Đề thi các nước và khu vực 2012 2013

Đề thi nước khu vực năm 2012-2013 Biên soạn: Đỗ Trọng Đạt Nguồn: Mathlinks.ro Austria Federal Competition For Advanced Students, Part 2012 Day 1 Determine the maximum value of m, such that the inequality (a2 + 4(b2 + c2 ))(b2 + 4(a2 + c2 ))(c2 + 4(a2 + b2 )) ≥ m holds for every a, b, c ∈ R \ {0} with Solve over Z: a + b + c ≤ When does equality occur? x4 y (y − x) = x3 y − 216 We call an isosceles trapezoid P QRS interesting, if it is inscribed in the unit square ABCD in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square Find all interesting isosceles trapezoids and their areas This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Austria Federal Competition For Advanced Students, Part 2012 Day Given a sequence < a1 , a2 , a3 , · · · > of real numbers, we define mn as the arithmetic mean of the numbers a1 to an for n ∈ Z+ If there is a real number C, such that (i − j)mk + (j − k)mi + (k − i)mj = C for every triple (i, j, k) of distinct positive integers, prove that the sequence < a1 , a2 , a3 , · · · > is an arithmetic progression We define N as the set of natural numbers n < 106 with the following property: There exists an integer exponent k with ≤ k ≤ 43, such that 2012|nk − Find |N | Given an equilateral triangle ABC with sidelength 2, we consider all equilateral triangles P QR with sidelength such that P lies on the side AB,[/*:m] Q lies on the side AC, and[/*:m] R lies in the inside or on the perimeter of ABC.[/*:m] Find the locus of the centroids of all such triangles P QR This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Brazil National Olympiad 2012 Day 1 In a culturing of bacteria, there are two species of them: red and blue bacteria When two red bacteria meet, they transform into one blue bacterium When two blue bacteria meet, they transform into four red bacteria When a red and a blue bacteria meet, they transform into three red bacteria Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria ABC is a non-isosceles triangle TA is the tangency point of incircle of ABC in the side BC (define TB ,TC analogously) IA is the ex-center relative to the side BC (define IB ,IC analogously) XA is the mid-point of IB IC (define XB ,XC analogously) Show that XA TA ,XB TB ,XC TC meet in a common point, colinear with the incenter and circumcenter of ABC Find the least non-negative integer n such that exists a non-negative integer k such that the last 2012 decimal digits of nk are all 1’s This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Brazil National Olympiad 2012 Day There exists some integers n, a1 , a2 , , a2012 such that n2 = pi 1≤i≤2012 where pi is the i-th prime (p1 = 2, p2 = 3, p3 = 5, p4 = 7, ) and > for all i? In how many ways we can paint a N × N chessboard using colours such that squares with a common side are painted with distinct colors and every × square (formed with squares in consecutive lines and columns) is painted with the four colors? Find all surjective functions f :]0, +∞[⇒]0, +∞[ such that 2x · f (f (x)) = f (x) · (x + f (f (x))) This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Brazil Olympic Revenge 2012 Let a and b real numbers Let f : [a, b] → R a continuous function We say that f is ”smp” if [a, b] = [c0 , c1 ] ∪ [c1 , c2 ] ∪ [cn−1 , cn ] satisfying c0 < c1 < cn and for each i ∈ {0, 1, n − 1}: ci < x < ci+1 ⇒ f (ci ) < f (x) < f (ci+1 ) or ci > x > ci+1 ⇒ f (ci ) > f (x) > f (ci+1 ) Prove that if f : [a, b] → R is continuous such that for each v ∈ R there are only finitely many x satisfying f (x) = v, then f is ”smp” We define (x1 , x2 , , xn )∆(y1 , y2 , , yn ) = ( where the indices are taken modulo n n i=1 xi y2−i , n i=1 xi y3−i , , n i=1 xi yn+1−i ), Besides this, if v is a vector, we define v k = v, if k = 1, or v k = v∆v k−1 , otherwise Prove that, if (x1 , x2 , , xn )k = (0, 0, , 0), for some natural number k, then x1 = x2 = = xn = Let G be a finite graph Prove that one can partition G into two graphs A ∪ B = G such that if we erase all edges conecting a vertex from A to a vertex from B, each vertex of the new graph has even degree Say that two sets of