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    China China Girls Math Olympiad 2003

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    PremMehta
    The document contains 4 geometry and number theory problems from the 2003 China Girls Math Olympiad: 1) The first problem involves proving properties of areas of triangles formed by points on the sides of another triangle. 2) The second problem calculates the maximum difference in possible sums of position values when students rearrange seats in a classroom. 3) The third problem proves that 4 points lie on a circle based on properties of angles and parallel lines in a figure involving a circle and its diameter. 4) The fourth problem involves arranging real numbers around a circle and proving properties about neighboring products being above or below a given value.

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    China China Girls Math Olympiad 2003

    Uploaded by

    PremMehta
    117 views3 pages
    The document contains 4 geometry and number theory problems from the 2003 China Girls Math Olympiad: 1) The first problem involves proving properties of areas of triangles formed by points on the sides of another triangle. 2) The second problem calculates the maximum difference in possible sums of position values when students rearrange seats in a classroom. 3) The third problem proves that 4 points lie on a circle based on properties of angles and parallel lines in a figure involving a circle and its diameter. 4) The fourth problem involves arranging real numbers around a circle and proving properties about neighboring products being above or below a given value.

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    The document contains 4 geometry and number theory problems from the 2003 China Girls Math Olympiad: 1) The first problem involves proving properties of areas of triangles formed by points on the sides of another triangle. 2) The second problem calculates the maximum difference in possible sums of position values when students rearrange seats in a classroom. 3) The third problem proves that 4 points lie on a circle based on properties of angles and parallel lines in a figure involving a circle and its diameter. 4) The fourth problem involves arranging real numbers around a circle and proving properties about neighboring products being above or below a given value.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    117 views3 pages

    China China Girls Math Olympiad 2003

    Uploaded by

    PremMehta
    The document contains 4 geometry and number theory problems from the 2003 China Girls Math Olympiad: 1) The first problem involves proving properties of areas of triangles formed by points on the sides of another triangle. 2) The second problem calculates the maximum difference in possible sums of position values when students rearrange seats in a classroom. 3) The third problem proves that 4 points lie on a circle based on properties of angles and parallel lines in a figure involving a circle and its diameter. 4) The fourth problem involves arranging real numbers around a circle and proving properties about neighboring products being above or below a given value.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    China

    China Girls Math Olympiad


    2003

    Day 1

    1 Let ABC be a triangle. Points D and E are on sides AB and AC, respectively, and point F
    is on line segment DE. Let AD AE DF
    AB = x, AC = y, DE = z. Prove that
    (1) S4BDF = (1 x)yS4ABC and S4CEF = x(1 y)(1 z)S4ABC ;
    p p p
    (2) 3 S4BDF + 3 S4CEF 3 S4ABC .

    2 There are 47 students in a classroom with seats arranged in 6 rows 8 columns, and the seat
    in the i-th row and j-th column is denoted by (i, j). Now, an adjustment is made for students
    seats in the new school term. For a student with the original seat (i, j), if his/her new seat
    is (m, n), we say that the student is moved by [a, b] = [i m, j n] and define the position
    value of the student as a + b. Let S denote the sum of the position values of all the students.
    Determine the difference between the greatest and smallest possible values of S.

    3 As shown in the figure, quadrilateral ABCD is inscribed in a circle with AC as its diameter,
    BD AC, and E the intersection of AC and BD. Extend line segment DA and BA through
    A to F and G respectively, such that DG k BF. Extend GF to H such that CH GH.
    Prove that points B, E, F and H lie on one circle.

    F
    H
    A

    E
    B D

    This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
    http://www.artofproblemsolving.com/
    China
    China Girls Math Olympiad
    2003

    4 (1) Prove that there exist five nonnegative real numbers a, b, c, d and e with their sum equal
    to 1 such that for any arrangement of these numbers around a circle, there are always two
    neighboring numbers with their product not less than 19 .
    (2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always
    possible to arrange them around a circle such that there are two neighboring numbers with
    their product not greater than 19 .

    This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
    http://www.artofproblemsolving.com/
    China
    China Girls Math Olympiad
    2003

    Day 2

    1 Let {an }
    1 be a sequence of real numbers such that a1 = 2, and

    an+1 = a2n an + 1, n N.

    Prove that
    2003
    1 X 1
    1 < < 1.
    20032003 ai
    i=1

    2 Let n 2 be an integer. Find the largest real number such that the inequality
    n1
    X
    a2n ai + 2 an .
    i=1

    holds for any positive integers a1 , a2 , . . . an satisfying a1 < a2 < . . . < an .

    3 Let the sides of a scalene triangle 4ABC be AB = c, BC = a, CA = b, and D, E, F be


    points on BC, CA, AB such that AD, BE, CF are angle bisectors of the triangle, respectively.
    Assume that DE = DF. Prove that
    a b c
    (1) b+c = c+a + a+b
    (2) BAC > 90 .

    4 Let n be a positive integer, and Sn , be the set of all positive integer divisors of n (including
    1 and itself). Prove that at most half of the elements in Sn have their last digits equal to 3.

    This file was downloaded from the AoPS Math Olympiad Resources Page Page 3
    http://www.artofproblemsolving.com/

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