, 32000 ] for which P (a) is divisible by 32000 (b) Find the number of integers a in [1, 32000 ] with the property from (a) AoPS:1293828 For every integer n ≥ and any given angle α with < α < π, let Pn (x) = xn sin α − x sin nα + sin(n − 1)α (a) Prove that there is a unique polynomial of the form f (x) = x2 + ax + b which divides Pn (x) for every n ≥ (b) Prove that there is no polynomial g(x) = x + c which divides Pn (x) for every n ≥ AoPS:1293833 Find all integers n ≥ such that there are n points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent AoPS:1293840 Let P (x) be a nonzero polynomial such that, for all real numbers x, P (x2 − 1) = P (x)P (−x) Determine the maximum possible number of real roots of P (x) AoPS:1293837 Vietnam National Olympiad 2001 A circle center O meets a circle center O at A and B The line T T touches the first circle at T and the second at T The perpendiculars from T and T meet the line OO at S and S The ray AS meets the first circle again at R, and the ray AS meets the second circle again at R Show that R, B and R are collinear AoPS:788932 Let N = 6n , where n is a positive integer, and let M = aN + bN , where a and b are relatively prime integers greater than 1.M has at least two odd divisors greater than are p, q Find the residue of pN + q N mod · 12n AoPS:788935 For real a, b define the sequence x0 , x1 , x2 , by x0 = a, xn+1 = xn + b sin xn If b = 1, show that the sequence converges to a finite limit for all a If b > 2, show that the sequence diverges for some a AoPS:788936 731 Find the maximum value of x12 + y22 + z32 , where x, y, z are positive reals satisfying √ √ √ √ √ √ √ min(x 2,y 3) , x + z ≥ 6, y + z 10 ≥ √1 ≤ z < AoPS:788939 2x Find all real-valued continuous functions defined on the interval (−1, 1) such that (1−x2 )f ( 1+x 2) = 2 (1 + x ) f (x) for all x AoPS:788940 (a1 , a2 , , a2n ) is a permutation of {1, 2, , 2n} such that |ai − ai+1 | = |aj − aj+1 | for i = j Show that a1 = a2n + n iff ≤ a2i ≤ n for i = 1, 2, , n AoPS:788942 Vietnam National Olympiad 2002 Solve the equation √ − 10 − 3x = x − AoPS:1293843 An isosceles triangle ABC with AB = AC is given on the plane A variable circle (O) with center O on the line BC passes through A and does not touch either of the lines AB and AC Let M and N be the second points of intersection of (O) with lines AB and AC, respectively Find the locus of the orthocenter of triangle AM N AoPS:1293848 Let be given two positive integers m, n with m < 2001, n < 2002 Let distinct real numbers be written in the cells of a 2001 × 2002 board (with 2001 rows and 2002 columns) A cell of the board is called bad if the corresponding number is smaller than at least m numbers in the same column and at least n numbers in the same row Let s denote the total number of bad cells Find the least possible value of s AoPS:1293853 Let a, b, c be real numbers for which the polynomial x3 + ax2 + bx + c has three real roots Prove that 12ab + 27c ≤ 6a3 + 10 a2 − 2b When does equality occur? AoPS:1293900 √ Determine for which n positive integer the equation: a + b + c + d = n abcd has positive integer solutions AoPS:137109 For a positive integer n, consider the equation x−1 + 4x−1 + ··· + k2 x−1 + ··· + n2 x−1 = 12 (a) Prove that, for every n, this equation has a unique root greater than 1, which is denoted by xn (b) Prove that the limit of sequence (xn ) is as n approaches infinity AoPS:1293911 Vietnam National Olympiad 2003 Let f : R → R is a function such that f (cot x) = cos 2x + sin 2x for all < x < π Define g(x) = f (x)f (1 − x) for −1 ≤ x ≤ Find the maximum and minimum values of g on the closed interval [−1, 1] AoPS:779769 732 The circles C1 and C2 touch externally at M and the radius of C2 is larger than that of C1 A is any point on C2 which does not lie on the line joining the centers of the circles B and C are points on C1 such that AB and AC are tangent to C1 The lines BM , CM intersect C2 again at E, F respectively D is the intersection of the tangent at A and the line EF Show that the locus of D as A varies is a straight line AoPS:112198 Let Sn be the number of permutations (a1 , a2 , , an ) of (1, 2, , n) such that ≤ |ak − k| ≤ for all k Show that 74 Sn−1 < Sn < 2Sn−1 for