positive integers S, T are k-equivalent if the sum of the ith powers of elements of S equals the sum of the ith powers of elements of T , for each i = 1, 2, , k Given k, prove that there are infinitely many numbers N such that {1, 2, , N k+1 } can be divided into N subsets, all of which are k-equivalent to each other Let x1 , x2 , , xn positive real numbers Prove that: x3 cyc i + xi−1 xi xi+1 ≤ cyc xi xi+1 (xi + xi+1 ) Let ABC be an scalene triangle and I and H its incenter, ortocenter respectively The incircle touchs BC, CA and AB at D, E an F DF and AC intersects at K while EF and BC intersets at M Shows that KM cannot be paralel to IH PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students Sorry if I had some English mistakes here This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Bulgaria National Olympiad 2012 Day 1 The sequence a1 , a2 , a3 , consisting of natural numbers, is defined by the rule: an+1 = an + 2t(n) for every natural number n, where t(n) is the number of the different divisors of n (including and n) Is it possible that two consecutive members of the sequence are squares of natural numbers? Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number p and every natural number n, the numbers pn , pn+1 and pn+2 not have the same colour 2) There does not exist an infinite geometric sequence of natural numbers of the same colour We are given a real number a, not equal to or Sacho and Deni play the following game First is Sasho and then Deni and so on (they take turns) On each turn, a player changes one of the * symbols in the equation: ∗x4 + ∗x3 + ∗x2 + ∗x1 + ∗ = with a number of the type an , where n is a whole number Sasho wins if at the end the equation has no real roots, Deni wins otherwise Determine (in term of a) who has a winning strategy This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Bulgaria National Olympiad 2012 Day Let n be an even natural number and let A be the set of all non-zero sequences of length n, consisting of numbers and (length n binary sequences, except the zero sequence (0, 0, , 0)) Prove that A can be partitioned into groups of three elements, so that for every triad {(a1 , a2 , , an ), (b1 , b2 , , bn ), (c1 , c2 , , cn )}, and for every i = 1, 2, , n, exactly zero or two of the numbers , bi , ci are equal to Let Q(x) be a quadratic trinomial Given that the function P (x) = x2 Q(x) is increasing in the interval (0, ∞), prove that: P (x) + P (y) + P (z) > for all real numbers x, y, z such that x + y + z > and xyz > We are given an acute-angled triangle ABC and a random point X in its interior, different from the centre of the circumcircle k of the triangle The lines AX, BX and CX intersect k for a second time in the points A1 , B1 and C1 respectively Let A2 , B2 and C2 be the points that are symmetric of A1 , B1 and C1 in respect to BC, AC and AB respectively Prove that the circumcircle of the triangle A2 , B2 and C2 passes through a constant point that does not depend on the choice of X This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page Canada National Olympiad 2012 Let x, y and z be positive real numbers Show that x2 + xy + xyz ≥ 4xyz − For any positive integers n and k, let L(n, k) be the least common multiple of the k consecutive integers n, n + 1, , n + k − Show that for any integer b, there exist integers n and k such that L(n, k) > bL(n + 1, k) Let ABCD be a convex quadrilateral and let P be the point of intersection of AC and BD Suppose that AC + AD = BC + BD Prove that the internal angle bisectors of ∠ACB, ∠ADB and ∠AP B meet at a common point A number of robots are placed on the squares of a finite, rectangular grid of squares A square can hold any number of robots Every edge of each square of the grid is classified as either passable or impassable All edges on the boundary of the grid are impassable You can give any of the commands up, down, left, or right All of the robots then simultaneously try to move in the specified direction If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square Otherwise, the robot remains on its current square You can then give another