IMO shortlist problems 1959-2009 (new)

Determine the locus of points in space which are vertices of right angles with one side passing through A, and the other side intersecting segment BC... 3 Given the tetrahedron ABCD whos

IMO ShortList Problems

1 Prove that the fraction 21n + 4

14n + 3 is irreducible for every natural number n.

2 For what real values of x is

q

x +√2x − 1 +

q

x −√2x − 1 = Agiven

a) A =√2;

b) A = 1;

c) A = 2,

where only non-negative real numbers are admitted for square roots?

3 Let a, b, c be real numbers Consider the quadratic equation in cos x

a cos x2+ b cos x + c = 0

Using the numbers a, b, c form a quadratic equation in cos 2x whose roots are the same asthose of the original equation Compare the equation in cos x and cos 2x for a = 4, b = 2,

c = −1

a) Prove that N and N0 coincide;

b) Prove that the straight lines M N pass through a fixed point S independent of the choice

of M ;

c) Find the locus of the midpoints of the segments P Q as M varies between A and B

6 Two planes, P and Q, intersect along the line p The point A is given in the plane P , andthe point C in the plane Q; neither of these points lies on the straight line p Construct

an isosceles trapezoid ABCD (with AB k CD) in which a circle can be inscribed, and withvertices B and D lying in planes P and Q respectively

1 Determine all three-digit numbers N having the property that N is divisible by 11, and N

11

is equal to the sum of the squares of the digits of N

2 For what values of the variable x does the following inequality hold:

4x2(1 −√2x + 1)2 < 2x + 9 ?

3 In a given right triangle ABC, the hypotenuse BC, of length a, is divided into n equal parts (nand odd integer) Let α be the acute angel subtending, from A, that segment which containsthe mdipoint of the hypotenuse Let h be the length of the altitude to the hypotenuse fo thetriangle Prove that:

tan α = 4nh

(n2− 1)a.

b) Find the locus of points Z which lie on the segment XY of part a) with ZY = 2XZ.

6 Consider a cone of revolution with an inscribed sphere tangent to the base of the cone Acylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone.let V1 be the volume of the cone and V2 be the volume of the cylinder

a) Prove that V1 6= V2;

b) Find the smallest number k for which V1 = kV2; for this case, construct the angle subtended

by a diamter of the base of the cone at the vertex of the cone

7 An isosceles trapezoid with bases a and c and altitude h is given

a) On the axis of symmetry of this trapezoid, find all points P such that both legs of thetrapezoid subtend right angles at P ;

b) Calculate the distance of p from either base;

c) Determine under what conditions such points P actually exist Discuss various cases thatmight arise

1 Solve the system of equations:

x + y + z = a

x2+ y2+ z2 = b2

xy = z2where a and b are constants Give the conditions that a and b must satisfy so that x, y, z aredistinct positive numbers

2 Let a, b, c be the sides of a triangle, and S its area Prove:

a2+ b2+ c2 ≥ 4S√3

In what case does equality hold?

3 Solve the equation cosnx − sinnx = 1 where n is a natural number

at least one is ≤ 2 and at least one is ≥ 2

5 Construct a triangle ABC if AC = b, AB = c and ∠AM B = w, where M is the midpoint ofthe segment BC and w < 90 Prove that a solution exists if and only if

b tanw

2 ≤ c < b

In what case does the equality hold?

6 Consider a plane  and three non-collinear points A, B, C on the same side of ; suppose theplane determined by these three points is not parallel to  In plane  take three arbitrarypoints A0, B0, C0 Let L, M, N be the midpoints of segments AA0, BB0, CC0; Let G be thecentroid of the triangle LM N (We will not consider positions of the points A0, B0, C0 suchthat the points L, M, N do not form a triangle.) What is the locus of point G as A0, B0, C0range independently over the plane ?