n > AoPS:779774 Find the largest positive integer n such that the following equations have integer solutions in x, y1 , y2 , , yn : (x + 1)2 + y12 = (x + 2)2 + y22 = · · · = (x + n)2 + yn2 AoPS:779779 Define p(x) = 4x3 − 2x2 − 15x + 9, q(x) = 12x3 + 6x2 − 7x + Show that each polynomial has just three distinct real roots Let A be the largest root of p(x) and B the largest root of q(x) Show that A2 + 3B = AoPS:779781 Let F be the set of all functions f : (0, ∞) → (0, ∞) such that f (3x) ≥ f (f (2x)) + x for all x Find the largest A such that f (x) ≥ Ax for all f ∈ F and all x AoPS:779784 Vietnam National Olympiad 2004 Solve the system of equations   x + x(y − z) = y + y(z − x)2 = 30   z + z(x − y)2 = 16 AoPS:1293960 In a triangle ABC, the bisector of ∠ACB cuts the side AB at D An arbitrary circle (O) passing through C and D meets the lines BC and AC at M and N (different from C), respectively (a) Prove that there is a circle (S) touching DM at M and DN at N (b) If circle (S) intersects the lines BC and CA again at P and Q respectively, prove that the lengths of the segments M P and N Q are constant as (O) varies AoPS:1293963 Let A be the set of the 16 first positive integers Find the least positive integer k satisfying the condition: In every k-subset of A, there exist two distinct a, b ∈ A such that a2 + b2 is prime AoPS:850156 (2+cos 2α)xn −cos2 α (2−2 cos 2α)xn +2−cos 2α , for all n ∈ N, where α The sequence (xn )∞ n=1 is defined by x1 = and xn+1 = n is a given real parameter Find all values of α for which the sequence (yn ) given by yn = k=1 2xk1+1 has a finite limit when n → +∞ and find that limit AoPS:1293969 Let x, y, z be positive reals satisfying (x + y + z) = 32xyz Find the minimum and the maximum of P = x4 +y +z (x+y+z)4 AoPS:307796 733 Let S(n) be the sum of decimal digits of a natural number n Find the least value of S(m) if m is an integral multiple of 2003 AoPS:1293971 Vietnam National Olympiad 2005 Let x, y be real numbers satisfying the condition: √ x − x + = y + − y Find the greatest value and the smallest value of: P =x+y AoPS:182624 Let (O) be a fixed circle with the radius R Let A and B be fixed points in (O) such that A, B, O are not collinear Consider a variable point C lying on (O) (C = A, B) Construct two circles (O1 ), (O2 ) passing through A, B and tangent to BC, AC at C, respectively The circle (O1 ) intersects the circle (O2 ) in D (D = C) Prove that: (a) CD ≤ R (b) The line CD passes through a point independent of C (i.e there exists a fixed point on the line CD when C lies on (O)) AoPS:182627 Let A1 A2 A3 A4 A5 A6 A7 A8 be convex 8-gon (no three diagonals concruent) The intersection of arbitrary two diagonals will be called “button”.Consider the convex quadrilaterals formed by four vertices of A1 A2 A3 A4 A5 A6 A7 A8 and such convex quadrilaterals will be called “sub quadrilaterals”.Find the smallest n satisfying: We can color n “button” such that for all i, k ∈ {1, 2, 3, 4, 5, 6, 7, 8}, i = k, s(i, k) are the same where s(i, k) denote the number of the “sub quadrilaterals” has Ai , Ak be the vertices and the intersection of two its diagonals is ”button” AoPS:182628 Find all function f : R → R satisfying the condition: f (f (x − y)) = f (x) · f (y) − f (x) + f (y) − xy AoPS:183462 Find all triples of natural (x, y, n) satisfying the condition: x! + y! = 3n n! AoPS:183463 Let {xn } be a real sequence defined by: x1 = a, xn+1 = 3x3n − 7x2n + 5xn for all n = 1, 2, 3, and a ∈ R Find all a such that {xn } has finite limit when n → +∞ and find the finite limit in that cases AoPS:183465 734 Vietnam National Olympiad 2007 Solve the system of equations: 12 3x + y 12 1− 3x + y l 1+ √ x = √ y = AoPS:750310 Let x, y be integer number with x, y = −1 so that by x + x −1 y+1 −1 + yx+1 ∈ Z Prove that x4 y 44 − is divisible AoPS:750312 Let B, C be fixed points and A be moving point Let H, G be orthocentre and centroid of triangle ABC Known midpoint of HG lies on BC, find locus of A AoPS:750355 Given a regular 2007-gon Find the minimal number k such that: Among every k vertexes of the polygon, there always exists vertexes forming a convex quadrilateral such that sides of the quadrilateral are also sides of the polygon AoPS:750364 