command of up, down, left, or right, then another, for as long as you want Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time A bookshelf contains n volumes, labelled to n, in some order The librarian wishes to put them in the correct order as follows The librarian selects a volume that is too far to the right, say the volume with label k, takes it out, and inserts it in the k-th position For example, if the bookshelf contains the volumes 1, 3, 2, in that order, the librarian could take out volume and place it in the second position The books will then be in the correct order 1, 2, 3, (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order (b) What is the largest number of steps that this process can take? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page China China Girls Math Olympiad 2012 Day - 10 August 2012 Let a1 , a2 , , an be non-negative real numbers Prove that a2 ···an−1 · · · + (1+a1a)(1+a ≤ )···(1+an ) a1 a2 a1 1+a1 + (1+a1 )(1+a2 ) + (1+a1 )(1+a2 )(1+a3 ) + Circles Q1 and Q2 are tangent to each other externally at T Points A and E are on Q1 , lines AB and DE are tangent to Q2 at B and D, respectively, lines AE and BD meet at point P ED ◦ Prove that (1) AB AT = ET ; (2) ∠AT P + ∠ET P = 180 P B A Q1 Q2 T E D Find all pairs (a, b) of integers satisfying: there exists an integer d ≥ such that an + bn + is divisible by d for all positive integers n There is a stone at each vertex of a given regular 13-gon, and the color of each stone is black or white Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the 13-gon This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 78 Spain Spain Mathematical Olympiad 2012 Day 1 Determine if the number λn = √ 3n2 + 2n + is irrational for all non-negative integers n Find all functions f : R → R such that (x − 2)f (y) + f (y + 2f (x)) = f (x + yf (x)) for all x, y ∈ R Let x and n be integers such that ≤ x ≤ n We have x + separate boxes and n − x identical balls Define f (n, x) as the number of ways that the n − x balls can be distributed into the x + boxes Let p be a prime number Find the integers n greater than such that the prime number p is a divisor of f (n, x) for all x ∈ {1, 2, , n − 1} This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 79 Spain Spain Mathematical Olympiad 2012 Day Find all positive integers n and k such that (n + 1)n = 2nk + 3n + A sequence (an )n≥1 of integers is defined by the recurrence a1 = 1, a2 = 5, an = a2n−1 + for n ≥ an−2 Prove that all terms of the sequence are integers and find an explicit formula for an Let ABC be an acute-angled triangle Let ω be the inscribed circle with centre I, Ω be the circumscribed circle with centre O and M be the midpoint of the altitude AH where H lies on BC The circle ω be tangent to the side BC at the point D The line M D cuts ω at a second point P and the perpendicular from I to M D cuts BC at N The lines N R and N S are tangent to the circle Ω at R and S respectively Prove that the points R, P, D and S lie on the same circle This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 80 The Philippines Mathematical Olympiad 2013 1 Determine, with proof, the least positive integer n for which there exist n distinct positive integers, − x1 1− x2 − xn = 15 2013 2 Let P be a point in the interior of triangle ABC Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC 3 Let n be a positive integer The numbers 1, 2, 3, , 2n are randomly assigned to 2n distinct points on a circle To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is n2 4 Let a, p and q be positive integers with p ≤ q Prove that if one of the numbers ap and aq is divisible by p , then the other number must also be divisible by p Let r and s be positive real numbers such that (r +s−rs)(r +s+rs) = rs Find the minimum value of r + s − rs and r + s + rs This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 81 Turkey Junior National Olympiad 2012 Let x, y be integers and p be a prime for which x2 − 3xy + p2 y = 12p Find all triples (x, y, p) In a convex quadrilateral ABCD, the diagonals are perpendicular to each other and they intersect at E Let P be a point on the