Đề luyện thi HSG Singapore

Singapore International Mathematical Olympiad Committee 22-9-2001 1. Consider the set of integers A = {2 a 3 b 5 c : 0 ≤ a, b, c ≤ 5}. Find the smallest number n such that whenever S is a subset of A with n elements, you can find two numbers p, q in A with p | q. 2. “Words” are formed with the letters A and B. Using the words x 1 , x 2 , . . ., x n we can form a new word if we write these words consecutively one next to another: x 1 x 2 . . . x n . A word is called a palindrome, if it is not changed after rewriting its letters in the reverse order. Prove that any word with 1995 letters A and B can b e formed with less than 800 palindromes. 3. Let n ≥ 2 be an integer and M = {1, 2, . . . , n}. For every k ∈ {1, 2, . . . , n −1}, let x k = 1 n + 1  A⊂M |A|=k (min A + max A). Prove that x 1 , x 2 , . . . , x n−1 are integers, not all divisible by 4. 4. The lattice frame construction of 2 ×2×2 cube is formed with 54 metal shafts of length 1 (points of shafts’ connection are called junctions). An ant starts from some junction A and creeps along the shafts in accordance with the following rule: when the ant reaches the next junction it turns to a perpendicular shaft. t some moment the ant reaches the initial junction A; there is no junction (except for A) where the ant has been twice. What is the maximum length of the ant’s path? 5. Let n black and n white objects be placed on the circumference of a circle, and define any set of m consecutive objects from this cyclic sequence to be an m-chain. (a) Prove that for each natural number k ≤ n, there exists a chain of 2k consecutive pieces on the circle of which exactly k are black. (b) Prove that there are at least two such chains that are disjoint if k ≤ √ 2n + 2 −2 . 1 Singapore International Mathematical Olympiad Committee 29-9-2001 6. We are given 1999 rectangles with sides of integer not exceeding 1998. Prove that among these 1999 rectangles there are rectangles, say A, B and C such that A will fit inside B and B will fit inside C. 7. We are given N lines (N > 1) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding N, such that the sum of the integers on either side of any of the given lines is equal to 0. 8. Let S be a set of 2n + 1 points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of S on its circumference, n − 1 points in its interior and n − 1 in its exterior. Prove that the number of good circles has the same parity as n. 2 Singapore International Mathematical Olympiad Committee 6-10-2001 9. Let ARBP CQ be a hexagon. Suppose that ∠AQR = ∠ARQ = 15 o , ∠BP R = ∠CP Q = 30 o and ∠BRP = ∠CQP = 45 o . Prove that AB is perpendicular to AC. 10. Γ 1 and Γ 2 are two circles on the plane such that Γ 1 and Γ 2 lie outside each other. An external common tangent to the two circles touches Γ 1 at A and Γ 2 at C and an internal common tangent to the two circles touches Γ 1 at B and Γ 2 at D. Prove that the intersection of AB and CD lie on the line joining the centres of Γ 1 and Γ 2 11. Let E and F be the midpoints of AC and AB of ABC respectively. Let D be a point on BC. Suppose that (i) P is a point on BF and DP is parallel to CF , (ii) Q is a point on CE and DQ is parallel to BE, (iii) P Q intersects BE and CF at R and S respectively. Prove that PQ = 3 RS. 12. Let AD, BE and CF be the altitudes of an acute-angled triangle ABC. Let P be a point on DF and K the point of intersection between AP and EF . Suppose Q is a point on EK such that ∠KAQ = ∠DAC. Prove that AP bisects ∠F P Q. 13. In a square ABCD , C is a circular arc centred at A with radius AB, P and M are points on CD and BC respectively such that PM is tangent to C. Let AP and AM intersect BD at Q and N respectively. Prove that the vertices of the pentagon P QNMC lie on a circle. 