Given a number b > 0, find all functions f : R → R such that: y f (x + y) = f (x).3b +f (y)−1 + bx 3b y +f (y)−1 − by ∀x, y ∈ R AoPS:750373 Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC P is a point lying outer segment BC P A cut (O) at N (that means P A isn’t tangent of (O)), the circle with diameter P D intersect (O) at E, DE meet BC at N Prove that M N always pass through a fixed point AoPS:750359 Vietnam National Olympiad 2008 Determine the number of solutions of the simultaneous equations x2 +y = 29 and log3 x·log2 y = AoPS:1236750 Given a triangle with acute angle ∠BEC, let E be the midpoint of AB Point M is chosen on the opposite ray of EC such that ∠BM E = ∠ECA Denote by θ the measure of angle ∠BEC C Evaluate M AB in terms of θ AoPS:1236754 Let m = 20072008 , how many natural numbers n are there such that n < m and n(2n + 1)(5n + 2) is divisible by m (which means that m | n(2n + 1)(5n + 2)) ? AoPS:1236757 he sequence of real number (xn ) is defined by x1 = 0, x2 = and xn+2 = 2−xn + Prove that the sequence has a limit as n approaches +∞ Determine the limit ∀n = 1, 2, AoPS:1236767 What is the total number of natural numbers divisible by the number of digits of which does not exceed 2008 and at least two of the digits are 9s? AoPS:1236769 735 Let x, y, z be distinct non-negative real numbers Prove that 1 + + ≥ (x − y)2 (y − z)2 (z − x)2 xy + yz + zx When does the equality hold? AoPS:1236774 Let AD is centroid of ABC triangle Let (d) is the perpendicular line with AD Let M is a point on (d) Let E, F are midpoints of M B, M C respectively The line through point E and perpendicular with (d) meet AB at P The line through point F and perpendicular with (d) meet AC at Q Let (d ) is a line through point M and perpendicular with P Q Prove (d ) always pass a fixed point AoPS:1020457 Vietnam National Olympiad 2009 Find all (x, y) such that: √ + + 2x2 x(1 − 2x) + 1+ 2y y(1 − 2y) = √ = + 2xy AoPS:1418664 The sequence {xn } is defined by x1 xn = x2n−1 + 4xn−1 + xn−1 = Prove that the sequence {yn }, where yn = n i=1 xi , has a finite limit and find that limit AoPS:1418806 Let A, B be two fixed points and C is a variable point on the plane such that ∠ACB = α (constant) (0◦ ≤ α ≤ 180◦ ) Let D, E, F be the projections of the incentre I of triangle ABC to its sides BC, CA, AB, respectively Denoted by M , N the intersections of AI, BI with EF , respectively Prove that the length of the segment M N is constant and the circumcircle of triangle DM N always passes through a fixed point AoPS:1418799 Let a, b, c be three real numbers For each positive integer number n, an + bn + cn is an integer number Prove that there exist three integers p, q, r such that a, b, c are the roots of the equation x3 + px2 + qx + r = AoPS:1418800 Let S = {1, 2, 3, , 2n} (n ∈ Z+ ) Determine the number of subsets T of S such that there are no element in T a, b such that |a − b| = {1, n} AoPS:1418805 Vietnam National Olympiad 2010 Solve the system equations x4 − y = 240 x − 2y = 3(x2 − 4y ) − 4(x − 8y) 3 AoPS:2103782 736 Let {an } be a sequence which satisfy a1 = and an= n n−1 + 2.3n−1 an−1 n−1 + ∀n ≥ (a) Find the general formula for an (b) Prove that {an } is decreasing sequences AoPS:2103783 In plane,let a circle (O) and two fixed points B, C lies in (O) such that BC not is the diameter.Consider a point A varies in (O) such that A = B, C and AB = AC Call D and E respective is intersect of BC and internal and external bisector of BAC,I is midpoint of DE.The line that pass through orthocenter of ABC and perpendicular with AI intersects AD, AE at M, N , respectively (a) Prove that M N pass through a fixed point (b) Determine the place of A such that SAM N has maxium value AoPS:2103785 Prove that for each positive integer n, the equation x2 + 15y = 4n has at least n integer solution (x, y) AoPS:2103786 Let a positive integer n Consider square table × One use n colors to color all cell of table such that each cell is colored by exactly one color Two colored table is same if we can receive them from other by a rotation through center of × table How many way to color this square table satifies above conditions? AoPS:2103784 Vietnam National Olympiad 2011 Prove that if x > and n ∈ N, then