India International Mathematical Olympiad Training Camp-2013

India
International Mathematical Olympiad Training Camp 2013
Practice Test
Day 1 - 05 May 2013

1 For a prime p, a natural number n and an integer a, we let Sn (a, p) denote the exponent of n p in the prime factorisation of ap 1. For example, S1 (4, 3) = 2 and S2 (6, 2) = 0. Find all pairs (n, p) such that Sn (2013, p) = 100. 2 Let ABCD by a cyclic quadrilateral with circumcenter O. Let P be the point of intersection of the diagonals AC and BD, and K, L, M, N the circumcenters of triangles AOP, BOP , COP, DOP , respectively. Prove that KL = M N . 3 We dene an operation on the set {0, 1} by 0 0 = 0,0 1 = 1,1 0 = 1,1 1 = 0. For two natural numbers a and b, which are written in base 2 as a = (a1 a2 . . . ak )2 and b = (b1 b2 . . . bk )2 (possibly with leading 0s), we dene a b = c where c written in base 2 is (c1 c2 . . . ck )2 with ci = ai bi , for 1 i k . For example, we have 7 3 = 4 since 7 = (111)2 and 3 = (011)2 . For a natural number n, let f (n) = n [n/2], where [x] denotes the largest integer less than or equal to x. Prove that f is a bijection on the set of natural numbers.
Day 2 - 10 May 2013

1 Let a, b, c be positive real numbers such that a + b + c = 1. If n is a positive integer then prove that (3a)n (3b)n (3c)n 27 + + . (b + 1)(c + 1) (c + 1)(a + 1) (a + 1)(b + 1) 16 2 In a triangle ABC with B = 90 , D is a point on the segment BC such that the inradii of triangles ABD and ADC are equal. If ADB = then prove that tan2 (/2) = tan(C/2). 3 A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers a, b, neither of which was chosen earlier by any player and move the marker by a units in the horizontal direction and b units in the
This le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/
Page 1
India
vertical direction. Hobbes wins if the marker is back at the origin any time after the rst turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).
Page 2
India
Team Selection Test
Day 1 - 15 May 2013

1 Let n 2 be an integer. There are n beads numbered 1, 2, . . . , n. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no ipping allowed). For example, with n 5, the necklace with four beads 1, 5, 3, 2 in the clockwise order is same as the one with 5, 3, 2, 1 in the clockwise order, but is dierent from the one with 1, 2, 3, 5 in the clockwise order. We denote by D0 (n) (respectively D1 (n)) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least 3. Prove that n 1 divides D1 (n) D0 (n). 2 In a triangle ABC , with A > 90 , let O and H denote its circumcenter and orthocenter, respectively. Let K be the reection of H with respect to A. Prove that K, O and C are collinear if and only if A B = 90 . 3 For a positive integer n, a cubic polynomial p(x) is said to be n-good if there exist n distinct integers a1 , a2 , . . . , an such that all the roots of the polynomial p(x) + ai = 0 are integers for 1 i n. Given a positive integer n prove that there exists an n-good cubic polynomial.
Day 2 - 16 May 2013

1 Find all functions f from the set of real numbers to itself satisfying f (x(1 + y )) = f (x)(1 + f (y )) for all real numbers x, y . 2 An integer a is called friendly if the equation (m2 + n)(n2 + m) = a(m n)3 has a solution over the positive integers. a) Prove that there are at least 500 friendly integers in the set {1, 2, . . . , 2012}. b) Decide whether a = 2 is friendly. 3 Players A and B play a game with N 2012 coins and 2012 boxes arranged around a circle. Initially A distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order B, A, B, A, . . . by the following rules: (a) On every move of his B passes 1 coin from every box to an adjacent box. (b) On every move of
Page 3
India
hers A chooses several coins that were not involved in B s previous move and are in dierent boxes. She passes every coin to and adjacent box. Player As goal is to ensure at least 1 coin in each box after every move of hers, regardless of how B plays and how many moves are made. Find the least N that enables her to succeed.
Day 3 - 22 May 2013

1 For a positive integer n, a sum-friendly odd partition of n is a sequence (a1 , a2 , . . . , ak ) of odd positive integers with a1 a2 ak and a1 + a2 + + ak = n such that for all positive integers m n, m can be uniquely written as a subsum m = ai1 + ai2 + + air . (Two subsums ai1 + ai2 + + air and aj1 + aj2 + + ajs with i1 < i2 < < ir and j1 < j2 < < js are considered the same if r = s and ail = ajl for 1 l r.) For example, (1, 1, 3, 3) is a sum-friendly odd partition of 8. Find the number of sum-friendly odd partitions of 9999. 2 In a triangle ABC , let I denote its incenter. Points D, E, F are chosen on the segments BC, CA, AB , respectively, such that BD + BF = AC and CD + CE = AB . The circumcircles of triangles AEF, BF D, CDE intersect lines AI, BI, CI , respectively, at points K, L, M (dierent from A, B, C ), respectively. Prove that K, L, M, I are concyclic. 3 Let h 3 be an integer and X the set of all positive integers that are greater than or equal to 2h. Let S be a nonempty subset of X such that the following two conditions hold: if a + b S with a h, b h, then ab S ; [/*:m] if ab S with a h, b h, then a + b S .[/*:m] Prove that S = X .
Day 4 - 23 May 2013

1 A positive integer a is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form a = a1 a2 ak a1 a2 ak with 0 ai 9 for 1 i k , and a1 = 0. For example, 283283 is a double number. Determine whether or not there are innitely many double numbers a such that a + 1 is a square and a + 1 is not a power of 10. 2 Let n 2 be an integer and f1 (x), f2 (x), . . . , fn (x) a sequence of polynomials with integer coecients. One is allowed to make moves M1 , M2 , . . . as follows: in the k -th move Mk one chooses an element f (x) of the sequence with degree of f at least 2 and replaces it with
Page 4
India
(f (x) f (k ))/(x k ). The process stops when all the elements of the sequence are of degree 1. If f1 (x) = f2 (x) = = fn (x) = xn + 1, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of n identical polynomials of degree 1. 3 In a triangle ABC , with AB = BC , E is a point on the line AC such that BE is perpendicular to AC . A circle passing through A and touching the line BE at a point P = B intersects the line AB for the second time at X . Let Q be a point on the line P B dierent from P such that BQ = BP . Let Y be the point of intersection of the lines CP and AQ. Prove that the points C, X, Y, A are concyclic if and only if CX is perpendicular to AB .
Page 5

India International Mathematical Olympiad Training Camp-2013

Uploaded by

Copyright:

Available Formats

India International Mathematical Olympiad Training Camp-2013

Uploaded by

Document Information

Original Description:

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

India International Mathematical Olympiad Training Camp-2013

Uploaded by

Copyright:

Available Formats

India

International Mathematical Olympiad Training Camp 2013

Day 1 - 05 May 2013

Day 2 - 10 May 2013

Team Selection Test

Day 1 - 15 May 2013

Day 2 - 16 May 2013

Day 3 - 22 May 2013

Day 4 - 23 May 2013

You might also like