side AD which is different from A such that P E = EC The circumcircle of triangle BCD intersects the side AD at Q where Q is also different from A The circle, passing through A and tangent to line EP at P , intersects the line segment AC at R If the points B, R, Q are concurrent then show that ∠BCD = 90◦ Let a, b, c be positive real numbers satisfying a3 + b3 + c3 = a4 + b4 + c4 Show that a2 a b c + + ≥1 3 +b +c a +b +c a + b3 + c2 We want to place 2012 pockets, including variously colored balls, into k boxes such that i) For any box, all pockets in this box must include a ball with the same color or ii) For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box Find the smallest value of k for which we can always this placement whatever the number of balls in the pockets and whatever the colors of balls This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 82 Undergraduate Competitions IMC 2012 Day - 28 July 2012 For every positive integer n, let p(n) denote the number of ways to express n as a sum of positive integers For instance, p(4) = because = + = + = + + = + + Also define p(0) = Prove that p(n) − p(n − 1) is the number of ways to express n as a sum of integers each of which is strictly greater than Let n be a fixed positive integer Determine the smallest possible rank of an n×n matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal Given an integer n > 1, let Sn be the group of permutations of the numbers 1, 2, 3, , n Two players, A and B, play the following game Taking turns, they select elements (one element at a time) from the group Sn It is forbidden to select an element that has already been selected The game ends when the selected elements generate the whole group Sn The player who made the last move loses the game The first move is made by A Which player has a winning strategy? Let f : R → R be a continuously differentiable function that satisfies f (t) > f (f (t)) for all t ∈ R Prove that f (f (f (t))) ≤ for all t ≥ Let a be a rational number and let n be a positive integer Prove that the polynomial n n X (X + a)2 + is irreducible in the ring Q[X] of polynomials with rational coefficients This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 83 Undergraduate Competitions IMC 2012 Day - 29 July 2012 Consider a polynomial f (x) = x2012 + a2011 x2011 + · · · + a1 x + a0 Albert Einstein and Homer Simpson are playing the following game In turn, they choose one of the coefficients a0 , a1 , , a2011 and assign a real value to it Albert has the first move Once a value is assigned to a coefficient, it cannot be changed any more The game ends after all the coefficients have been assigned values Homer’s goal is to make f (x) divisible by a fixed polynomial m(x) and Albert’s goal is to prevent this (a) Which of the players has a winning strategy if m(x) = x − 2012? (b) Which of the players has a winning strategy if m(x) = x2 + 1? Define the sequence a0 , a1 , inductively by a0 = 1, a1 = 12 , and an+1 = ∞ Show that the series k=0 na2n , + (n + 1)an ∀n ≥ ak+1 converges and determine its value ak Is the set of positive integers n such that n! + divides (2012n)! finite or infinite? Let n ≥ be an integer x1 , x2 , , xn satisfying Find all real numbers a such that there exist real numbers x1 (1 − x2 ) = x2 (1 − x3 ) = · · · = xn (1 − x1 ) = a Let c ≥ be a real number Let G be an Abelian group and let A ⊂ G be a finite set satisfying |A + A| ≤ c|A|, where X + Y := {x + y|x ∈ X, y ∈ Y } and |Z| denotes the cardinality of Z Prove that | A + A + · · · + A | ≤ ck |A| k for every positive integer k This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 84 Undergraduate Competitions Putnam 2012 A Let d1 , d2 , , d12 be real numbers in the open interval (1, 12) Show that there exist distinct indices i, j, k such that di , dj , dk are the side lengths of an acute triangle Let ∗ be a commutative and associative binary operation on a set S Assume that for every x and y in S, there exists z in S such that x ∗ z = y (This z may depend on x and y.) Show that if a, b, c are in S and a ∗ c = b ∗ c, then a = b Let f : [−1, 1] → R be a continuous function