we have xn (xn+1 + 1) ≤ xn + 2n+1 x+1 AoPS:2143336 Let xn be a sequence of real numbers defined as x1 = 1; xn = 2n (n − 1)2 n−1 xi i=1 Show that the sequence yn = xn+1 − xn has finite limits as n → ∞ AoPS:2143337 Let AB be a diameter of a circle (O) and let P be any point on the tangent drawn at B to (O) Define AP ∩ (O) = C = A, and let D be the point diametrically opposite to C If DP meets (O) second time in E, then, (a) Prove that AE, BC, P O concur at M (b) If R is the radius of (O), find P such that the area of area in terms of R AM B is maximum, and calculate the AoPS:2143335 737 √ A convex pentagon ABCDE satisfies that the sidelengths and AC, AD ≤ Let us choose 2011 distinct points inside this pentagon Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least 403 points out of the 2011 given points AoPS:2143338 Define the sequence of integers an as; a0 = 1, a1 = −1, and an = 6an−1 + 5an−2 ∀n ≥ Prove that a2012 − 2010 is divisible by 2011 AoPS:2144391 Let ABC be a triangle such that ∠C and ∠B are acute Let D be a variable point on BC such that D = B, C and AD is not perpendicular to BC Let d be the line passing through D and perpendicular to BC Assume d ∩ AB = E, d ∩ AC = F If M, N, P are the incentres of AEF, BDE, CDF Prove that A, M, N, P are concyclic if and only if d passes through the incentre of ABC AoPS:2144390 Let n ∈ N and define P (x, y) = xn + xy + y n Show that we cannot obtain two non-constant polynomials G(x, y) and H(x, y) with real coefficients such that P (x, y) = G(x, y) · H(x, y) AoPS:2144389 Vietnam National Olympiad 2012 Define a sequence {xn } as:   amp; x1 =  amp; xn = n + (xn−1 + 2) for n ≥ 3n Prove that this sequence has a finite limit as n → +∞ Also determine the limit AoPS:2594455 Let an and bn be two arithmetic sequences of numbers, and let m be an integer greater than Define Pk (x) = x2 + ak x + bk , k = 1, 2, · · · , m Prove that if the quadratic expressions P1 (x), Pm (x) not have any real roots, then all the remaining polynomials also don’t have real roots AoPS:2594456 Let ABCD be a cyclic quadrilateral with circumcentre O, and the pair of opposite sides not parallel with each other Let M = AB ∩ CD and N = AD ∩ BC Denote, by P, Q, S, T ; the intersection of the internal angle bisectors of ∠M AN and ∠M BN ; ∠M BN and ∠M CN ; ∠M DN and ∠M AN ; ∠M CN and ∠M DN Suppose that the four points P, Q, S, T are distinct (a) Show that the four points P, Q, S, T are concyclic Find the centre of this circle, and denote it as I (b) Let E = AC ∩ BD Prove that E, O, I are collinear AoPS:2594459 Let n be a natural number There are n boys and n girls standing in a line, in any arbitrary order A student X will be eligible for receiving m candies, if we can choose two students of opposite sex with X standing on either side of X in m ways Show that the total number of candies does not exceed 31 n(n2 − 1) AoPS:2594463 For a group of girls, denoted as G1 , G2 , G3 , G4 , G5 and 12 boys There are 17 chairs arranged in a row The students have been grouped to sit in the seats such that the following conditions are simultaneously met: 738 (a) Each chair has a proper seat (b) The order, from left to right, of the girls seating is G1 ; G2 ; G3 ; G4 ; G5 (c) Between G1 and G2 there are at least three boys (d) Between G4 and G5 there are at least one boy and most four boys How many such arrangements are possible? AoPS:2594466 Consider two odd natural numbers a and b where a is a divisor of b2 + and b is a divisor of a2 + Prove that a and b are the terms of the series of natural numbers defined by v1 = v2 = 1; = 4vn−1 − vn−2 for n ≥ AoPS:2594467 Find all f : R → R such that: (a) For every real number a there exist real number b:f (b) = a (b) If x > y then f (x) > f (y) (c) f (f (x)) = f (x) + 12x AoPS:2570340 Vietnam National Olympiad 2013 Solve with full solution:   (sin x)2 +  (sin y)2 + Define a sequence {an } as: (sin x)2 + (cos y)2 + (cos y)2 = 20y x+y (sin y)2 + (cos x)2 + (cos x)2 = 20x x+y AoPS:2901354   a1 =  an+1 = − an + for n ≥ 2an Prove that this sequence has a finite limit as n → +∞ Also determine the limit AoPS:2901355 Let ABC be a triangle such that ABC isn’t a isosceles triangle (I) is incircle of triangle touches BC, CA, AB