1 Find the smallest natural number n which has the following properties:

a) Its decimal representation has a 6 as the last digit

b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number

is four times as large as the original number n

2 Determine all real numbers x which satisfy the inequality:

Day 2

4 Solve the equation cos2x + cos22x + cos23x = 1

5 On the circle K there are given three distinct points A, B, C Construct (using only a edge and a compass) a fourth point D on K such that a circle can be inscribed in the quadri-lateral thus obtained

straight-6 Consider an isosceles triangle let R be the radius of its circumscribed circle and r be theradius of its inscribed circle Prove that the distance d between the centers of these two circleis

d =pR(R − 2r)

7 The tetrahedron SABC has the following property: there exist five spheres, each tangent tothe edges SA, SB, SC, BC, CA, AB, or to their extensions

a) Prove that the tetrahedron SABC is regular

b) Prove conversely that for every regular tetrahedron five such spheres exist

1 Find all real roots of the equation

p

x2− p + 2px2− 1 = x

where p is a real parameter

2 Point A and segment BC are given Determine the locus of points in space which are vertices

of right angles with one side passing through A, and the other side intersecting segment BC

3 In an n-gon A1A2 An, all of whose interior angles are equal, the lengths of consecutivesides satisfy the relation

a1 ≥ a2≥ · · · ≥ an.Prove that a1 = a2= = an

5 Prove that cosπ7 − cos2π

7 + cos3π7 = 12

6 Five students A, B, C, D, E took part in a contest One prediction was that the contestantswould finish in the order ABCDE This prediction was very poor In fact, no contestantfinished in the position predicted, and no two contestants predicted to finish consecutivelyactually did so A second prediction had the contestants finishing in the order DAECB Thisprediction was better Exactly two of the contestants finished in the places predicted, andtwo disjoint pairs of students predicted to finish consecutively actually did so Determine theorder in which the contestants finished

2 Suppose a, b, c are the sides of a triangle Prove that

Day 2

4 Seventeen people correspond by mail with one another-each one with all the rest In theirletters only three different topics are discussed each pair of correspondents deals with onlyone of these topics Prove that there are at least three people who write to each other aboutthe same topic

5 Supppose five points in a plane are situated so that no two of the straight lines joining themare parallel, perpendicular, or coincident From each point perpendiculars are drawn to allthe lines joining the other four points Determine the maxium number of intersections thatthese perpendiculars can have

6 In tetrahedron ABCD, vertex D is connected with D0, the centrod if 4ABC Line parallel

to DD0 are drawn through A, B and C These lines intersect the planes BCD, CAD andABD in points A2, B1, and C1, respectively Prove that the volume of ABCD is one thirdthe volume of A1B1C1D0 Is the result if point Do is selected anywhere within 4ABC?

1 Determine all values of x in the interval 0 ≤ x ≤ 2π which satisfy the inequality

2 cos x ≤√1 + sin 2x −√1 − sin 2x ≤√2

2 Consider the sytem of equations

a11x1+ a12x2+ a13x3 = 0

a21x1+ a22x2+ a23x3 = 0

a31x1+ a32x2+ a33x3 = 0with unknowns x1, x2, x3 The coefficients satisfy the conditions:

a) a11, a22, a33 are positive numbers;

b) the remaining coefficients are negative numbers;

c) in each equation, the sum ofthe coefficients is positive

Prove that the given system has only the solution x1 = x2= x3= 0

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 ω.Tetrahedron ABCD is divided into two solids by plane , parallel to lines AB and CD Theratio of the distances of  from AB and CD is equal to k Compute the ratio of the volumes

of the two solids obtained

Day 2

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 equal to 2

5 Consider 4OAB with acute angle AOB Thorugh a point M 6= O perpendiculars are drawn

to OA and OB, the feet of which are P and Q respectively The point of intersection of thealtitudes of 4OP Q is H What is the locus of H if M is permitted to range over

a) the side AB;

b) the interior of 4OAB

6 In a plane a set of n points (n ≥ 3) is give Each pair of points is connected by a segment.Let d be the length of the longest of these segments We define a diameter of the set to beany connecting segment of length d Prove that the number of diameters of the given set is

at most n

exist a circle passing through (at least) 3 of the given points and not containing any other ofthe n points in its interior ?