such that (i) f (x) = 2−x2 f x2 2−x2 for every x in [−1, 1], (ii) f (0) = 1, and (iii) limx→1− f (x) √ 1−x exists and is finite Prove that f is unique, and express f (x) in closed form Let q and r be integers with q > 0, and let A and B be intervals on the real line Let T be the set of all b + mq where b and m are integers with b in B, and let S be the set of all integers a in A such that is in T Show that if the product of the lengths of A and B is less than q, then S is the intersection of A with some arithmetic progression Let Fp denote the field of integers modulo a prime p, and let n be a positive integer Let v be a fixed vector in Fnp , let M be an n × n matrix with entries in Fp , and define G : Fnp → Fnp by G(x) = v+M x Let G(k) denote the k-fold composition of G with itself, that is, G(1) (x) = G(x) and G(k+1) (x) = G(G(k) (x)) Determine all pairs p, n for which there exist v and M such that the pn vectors G(k) (0), k = 1, 2, , pn are distinct Let f (x, y) be a continuous, real-valued function on R2 Suppose that, for every rectangular region R of area 1, the double integral of f (x, y) over R equals Must f (x, y) be identically 0? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 85 Undergraduate Competitions Putnam 2012 B Let S be a class of functions from [0, ∞) to [0, ∞) that satisfies: (i) The functions f1 (x) = ex − and f2 (x) = ln(x + 1) are in S; (ii) If f (x) and g(x) are in S, the functions f (x) + g(x) and f (g(x)) are in S; (iii) If f (x) and g(x) are in S and f (x) ≥ g(x) for all x ≥ 0, then the function f (x) − g(x) is in S Prove that if f (x) and g(x) are in S, then the function f (x)g(x) is also in S Let P be a given (non-degenerate) polyhedron Prove that there is a constant c(P ) > with the following property: If a collection of n balls whose volumes sum to V contains the entire surface of P, then n > c(P )/V A round-robin tournament among 2n teams lasted for 2n − days, as follows On each day, every team played one game against another team, with one team winning and one team losing in each of the n games Over the course of the tournament, each team played every other team exactly once Can one necessarily choose one winning team from each day without choosing any team more than once? Suppose that a0 = and that an+1 = an + e−an for n = 0, 1, 2, Does an − log n have a finite limit as n → ∞? (Here log n = loge n = ln n.) Prove that, for any two bounded functions g1 , g2 : R → [1, ∞), there exist functions h1 , h2 : R → R such that for every x ∈ R, sup (g1 (s)x g2 (s)) = max (xh1 (t) + h2 (t)) s∈R t∈R Let p be an odd prime number such that p ≡ (mod 3) Define a permutation π of the residue classes modulo p by π(x) ≡ x3 (mod p) Show that π is an even permutation if and only if p ≡ (mod 4) This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 86 USA USAMO 2012 Day - 24 April 2012 Find all integers n ≥ such that among any n positive real numbers a1 , a2 , , an with max(a1 , a2 , , an ) ≤ n · min(a1 , a2 , , an ), there exist three that are the side lengths of an acute triangle A circle is divided into 432 congruent arcs by 432 points The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent Determine which integers n > have the property that there exists an infinite sequence a1 , a2 , a3 , of nonzero integers such that the equality ak + 2a2k + + nank = holds for every positive integer k This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 87 USA USAMO 2012 Day - 25 April 2012 Find all functions f : Z+ → Z+ (where Z+ is the set of positive integers) such that f (n!) = f (n)! for all positive integers n and such that m − n divides f (m) − f (n) for all distinct positive integers m, n Let P be a point in the plane of ABC, and γ a line passing through P Let A , B , C be the points where the reflections of lines P A, P B, P C with respect to γ intersect lines BC, AC, AB respectively Prove that A , B , C are collinear For integer n ≥ 2, let x1 , x2 , , xn be real numbers satisfying x1 + x2 + + xn = 0, x21 + x22 + + x2n = and For each subset A ⊆ {1, 2, , n}, define SA = xi i∈A (If A is the empty set, then SA = 0.) Prove that for any positive number λ, the number of sets A satisfying SA ≥ λ is at most 2n−3 /λ2 For which choices of x1 , x2 , , xn , λ does equality hold? