About the Author
István Reiman was formerly Leader of the Chair of Geometry at the Budapest University of Technology. He has guided the Youth Mathematical Circle of the J Bolyai Mathematical Society and directed the preparation of Hungarian students for the annual International Maths Olympiad for 40 years. He was awarded the Paul Erdos Award by the World Federation of Mathematics Competitions in early 2000.
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International Mathematical Olympiad Volume I
By Istvan Reiman, János Pataki, András Stipsitz, Csaba Szabó
Wimbledon Publishing CompanyCopyright © 2016 Typotex Ltd.
All rights reserved.
International Mathematical Olympiad Problems
1959/1. Prove that the expression 21n+4 / 14n+3 is irreducible for every positive integer n.
1959/2. Determine the real solutions of the following equations.
a) [MATHEMATICAL EXPRESSION OMITTED]
b) [MATHEMATICAL EXPRESSION OMITTED]
c) [MATHEMATICAL EXPRESSION OMITTED]
1959/3. Suppose that x satisfies
(1) a cos2x + b cos x + c = 0.
Form a quadratic equation in cos 2x whose roots are the same values of x. Apply your result for the special case a = 4, b = 2, c = [perpendicular to] l.
1959/4. Construct a right triangle with a given hypotenuse, if we know that the median corresponding to the hypotenuse is equal to the geometric mean of the two adjacent sides.
1959/5. A point M is moving on the interval AB. Squares AMCD and BMEF over the subintervals AM and MB (on the same side of the line AB) are constructed The circumcircles of these squares intersect each other in the points M and N. Verify that the lines AE and BC pass through N. Show that for any choice of M the line M N passes through a fixed point. Find the locus of the midpoints of the intervals joining the centers of the two squares.
1959/6. The planes P and Q intersect in the line p. A and TC1[C]TC1 are points of the planes P and Q respectively, neither of them is on p. Construct that symmetric trapezium ABCD (with AB CD) for which its vertices B and D are on the planes P and Q and ABCD admits an incircle.
1960/1. Determine all three digit numbers which are equal to 11 times the sum of the squares of their digits.
1960/2. Determine the real solutions of the inequality
[MATHEMATICAL EXPRESSION OMITTED]
1960/3. The hypotenuse BC = a of the right triangle ABC has been divided into n equal intervals with n an odd integer. Let h denote the altitude corresponding to the hypotenuse; and the central interval subtends an angle α at A. Show that
tan α = 4nh/(n2 [perpendicular to] 1)a
1960/4. Construct the triangle ABC if its two altitudes ma and mb (corresponding to the vertices A and B) and the median corresponding to A is given.
1960/5. For a given cube ABCDA'B'C'D' let X be a point of the face diagonal AC, and Y a point of B'D'.
a) Find the locus of the midpoints of the intervals XY for all possible choices of X and Y.
b) Consider the point Z [member of] XY satisfying the equality ZY = 2XZ and determine the locus of these points for all choices of X and Y.
1960/6. A cone of revolution has an inscribed sphere tangent to the base of the cone (and to the sloping surface of the cone). A cylinder is circumscribed about the sphere so that its base lies in the base of the cone. Let V1 and V2 denote the volume of the cone and the resulting cylinder, respectively.
a) Show that V1 is not equal to V2.
b) Determine the smallest possible value of k = V1/V2, and for this minimal k construct the half angle of the symmetric cone.
1960/7. The parallel sides of a symmetric trapezium are of length a and b, while its altitude is m.
a) Construct the point P on the symmetry axis of the trapezium which is on the Thales circles over the legs of the trapezium.
b) Determine the distance of P from one of the parallel sides.
c) Under what assumption does such P exist?
1961/1. Solve the following system of equations for x, y and z:
(1) x + y + z = a,
(2) x2 + y2 + z2 = b2.
(3) xy = z2
where a and b are given real numbers. What conditions must a and b satisfy for x, y and z to be all positive and distinct?