2 Given n positive real numbers a1, a2, , an such that a1a2· · · an= 1, prove that

(1 + a1)(1 + a2) · · · (1 + an) ≥ 2n

3 A regular triangular prism has the altitude h, and the two bases of the prism are equilateraltriangles with side length a Dream-holes are made in the centers of both bases, and thethree lateral faces are mirrors Assume that a ray of light, entering the prism through thedream-hole in the upper base, then being reflected once by any of the three mirrors, quits theprism through the dream-hole in the lower base Find the angle between the upper base andthe light ray at the moment when the light ray entered the prism, and the length of the way

of the light ray in the interior of the prism

4 Given 5 points in the plane, no three of them being collinear Show that among these 5points, we can always find 4 points forming a convex quadrilateral

5 Prove the inequality

tanπ sin x

4 sin α + tan

π cos x

4 cos α > 1for any x, α with 0 ≤ x ≤ π2 and π6 < y < π3

6 Let m be a convex polygon in a plane, l its perimeter and S its area Let M (R) be the locus

of all points in the space whose distance to m is ≤ R, and V (R) is the volume of the solid

Hereby, we say that the distance of a point C to a figure m is ≤ R if there exists a point D

of the figure m such that the distance CD is ≤ R (This point D may lie on the boundary ofthe figure m and inside the figure.)

additional question:

b.) Find the area of the planar R-neighborhood of a convex or non-convex polygon m

7 For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

8 We are given a bag of sugar, a two-pan balance, and a weight of 1 gram How do we obtain

1 kilogram of sugar in the smallest possible number of weighings?

9 Find x such that trigonometric

sin 3x cos(60◦− x) + 1sin(60◦− 7x) − cos(30◦+ x) + m = 0where m is a fixed real number

10 How many real solutions are there to the equation x = 1964 sin x − 189 ?

11 Does there exist an integer z that can be written in two different ways as z = x! + y!, where

x, y are natural numbers with x ≤ y ?

12 Find digits x, y, z such that the equality

X

Posted already on the board I think

15 Given four points A, B, C, D on a circle such that AB is a diameter and CD is not a diameter.Show that the line joining the point of intersection of the tangents to the circle at the points

C and D with the point of intersection of the lines AC and BD is perpendicular to the lineAB

16 We are given a circle K with center S and radius 1 and a square Q with center M and side 2.Let XY be the hypotenuse of an isosceles right triangle XY Z Describe the locus of points

Z as X varies along K and Y varies along the boundary of Q

17 Let ABCD and A0B0C0D0 be two arbitrary parallelograms in the space, and let M, N, P, Q

be points dividing the segments AA0, BB0, CC0, DD0 in equal ratios

a.) Prove that the quadrilateral M N P Q is a parallelogram

b.) What is the locus of the center of the parallelogram M N P Q, when the point M moves

on the segment AA0 ?

(Consecutive vertices of the parallelograms are labelled in alphabetical order

18 Solve the equation sin x1 +cos x1 = 1p where p is a real parameter Discuss for which values of pthe equation has at least one real solution and determine the number of solutions in [0, 2π)for a given p

19 Construct a triangle given the radii of the excircles

20 Given three congruent rectangles in the space Their centers coincide, but the planes they lie

in are mutually perpendicular For any two of the three rectangles, the line of intersection ofthe planes of these two rectangles contains one midparallel of one rectangle and one midparallel

of the other rectangle, and these two midparallels have different lengths Consider the convexpolyhedron whose vertices are the vertices of the rectangles

a.) What is the volume of this polyhedron ?

b.) Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the conditionfor this polyhedron to be regular ?