3 Singapore International Mathematical Olympiad Committee 13-10-2001 14. In triangle ABC, the angle bisectors of angle B and C meet the median AD at points E and F respectively. If BE = CF , prove that ABC is isosceles. 15. Let ABCD be a convex quadrilateral. Prove that there exists a point E in the plane of ABCD such that ABE is similar to CDE. 16. Let P, Q be points taken on the side BC of a triangle ABC, in the order B, P, Q, C. Let the circumcircles of PAB, QAC intersect at M (= A) and those of P AC, QAB at N (= A). Prove that A, M, N are collinear if and only if P, Q are symmetric in the midpoint A  of BC. 17. About a set of four concurrent circles of the same radius r, four of the common tangents are drawn to determine the circumscribing quadrilateral ABCD. Prove that ABCD is a cyclic quadrilateral. 18. Three circles of the same radius r meet at common point. Prove that the triangle having the other three points of intersections as vertices has circumradius equal to r 4 Singapore International Mathematical Olympiad Committee 27-10-2001 19. For any positive real numbers a, b, c, a b + c + b c + a + c a + b ≥ 3 2 . 20. Let a, b, c be positive numbers such that a + b + c ≤ 3. Prove that 1 1 + a + 1 1 + b + 1 1 + c ≥ 3 2 . 21. Prove that for any positive real numbers a, b, c, a 10b + 11c + b 10c + 11a + c 10a + 11b ≥ 1 7 . 22. Prove that for any positive real numbers a, b, c, d, e, a b + 2c + 3d + 4e + b c + 2d + 3e + 4a + c d + 2e + 3a + 4b + d e + 2a + 3b + 4c + e a + 2b + 3c + 4d ≥ 1 2 . 23. Let n be a positive integer and let a 1 , a 2 , . . ., a n be n positive real numbers such that a 1 + a 2 + ···+ a n = 1. Is it true that a 4 1 a 2 1 + a 2 2 + a 4 2 a 2 2 + a 2 3 + ···+ a 4 n a 2 n + a 2 1 ≥ 1 2n ? 24. Let a, b, c, d be nonnegative real numbers such that ab + bc + cd + da = 1. Prove that a 3 b + c + d + b 3 a + c + d + c 3 a + b + d + d 3 a + b + c ≥ 1 3 . 25. (IMO 95) Let a, b and c be positive real numbers such that abc = 1. Prove that 1 a 3 (b + c) + 1 b 3 (a + b) + 1 c 3 (a + b) ≥ 3 2 . Cauchy’s inequality is useful for these questions. (x 2 1 + x 2 2 + ···+ x 2 i )(y 2 1 + y 2 2 + ···+ y 2 i ) ≥ (x 1 y 1 + x 2 y 2 + ···+ x i y i ) 2 . Equality holds iff x j = ty j for all j, where t is some constant. 5 Singapore International Mathematical Olympiad Committee 3-11-2001 26. A sequence of natural numbers {a n } is defined by a 1 = 1, a 2 = 3 and a n = (n + 1)a n−1 − na n−2 (n ≥ 2). Find all values of n such that 11|a n . 27. Let{x n }, n ∈ N be a sequence of numbers satisfying the condition |x 1 | < 1, x n+1 = −x n +  3 −3x 2 n 2 , (n ≥ 1). (a) What other condition does x 1 need to satisfy so that all the numbers of the sequence are positive? (b) Is the given sequence periodic? 28. Suppose that a function f defined on the positive integers satisfies f (1) = 1, f(2) = 2, and f(n + 2) = f(n + 2 −f(n + 1)) + f(n + 1 −f(n)), (n ≥ 1). (a) Show that 0 ≤ f(n + 1) − f(n) ≤ 1. (b) Show that if f (n) is odd, then f (n + 1) = f(n) + 1. (c) Determine, with justification, all values of n for which f(n) = 2 10 + 1. 29. Determine the number of all sequences {x 1 , x 2 , . . . , x n }, with x i ∈ {a, b, c} for i = 1, 2, . . . , n that satisfy x 1 = x n = a and x i = x i+1 for i = 1, 2, . . . , n − 1. 30. Given is a prime p > 3. Set q = p 3 . Define the sequence {a n } by: a n =  n for n = 0, 1, 2, . . . , p − 1, a n−1 + a n−p for n > p −1. Determine the remainder when a q is divided by p. 31. A and B are two candidates taking part in an election. Assume that A receives m votes and B receives n votes, where m, n ∈ N and m > n. Find the number of ways in which the ballots can be arranged in order that when they are counted, one at a time, the number of votes for A will always be more than that for B at any time during the counting process. 