1961/2. Let a, b and c be the sides of a given triangle while t is its area. Show that
(1) a2 + b2 + c2 ≥ 4t [square root of 3]
When does equality hold?
1961/3. Solve the equation
cosnx [perpendicular to] sinnx = 1
where n is a positive integer.
1961/4.P is a point inside the triangle P1P2P3. The intersections of the lines P1P, P2P and P3P with the opposite sides are denoted by Q1, Q2 and Q3, respectively. Show that there is one among the ratios
P1P/PQ1, P2P/PQ2, P3P/PQ3
which is not less, and one which is not more than 2.
1961/5. Construct a triangle ABC if the length of the two sides AC = b and AB = c and the acute angle AMB = ω is given — here M is the midpoint of BC. Show also that the problem admits a solution if and only if
b tan ω/2 ≤ c< b.
1961/6. Let ε be a given plane and A, B, C three non-collinear points on one side of ε. Suppose furthermore that the plane determined by these points is parallel toε. Let A', B' and C' be three arbitrary points onε. The midpoints of AA', BB' and CC' are denoted by L, M and N respectively. The centre of gravity of the triangle LMN is denoted by G. (Those triples A', B', C' for which L, M, N do not form a triangle, are disregarded.) Determine the locus of G for any possible choice of the triple A', B', C' on the planeε.
1962/1. Determine the smallest possible positive integer x whose last decimal digit is 6, and if we erase this last 6 and put it in front of the remaining digits, we get four times x.
1962/2. Determine all real x satisfying
[square root of 3 [perpendicular to] x] [perpendicular to] [square root of x + 1 > 1/2]
1962/3. The cube ABCDA'B'C'D' with upper face ABCD and lower face A'B'C'D' (AA' BB' CC' DD') is given. A point X runs along the perimeter of ABCD (in the direction given by the above order) with constant speed, while a point Y does the same (with equal speed) along the perimeter of the square B'C'CB. X and Y start in the same instant from A and B', respectively. Determine the locus of the midpoint Z of the interval XY.
1962/4. Solve the following equation:
cos2x + cos2 2x + cos2 3x = 1.
1962/5. Three distinct points A, B and C on a circle k are given. Construct the point D on the circle for which the quadrilateral ABCD admits an incircle.
1962/6. Let R and r denote the radii of the circumcircle and the incircle of an isosceles triangle. Show that the distance d between the centres of the two circles is
(1) d = [square root of R(R [perpendicular to] 2r)].
1962/7. There are five spheres which are tangent to all extended edges of a tetrahedron SABC. Show that
a) SABC is a regular tetrahedron;
b) conversely: a regular tetrahedron admits five spheres with the properties described above.
1963/1. Determine the real solutions of the following equality (p denotes a real parameter)
[square root of x2 [perpendicular to] p + 2 [square root of x2 [perpendicular to] 1 = x.
1963/2. Given a point A and a segment BC in the 3-dimensional space, determine the locus of those points, P, for which the angle [angle] APX is a right angle for some X on the segment BC.
1963/3. Consider a convex n-gon with equal angles and with consecutive sides a1, a2, ..., an satisfying
(1) a1 ≥ a2 ... ≥ an.
Show that under the above conditions we have
(2) a1 = a2 = ... = an.
1963/4. Determine the values x1, x2, x3, x4, x5 satisfying
(1) x5 + TC1[x]TC12 = yx1,
(2) x1 + x3 = yx2,
(3) x2 + x4 = yx3,
(4) x3 + x5 = yx4,
(5) x4 + x1 = yx5
where y is a given parameter.
1963/5. Show that
(1) cos π/7 [perpendicular to] cos 2π/7 + cos 3π/7 = 1/2.
1963/6. Five students, A, B, C, D and E were placed 1 to 5 in a contest. Someone made the initial guess that the final result would be the order ABCDE, but — as it turned out — this person was wrong on the final position of all the contestants; moreover no two students predicted to finish consecutively did so. A second person guessed DAECB, which was much better, since exactly two contestants finished in the place predicted, and two disjoint pairs predicted to finish consecutively did so. Determine the outcome of the contest.