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 88 Uzbekistan National Olympiad 2012 Given a digits 0, 1, 2, , Find the number of numbers of digits which cantain or 7’s digit and they is permulated(For example 137456 and 314756 is one numbers) For any positive integers n and m satisfying the equation n3 + (n + 1)3 + (n + 2)3 = m3 , prove that | n + The inscribed circle ω of the non-isosceles acute-angled triangle ABC touches the side BC at the point D Suppose that I and O are the centres of inscribed circle and circumcircle of triangle ABC respectively The circumcircle of triangle ADI intersects AO at the points A and E Prove that AE is equal to the radius r of ω Given a, b and c positive real numbers with ab + bc + ca = Then prove that b3 1+9c2 ab + c3 1+9a2 bc ≥ (a+b+c)3 18 a3 1+9b2 ac + Given points A, B, C and D lie a circle AC ∩ BD = K I1 , I2 , I3 and I4 incenters of ABK, BCK, CDK, DKA M1 , M2 , M3 , M4 midpoints of arcs AB, BC, CA, DA Then prove that M1 I1 , M2 I2 , M3 I3 , M4 I4 are concurrent This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 89 Uzbekistan National Olympiad 2012 Given a digits 0, 1, 2, , Find the number of numbers of digits which cantain or 7’s digit and they is permulated(For example 137456 and 314756 is one numbers) For any positive integers n and m satisfying the equation n3 + (n + 1)3 + (n + 2)3 = m3 , prove that | n + The inscribed circle ω of the non-isosceles acute-angled triangle ABC touches the side BC at the point D Suppose that I and O are the centres of inscribed circle and circumcircle of triangle ABC respectively The circumcircle of triangle ADI intersects AO at the points A and E Prove that AE is equal to the radius r of ω Given a, b and c positive real numbers with ab + bc + ca = Then prove that b3 1+9c2 ab + c3 1+9a2 bc ≥ (a+b+c)3 18 a3 1+9b2 ac + Given points A, B, C and D lie a circle AC ∩ BD = K I1 , I2 , I3 and I4 incenters of ABK, BCK, CDK, DKA M1 , M2 , M3 , M4 midpoints of arcs AB, BC, CA, DA Then prove that M1 I1 , M2 I2 , M3 I3 , M4 I4 are concurrent This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 90 Vietnam National Olympiad 2012 Day - 11 January 2012   x1 = Define a sequence {xn } as: Prove that this sequence has  xn = n + (xn−1 + 2) for n ≥ 3n a finite limit as n → +∞ Also determine the limit 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 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 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 13 n(n2 − 1) This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 91 Vietnam National Olympiad 2012 Day - 12 January 2012 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: (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? 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 ≥ 3 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 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 92 Albania National Olympiad 2012 Find all primes p such that p + and p2 + 2p − are also primes The trinomial f (x) is such that (f (x))3 − f (x) = has three real roots Find the y-coordinate of the vertex of f (x) Let Si be the sum of the first i terms of the arithmetic sequence a1 , a2 , a3 Show that the value of the expression Sj Si Sk (j − k) + (k − i) + (i − j) i j k does not depend on the numbers i, j, k nor on the choice of the arithmetic sequence a1 , a2 , a3 , Find all functions f : R → R such that f (x3 ) + f (y ) = (x + y)f (x2 ) + f (y ) − f (xy) for all x ∈ R Let ABC be a triangle where AC = BC Let P be the foot of the altitude taken from C to AB; and let V be the orthocentre, O the circumcentre of ABC, and D the point of intersection between the radius OC and the side AB The midpoint of CD is E a) Prove that the reflection V of V in AB is on the circumcircle of the triangle ABC b) In what ratio does the segment EP divide the segment OV ? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page [...]