21 Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality

 6Vπ

2

≤

2S

π√3

3

When does equality occur?

22 Let P and P0 be two parallelograms with equal area, and let their sidelengths be a, b and a0,

b0 Assume that a0 ≤ a ≤ b ≤ b0, and moreover, it is possible to place the segment b0 such that

it completely lies in the interior of the parallelogram P

Show that the parallelogram P can be partitioned into four polygons such that these fourpolygons can be composed again to form the parallelogram P0

23 Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle

24 There are n ≥ 2 people at a meeting Show that there exist two people at the meeting whohave the same number of friends among the persons at the meeting (It is assumed that if A

is a friend of B, then B is a friend of A; moreover, nobody is his own friend.)

b.) Using the inequality (1), show that if the real numbers a1, a2, , ansatisfy the inequality

a1+ a2+ + an≥

q(n − 1) a2

1+ a2

2+ + a2

n
,then all of these numbers a1, a2, , an are non-negative

27 Given a point P lying on a line g, and given a circle K Construct a circle passing throughthe point P and touching the circle K and the line g

28 In the plane, consider a circle with center S and radius 1 Let ABC be an arbitrary trianglehaving this circle as its incircle, and assume that SA ≤ SB ≤ SC Find the locus of

a.) all vertices A of such triangles;

b.) all vertices B of such triangles;

c.) all vertices C of such triangles

29 A given natural number N is being decomposed in a sum of some consecutive integers.a.) Find all such decompositions for N = 500

b.) How many such decompositions does the number N = 2α3β5γ (where α, β and γ arenatural numbers) have? Which of these decompositions contain natural summands only?c.) Determine the number of such decompositions (= decompositions in a sum of consecutiveintegers; these integers are not necessarily natural) for an arbitrary natural N

Note by Darij: The 0 is not considered as a natural number

30 Let n be a positive integer, prove that :

(a) log10(n + 1) > 10n3 + log10n;

(b) log n! > 3n10 12 +13 + · · · +n1 − 1

31 Solve the equation |x2− 1| + |x2− 4| = mx as a function of the parameter m Which pairs(x, m) of integers satisfy this equation ?

32 The side lengths a, b, c of a triangle ABC form an arithmetical progression (such that b − a =

c − b) The side lengths a1, b1, c1 of a triangle A1B1C1 also form an arithmetical progression(with b1 − a1 = c1 − b1) [Hereby, a = BC, b = CA, c = AB, a1 = B1C1, b1 = C1A1,

c1 = A1B1.] Moreover, we know that ]CAB = ]C1A1B1

Show that triangles ABC and A1B1C1 are similar

Note by Darij: I have changed the wording of the problem since the conditions given in theoriginal were not sufficient, at least not in the form they were written

33 Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle.From each of the vertices of this triangle, we draw a tangent to the smaller circle Prove thatthe length of one of these tangents equals the sum of the lengths of the two other tangents

34 Find all pairs of positive integers (x; y) satisfying the equation 2x = 3y+ 5

35 Let ax3+ bx2+ cx + d be a polynomial with integer coefficients a, b, c, d such that ad is anodd number and bc is an even number Prove that (at least) one root of the polynomial isirrational

36 Let ABCD be a quadrilateral inscribed in a circle Show that the centroids of triangles ABC,CDA, BCD, DAB lie on one circle

37 Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic lateral to the respective opposite sides are concurrent

quadri-Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle

38 Two concentric circles have radii R and r respectively Determine the greatest possible number

of circles that are tangent to both these circles and mutually nonintersecting Prove that thisnumber lies between 32 ·

√ R+√r

√ R−√r − 1 and 6320·R+rR−r

39 Consider a circle with center O and radius R, and let A and B be two points in the plane ofthis circle

a.) Draw a chord CD of the circle such that CD is parallel to AB, and the point of theintersection P of the lines AC and BD lies on the circle

40 For a positive real number p, find all real solutions to the equation

p

x2+ 2px − p2−px2− 2px − p2 = 1

41 Given a regular n-gon A1A2 An (with n ≥ 3) in a plane How many triangles of the kind

AiAjAk are obtuse ?