6 Singapore International Mathematical Olympiad Committee 10-11-2001 32. Find all prime numb ers p for which the number p 2 +11 has exactly 6 different divisors (including 1 and the number itself.) 33. Determine all pairs (a, b) of positive integers such that ab 2 + b + 7 divides a 2 b + a + b. 34. Let p be an odd prime. Prove that 1 p−2 + 2 p−2 + 3 p−2 + ···+  p −1 2  p−2 ≡ 2 −2 p p (mod p). 35. (10th grade) Let d(n) denote the greatest odd divisor of the natural number n. Define the function f : N → N by f(2n − 1) = 2 n , f (2n) = n + 2n/d(n) for all n ∈ N. Find all k such that f(f(. . . (1) . . .)) = 1997 where f is iterated k times. 36. Given three real numbers such that the sum of any two of them is not equal to 1, prove that there are two numbers x and y such that xy/(x + y −1) does not belong to the interval (0, 1). 7 Singapore International Mathematical Olympiad Committee 17-11-2001 37. In the parliament of country A, each MP has at most 3 enemies. Prove that it is always possible to separate the parliament into two houses so that every MP in each house has at most one enemy in his own house. 38. Let T (x 1 , x 2 , x 3 , x 4 ) = (x 1 −x 2 , x 2 −x 3 , x 3 −x 4 , x 4 −x 1 ). If x 1 , x 2 , x 3 , x 4 are not equal integers, show that there is no n such that T n (x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 , x 3 , x 4 ). 39. Start with n pairwise different integers x 1 , x 2 , x 3 , x n , n > 2 and repeat the following step: T : (x 1 , x 2 , ···, x n ) → ( x 1 + x 2 2 , x 2 + x 3 2 , ···, x n + x 1 2 ). Show that T, T 2 , ···, finally leads to nonintegral component. 40. Is it possible to transform f(x) = x 2 + 4x + 3 into g(x) = x 2 + 10x + 9 by a sequence of transformations of the form f(x) → x 2 f(1/x + 1) or f(x) → (x −1) 2 f(1/(x − 1))? 41. Is it possible to arrange the integers 1, 1, 2, 2, ···, 1998, 1998, such that there are exactly i −1 other numbers between any two i’s? 42. A rectangular floor can be covered by n 2 ×2 and m 1 ×4 tiles, one tile got smashed. Show that one can not substitute that tile by the other type (2 ×2 or 1 × 4). 43. In how many ways can you tile a 2 × n rectangle by 2 ×1 dominoes? 44. In how many ways can you tile a 2 × n rectangle by 1 ×1 squares and L trominoes? 45. In how many ways can you tile a 2 × n rectangle by 2 ×2 squares and L trominoes? 46. Let a 1 = 0, |a 2 | = |a 1 + 1|, ···, |a n | = |a n−1 + 1|. Prove that a 1 + a 2 + ··· + a n n ≥ − 1 2 . 47. Find the number a n of all permutations σ of {1, 2, . . . , , n} with |σ(i) −i| ≤ 1 for all i. 48. Can you select from 1, 1 2 , 1 4 , 1 8 , . . . an infinite geometric sequence with sum (a) 1 5 ? (b) 1 7 ? 49. Let x 0 , a > 0, x n+1 = 1 2 (x n + a x n ). Find lim n→∞ a n . 8 50. Let 0 < a < b, a 0 = a and b 0 = b. For n ≥ 0, define a n+1 =  a n b n , b n+1 = a n + b n 2 . Show that lim n→∞ a n = lim n→∞ b n . 51. Let a 0 , a 1 = 1, a n = 2a n−1 + a n−2 , n > 1. Show that 2 k |a n if and only if 2 k |n. 9 Singapore International Mathematical Olympiad Committee 24-11-2001 52. Determine all α ∈ R such that there exists a nonconstant function f : R → R such that f(α(x + y)) = f(x) + f(y). 53. Let f : N → N be a function satisfying (a) For every n ∈ N, f(n + f(n)) = f(n). (b) For some n 0 ∈ N, f(n 0 ) = 1. Show that f(n) = 1 for all n ∈ N. 54. Suppose that f : N → N satisfies f(1) = 1 and for all n, (a) 3f(n)f(2n + 1) = f(2n)(1 + 3f(n)), (b) f(2n) < 6f (n). Find all (k, m) such that f(k) + f(m) = 2001. 55. Let f : R → R be a function such that for all x, y ∈ R, f(x 3 + y 3 ) = (x + y)((f(x)) 2 − f(x)f (y) + (f(y)) 2 ). Prove that for all x ∈ R, f(2001x) = 2001f(x). 