at D, E, F respectively The line through E perpendicular to BI cuts (I) again at K The line through F perpendicular to CI cuts (I) again at L.J is midpoint of KL (a) Prove that D, I, J collinear (b) B, C are fixed points,A is moved point such that AB AC = k with k is constant.IE, IF cut (I) again at M, N respectively.M N cuts IB, IC at P, Q respectively Prove that bisector perpendicular of P Q through a fixed point AoPS:2901351 Write down some numbers a1 , a2 , , an from left to right on a line Step 1, we write a1 + a2 between a1 , a2 ; a2 + a3 between a2 , a3 , , an−1 + an between an−1 , an , and then we have new sequence b = (a1 , a1 + a2 , a2 , a2 + a3 , a3 , , an−1 , an−1 + an , an ) Step 2, we the same thing with sequence b to have the new sequence c again, and so on If we 2013 steps, count the number of the number 2013 appear on the line if (a) n = 2, a1 = 1, a2 = 1000 (b) n = 1000, = i, i = 1, , 1000 739 AoPS:2901397 Find all f : R → R satisfy: f (0) = 0, f (1) = 2013 and (x − y)(f (f (x)2 ) − f (f (y)2 )) = (f (x) − f (y))(f (x)2 − f (y)2 ) AoPS:2902445 Let ABC be an acute angled triangle (O) is circumcircle of ABC D is on arc BC not containing A Line moved through H (H is orthocentre of ABC cuts circumcircle of ABH, circumcircle ACH again at M, N respectively (a) Find the triangle such that SAM N is maximum (b) d1 , d2 are the line through M perpendicular to DB,the line through N perpendicular to DC respectively d1 cuts d2 at P Prove that P move on a fixed circle AoPS:2902439 Find all ordered 6-tuples satisfy following system of modular equation: ab + a b ≡ 1(mod 15) bc + b c ≡ 1(mod 15) ca + c a ≡ 1(mod 15) Given that a, b, c, a , b , c (0; 1; 2; ; 14) AoPS:2902761 Vietnam National Olympiad 2014 Let (xn ), (yn ) be two positive sequences defined by x1 = 1, y1 = √ and xn+1 yn+1 − xn = x2n+1 + yn = for all n = 1, 2, 3, Prove that they are converges and find their limits AoPS:3342960 Given the polynomial P (x) = (x2 − 7x + 6)2n + 13 where n is a positive integer Prove that P (x) can’t be written as a product of n + non-constant polynomials with integer coefficients AoPS:3343025 Given a regular 103-sided polygon 79 vertices are coloured red and the remaining vertices are coloured blue Let A be the number of pairs of adjacent red vertices and B be the number of pairs of adjacent blue vertices (a) Find all possible values of pair (A, B) (b) Determine the number of pairwise non-similar colourings of the polygon satisfying B = 14 Two colourings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon AoPS:3343022 Let ABC be an acute triangle, (O) be the circumcircle, and AB < AC Let I be the midpoint of arc BC (not containing A) K lies on AC, K = C such that IK = IC BK intersects (O) at the second point D, D = B and intersects AI at E DI intersects AC at F (a) Prove that EF = BC (b) M lies on DI such that CM is parallel to AD KM intersects BC at N The circumcircle of triangle BKN intersects (O) at the second point P Prove that P K passes through the midpoint of segment AD AoPS:3343034 740 Given a circle (O) and two fixed points B, C on (O), and an arbitrary point A on (O) such that the triangle ABC is acute M lies on ray AB, N lies on ray AC such that M A = M C and N A = N B Let P be the intersection of (AM N ) and (ABC), P = A M N intersects BC at Q (a) Prove that A, P, Q are collinear (b) D is the midpoint of BC Let K be the intersection of (M, M A) and (N, N A), K = A d is the line passing through A and perpendicular to AK E is the intersection of d and BC (ADE) intersects (O) at F, F = A Prove that AF passes through a fixed point AoPS:3344340 Find the maximum of P = (x4 y z x3 z x4 y x3 y z + + 4 + y )(xy + z ) (y + z )(yz + x ) (z + x4 )(zx + y )3 where x, y, z are positive real numbers AoPS:3344230 Find all sets of not necessary distinct 2014 rational numbers such that if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same AoPS:3345557 741 Glossary Z, N, Q, R, Q+ , R+ , excircle, circumcircle, incircle, orthocentre, centroid, inradius, exradius, circumradius, incentre, excentre, circumcentre, x , x , cevian, arithmetic mean, geometric mean, harmonic mean, intervals, |x|, great circle on a sphere, mean, standard deviation, n! 742