1964/1. a) Find all positive integers n for which 7 divides 2n [perpendicular] 1.
b) Show that there is no positive integer n for which 7 divides 2n + 1.
1964/2. Let a, b and c denote the lengths of the sides of a triangle. Show that
(1) [MATHEMATICAL EXPRESSION OMITTED]
1964/3. Let a, b, c denote the lengths of the sides of the triangle ABC. Tangents to the inscribed circle are constructed parallel to the sides. Each tangent forms a triangle with the other two sides of the triangle, and a circle is inscribed in each of these three triangles. Find the total area of all four inscribed circles.
Consider the tangents of the incircle which are parallel to the sides. These tangents give rise to three subtriangles of ABC, consider the incircles of these subtriangles. Determine the sum of the areas of the four incircles.
1964/4. Each pair from 17 scientists exchange letters on one of three topics. Prove that there are at least three scientists who write to each other on the same topic.
1964/5. Five points on the plane are situated so that no two of the lines joining a pair of points are coincident, parallel or perpendicular. Through each point lines are drawn perpendicular to each of the lines through two of the other four points. Give the best possible upper bound for the number of intersection points of these orthogonals, disregarding the given 5 points.
1964/6. Let ABCD be a given tetrahedron and D1 the centroid of the face ABC. The parallels to DD1 passing through the vertices A, B and C intersect the opposite faces in A1, B1 and C1, respectively.
a) Show that the volume of ABCD is one-third the volume of A1B1C1D1.
b) Is the result valid for any choice of D1 in the interior of ABC?
1965/1. Find all x in the interval [0,2π] which satisfy
(1) [MATHEMATICAL EXPRESSION OMITTED]
1965/2. The coefficients of the system of equations
[MATHEMATICAL EXPRESSION OMITTED],
[MATHEMATICAL EXPRESSION OMITTED],
[MATHEMATICAL EXPRESSION OMITTED]
are subject to the following constraints:
a) a11, a22 and a33 are all positive,
b) all other coefficients are negative,
c) the sum of coefficients in each equation is positive.
Verify that the only solution of the system is
x1 = x2 = x3 = 0.
1965/3. The length of the edge AB in the tetrahedron ABCD is a, while the length of CD is b. The distance between the skew lines AB and CD is d, the angle determined by them is ω. The tetrahedron is divided into two parts by a plane εparallel to AB and CD. We also know that k times the distance between AB and ε equals the distance between CD and ε. Determine the ratio of the volumes of the parts of the tetrahedron.
1965/4. Find all sets of four real numbers x1, x2, x3, x4 such that the sum of any one and the product of the other three is 2.
1965/5. The triangle OAB has angle [angle] AOB acute. M is an arbitrary point in OAB different from O. The points P and Q are the feet of the perpendiculars from M to OA and OB, respectively. Determine the locus of the orthocentre H of the triangle OPQ if M is
a) on AB;
b) in the interior of OAB.
1965/6. For n ≥ 3 points in the plane denote the maximal distance of pairs of points by d. Prove that at most n pairs of points are of distance d apart.
1966/1. Problems A, B and C have been posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as C. The number of competitors solving only A was one more than the number of competitors solving A and at least one other problem. The number of competitors solving A equalled the number solving just B plus the number of competitors solving just C. How many competitors solved just B?
1966/2. Let a, b, c denote the sides of a triangle, while the opposite angles are denoted by α, β, γ. Prove that if
(1) a + b = tan γ/2 (a tan α + b tan β)
then the triangle is isosceles.
1966/3. Prove that a point in the space has the smallest sum of distances to vertices of a regular tetrahedron if and only if it is the centre of the tetrahedron.
Excerpted from International Mathematical Olympiad Volume I by Istvan Reiman, János Pataki, András Stipsitz, Csaba Szabó. Copyright © 2016 Typotex Ltd.. Excerpted by permission of Wimbledon Publishing Company.
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Table of Contents
Preface; Problems 1959-1975; A Glossary of Theorems