56. Find all functions f : N → N with the prop erty that for all n ∈ N, 1 f(1)f(2) + 1 f(2)f(3) + ···+ 1 f(n)f(n + 1) = f(f(n)) f(n + 1) . 57. Define f : N → N and g : N → Z such that (a) f(x, x) = x, (b) f(x, y) = f (y, x), (c) f(x, y) = f (x, x + y), (d) g(2001) = 2002, (e) g(xy) = g(x) + g(y) + mg(f(x, y)). Determine all integers m for which g exists. 10 [...]... if n = 2k + 2k − 1 = 2k+1 − 1, then f (n) = 2k This can be done using strong induction on k Notice that this helps us to express all the values of f (n) from n = 2k to 2k+1 , in terms of the values of f (n) from n = 0 to n = 2k − 1 (We can define 26 f (0) = 0 for convenience; this definition is consistent with the recurrence relation.) we have established this claim, parts (a) and (b) follow essentially... on this property of non-cycles, we can deduce that the number of non-cycles 10 wi li is i=1 2 + 2 Clearly, we have wi + li = 9, and rearranging the formula gives 10 10 wi li 2 i=1 2 + 2 = i=1 wi − 9wi + 36 And the number of non-cycles and the number of cycles sum up to 120 We aim to minimise the number of non-cycles This can be achieved if wi = 4 or 5, and is certainly attainable (The answer to this... 1989 Q5) First note the regions can be painted in two colours so that two regions sharing a common side have different colours This is 19 trivial for N = 1 Assume that it is true for N = k When the (k + 1)st line is drawn we simply reverse the colours on exactly one side of this line Assign to each region an integer whose magnitude is equal to the number of vertices of that region The sign is + is the... , C2 , D1 are on a sphere 12 Singapore International Mathematical Olympiad Committee 15-12-2001 68 The nonnegative real numbers a, b, c, A, B, C and k satisfy a + A = b + B = c + C = k Prove that aB + bC + cA ≤ k 2 69 Find the least constant C such that the inequality x1 x2 + x2 x3 + + x2000 x2001 + x2001 x1 ≤ C 2001 holds for any i=1 xi = 2001, x1 , , x2001 ≥ 0 For this constant C, determine the... or c Having established this simultaneous system of recurrence relations, we substitute and solve for Bn = Bn−1 + 2Bn−2 , B1 = 0, B2 = 1 Solving, we 2 obtain that An = 2Bn−1 = 3 [ 2n−2 + (−1)n−1 ] 30 Given is a prime p > 3 Set q = p3 Define the sequence {an } by: an = n an−1 + an−p for n = 0, 1, 2, , p − 1, for n > p − 1 Determine the remainder when aq is divided by p Soln (This remains an open question... we move to (1, −1) if the first vote goes to B instead Notice that this walk will end at (m + n, m − n) The number of ways of counting in which the number of votes A gets is always more than that which B has is equal to the number of paths on the plane from (0, 0) to (m+n, m−n) that do not cross the x-axis We count the complement of this set, ie the number of paths that do cross the x-axis Notice that... that cross the 27 x-axis and end at (m+n, m−n) is equal to the number of paths (unrestricted) from (1, −1) to (m + n, m − n) To see this, take any given path from (1, 1) to (m + n, m − n) and let X be the first point of intersection with the x-axis Reflect the portion of this path to the left of X about the x-axis and we have a path from (1, −1) to (m + n, m − n) Vice versa, we can take any such path... 1)2 (b − 1)2 (c − 1)2 . p > 3. Set q = p 3 . De ne the sequence {a n } by: a n =  n for n = 0, 1, 2, . . . , p − 1, a n−1 + a n−p for n > p −1. Determine the remainder when a q is divided by p. 31. A and B are. point E in the plane of ABCD such that ABE is similar to CDE. 16. Let P, Q be points taken on the side BC of a triangle ABC, in the order B, P, Q, C. Let the circumcircles of PAB, QAC intersect. 3 p−2 + ···+  p −1 2  p−2 ≡ 2 −2 p p (mod p). 35. (10th grade) Let d(n) denote the greatest odd divisor of the natural number n. De ne the function f : N → N by f(2n − 1) = 2 n , f (2n) = n