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International mathematical olimpiad(6th to tenth)

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International mathematical olimpiad(6th to tenth)

    6th International Mathematical Olympiad

    1964 USSR

    n1. (a) Find all positive integers n for which is divisible by 7. 21

    n(b) Prove that there is no positive integer n such that is divisible by 7. 21

    222abcabcabcabcabc()()()?????????32. Let a, b, c be the sides of a triangle. Prove that .

3. A circle is inscribed in triangle ABC with sides a, b, c. Tangents to the circle parallel to the sides of

    the triangle are constructed. Each of these tangents cuts off a triangle from (ABC. In each of these

    triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b,

    c).

4. Seventeen people correspond by mail with one another each one with all the rest. In their letters

    only three different topics are discussed. Each pair of correspondents deals with only one of these

    topics. Prove that there are at least three people who write to each other about the same topic.

    5. Suppose five points in a plane are situated so that no two of the straight lines joining them are

    parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines

    joining the other four points. Determine the maximum number of intersections that these

    perpendiculars can have.

    DDD6. In tetrahedron ABCD, vertex D is connected with the centroid of (ABC. Lines parallel to 00

    ABare drawn through A, B and C. These lines intersect the planes BCD, CAD and ABD in points , 11

    ABCDCand , respectively. Prove that the volume of ABCD is one third the volume of . Is the 11101

    Dresult true if point is selected anywhere within (ABC? 0

    7th International Mathematical Olympiad

    1965 West Germany

1. Determine all values x in the interval which satisfy the inequality 02??x~

    2cos|1sin21sin2|2xxx?????.

2. Consider the system of equations

    axaxax??!0111122133

     axaxax??!0211222233

    axaxax??!0311322333

    xwith unknowns x, x, . The coefficients satisfy the conditions: 312

    a(a) a, a, are positive numbers; 331122

    (b) the remaining coefficients are negative numbers; (c) in each equation, the sum of the coefficients is positive.

    xxx!!!0Prove that the given system has only the solution . 123

    3. Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively. The

    ?distance between the skew lines AB and CD is d, and the angle between them is . Tetrahedron

    ABCD is divided into two solids by plane , parallel to lines AB and CD. The ratio of the distances

    of from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained.

    xxxx4. Find all sets of four real numbers , , , such that the sum of any one and the product of 3124

    the other three is equal to 2.

    5. Consider (OAB with acute angle )AOB. Through a point perpendiculars are drawn to OA MO(

    and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of

    (OPQ is H. What is the locus of H if M is permitted to range over (a) the side AB, (b) the interior of (OAB?

    6. In a plane a set of n points () is given. Each pair of points is connected by a segment. Let d be n?3

    the length of the longest of these segments. We define a diameter of the set to be any connecting

    segment of length d. Prove that the number of diameters of the given set is at most n.

    8th International Mathematical Olympiad

    1966 Bulgaria

    1. In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A,

    the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many

    students solved only problem B?

    2. Let a, b, c be the lengths of the sides of a triangle, and , , , respectively, the angles opposite

    these sides. Prove that if , the triangle is isosceles. abab?!?tan(tantan);,2

3. Prove: The sum of the distances of the vertices of a regular tetrahedron from the centre of its

    circumscribed sphere is less than the sum of the distances of these vertices from any other point in

    space.

    k~4. Prove that for every natural number n, and for every real number (t = 0,1, , n; k any x(t2

    111ninteger), . ???!?xxcotcot2nxxxsin2sin4sin2

5. Solve the system of equations

    caaa!???1128

    222caaa!???2128

    nnncaaa!???n128

    aaaawhere , , , are four different real numbers. 3124

    6. In the interior of sides BC, CA, AB of triangle ABC, any points K, L, M, respectively, are selected.

    Prove that the area of at least one of the triangles AML, BKM, CLK is less than or equal to one

    quarter of the area of triangle ABC.

    9th International Mathematical Olympiad

    1967 Yugoslavia

1. Let ABCD be a parallelogram with side lengths , , and with . If (ABD is AD1ABa)!BAD

    acute, prove that the four circles of radius 1 with centres A, B, C, D cover the parallelogram if and

    a??cos3sin;;only if .

2. Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is smaller than

    1or equal to . 8

    css!?(1)3. Let k, m, n be natural numbers such that is a prime greater than . Let . mk??1n1s

    cccProve that the product is divisible by the product . cccccc???;;;;;;12nmkmkmnk???12

    ABC4. Let and be any two acute-angled triangles. Consider all triangles ABC that are ABC000111

    Asimilar to (so that vertices , , correspond to vertices A, B, C, respectively) and (ABCAA311121

    ABCABCcircumscribed about triangle (where lies on BC, on CA, and on AB). Of all 000000

    such possible triangles, determine the one with maximum area, and construct it.

    {}c5. Consider the sequence , where n

    caaa!???1128

    222caaa!???2128

    nnncaaa!???n128

    aaain which , , …, are real numbers not all equal to zero. Suppose that an infinite number of 812

    {}cc0terms of the sequence are equal to zero. Find all natural numbers n for which . nn

6. In a sports contest, there were m medals awarded on n successive days (). On the first day, one n1

    1medal and of the remaining medals were awarded. On the second day, two medals and m17

    1 of the now remaining medals were awarded; and so on. On the n-th and last day, the remaining n 7

    medals were awarded. How many days did the contest last, and how many medals were awarded

    altogether?

    10th International Mathematical Olympiad

    1968 USSR

1. Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of

    whose angles is twice as large as another.

2. Find all natural numbers x such that the product of their digits (in decimal notation) is equal to

    2xx??1022.

3. Consider the system of equations

    2axbxcx??!112

    2axbxcx??!223

    2axbxcx??!nnn??11

    2axbxcx??!nn1

    2x(!??(1)4bacwith unknowns x, x, …, , where a, b, c are real and . Let . Prove that a(0n12

    for this system

    (a) if , there is no solution, (,0

    (b) if , there is exactly one solution, (!0

    (c) if , there is more than one solution. (,0

4. Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths

    which are the sides of a triangle.

5. Let f be a real-valued function defined for all real numbers x such that, for some positive constant

    12a, the equation holds for all x. fxafxfx()()[()]?!??2

    (a) Prove that the function f is periodic.

     (i.e. there exists a positive number b such that fxbfx()()?! for all x.)

    (b) For , give an example of a non-constant function with the required properties. a1

    6. For every natural number n, evaluate the sum

    kk?,,,,nnnn????2122,,,,. !?????????11kk??????2242:?:?0k:?:?

    (The symbol [x] denotes the greatest integer not exceeding x.) 7.

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