... such a sequence contains infinitely many powers of 2 if and only if a1 is not divisible by 5 2 Find all quadruples (a, b, c, d) of positive real numbers such that abcd = 1, a2012 + 2012b = 2012c + d2012 and 2012a + b2012 = c2012 + 2012d 3 In triangle ABC the midpoint of BC is called M Let P be a variable interior point of the triangle such that ∠CP M = ∠P AB Let Γ be the circumcircle of triangle ABP... and Fajar Yuliawan 4 Given 2012 distinct points A1 , A2 , , A2012 on the Cartesian plane For any permutation B1 , B2 , , B2012 of A1 , A2 , , A2012 define the shadow of a point P as follows: Point P is rotated by 180◦ around B1 resulting P1 , point P1 is rotated by 180◦ around B2 resulting P2 , , point P2011 is rotated by 180◦ around B2012 resulting P2012 Then, P2012 is called the shadow of... there exists a positive integer s such that ass = k 8 Find the number of integers k in the set {0, 1, 2, · · · , 2012} such that 2012 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ 2012 k is a multiple of Page 2 11 China National Olympiad 2012 Day 1 1 In the triangle ABC, ∠A is biggest On the circumcircle of ABC, let D be the midpoint of ABC and... with respect to the permutation B1 , B2 , , B2012 Let N be the number of different shadows of P up to all permutations of A1 , A2 , , A2012 Determine the maximum value of N Proposer: Hendrata Dharmawan This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 17 Indonesia National Science Olympiad 2012 Day 2 1 Given positive integers m and n... Prove that k = 1 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 21 Baltic Way 2012 17 Let d(n) denote the number of positive divisors of n Find all triples (n, k, p), where n and k are positive integers and p is a prime number, such that nd(n) − 1 = pk 18 Find all triples (a, b, c) of integers satisfying a2 + b2 + c2 = 20122 012 19 Show that... any subset A of S = {1, 2, , 2012} with |A| = k, there exist three elements x, y, z in A such that x = a + b, y = b + c, z = c + a, where a, b, c are in S and are distinct integers Proposed by Huawei Zhu This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 13 France Team Selection Test 2012 Day 1 - 10 March 2012 1 Let n and k be two positive... with respect to BG) Show that ∠EHG = ∠BAC 2 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 25 Cono Sur Olympiad 2012 1 1 Around a circumference are written 2012 number, each of with is equal to 1 or −1 If there are not 10 consecutive numbers that sum 0, find all possible values of the sum of the 2012 numbers 2 2 In a square ABCD, let P be... its digits Show that for every positive integer n there are n consecutive good numbers.) This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 30 IMO 2012 Day 1 - 10 July 2012 1 Given triangle ABC the point J is the centre of the excircle opposite the vertex A This excircle is tangent to the side BC at M , and to the lines AB and AC at K and L,... Show that none of the positive real solutions of this equation is rational 4 Prove that for infinitely many pairs (a, b) of integers the equation x2012 = ax + b has among its solutions two distinct real numbers whose product is 1 5 Find all functions f : R → R for which f (x + y) = f (x − y) + f (f (1 − xy)) holds for all real numbers x and y 6 There are 2012 lamps arranged on a table Two persons play... a+b + Show that a+b+c− 1 b+c + 1 a+c = 1 and ab + bc + ac > 0 abc ≥4 ab + bc + ac This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 24 CentroAmerican 2012 Day 2 1 Trilandia is a very unusual city The city has the shape of an equilateral triangle of side lenght 2012 The streets divide the city into several blocks that are shaped like equilateral ... that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order (b) What is the largest number of steps that this process... absolute values not exceeding 2012 can be arranged on the arcs of this graph, so that the weight of each vertex is zero Proposed by W Tutte This file was downloaded from the AoPS Math Olympiad Resources... Every person, except the first, indicates a person in front of him/her and says ”This person is a scoundrel” or ”This person is a knight.” Knowing that there are strictly more scoundrel than knights,