42 Given a finite sequence of integers a1, a2, , anfor n ≥ 2 Show that there exists a subsequence

ak1, ak2, , akm, where 1 ≤ k1 ≤ k2≤ ≤ km≤ n, such that the number a2

k 1+ a2k

2+ + a2k

m

is divisible by n

Note by Darij: Of course, the 1 ≤ k1 ≤ k2 ≤ ≤ km ≤ n should be understood as

1 ≤ k1 < k2 < < km ≤ n; else, we could take m = n and k1 = k2 = = km, so that thenumber a2k1+ a2k2 + + a2km = n2a2k1 will surely be divisible by n

43 Given 5 points in a plane, no three of them being collinear Each two of these 5 points arejoined with a segment, and every of these segments is painted either red or blue; assume thatthere is no triangle whose sides are segments of equal color

a.) Show that:

(1) Among the four segments originating at any of the 5 points, two are red and two are blue.(2) The red segments form a closed way passing through all 5 given points (Similarly for theblue segments.)

b.) Give a plan how to paint the segments either red or blue in order to have the condition(no triangle with equally colored sides) satisfied

44 What is the greatest number of balls of radius 1/2 that can be placed within a rectangularbox of size 10 × 10 × 1 ?

45 An alphabet consists of n letters What is the maximal length of a word if we know that anytwo consecutive letters a, b of the word are different and that the word cannot be reduced to

a word of the kind abab with a 6= b by removing letters

46 Let a, b, c be reals and

f (a, b, c) =

+ |b − a|

|ab| +

b + a

ab +

2cProve that f (a, b, c) = 4 max{1a,1b,1c}

47 Consider all segments dividing the area of a triangle ABC in two equal parts Find the length

of the shortest segment among them, if the side lengths a, b, c of triangle ABC are given.How many of these shortest segments exist ?

48 For which real numbers p does the equation x2+ px + 3p = 0 have integer solutions ?

49 Two mirror walls are placed to form an angle of measure α There is a candle inside theangle How many reflections of the candle can an observer see?

50 Solve the equation sin x1 +cos x1 = 1p where p is a real parameter Discuss for which values of pthe equation has at least one real solution and determine the number of solutions in [0, 2π)for a given p

51 Consider n students with numbers 1, 2, , n standing in the order 1, 2, , n Upon a mand, any of the students either remains on his place or switches his place with anotherstudent (Actually, if student A switches his place with student B, then B cannot switch hisplace with any other student C any more until the next command comes.)

com-Is it possible to arrange the students in the order n, 1, 2, , n − 1 after two commands ?

52 A figure with area 1 is cut out of paper We divide this figure into 10 parts and color them

in 10 different colors Now, we turn around the piece of paper, divide the same figure on theother side of the paper in 10 parts again (in some different way) Show that we can color thesenew parts in the same 10 colors again (hereby, different parts should have different colors)such that the sum of the areas of all parts of the figure colored with the same color on bothsides is ≥ 101

53 Prove that in every convex hexagon of area S one can draw a diagonal that cuts off a triangle

of area not exceeding 16S

54 We take 100 consecutive natural numbers a1, a2, , a100 Determine the last two digits ofthe number a8

59 Let a, b, c be the lengths of the sides of a triangle, and α, β, γ respectively, the angles oppositethese sides Prove that if

a + b = tanγ

2(a tan α + b tan β)the triangle is isosceles

60 Prove that the sum of the distances of the vertices of a regular tetrahedron from the center

of its circumscribed sphere is less than the sum of the distances of these vertices from anyother point in space

61 Prove that for every natural number n, and for every real number x 6= kπ2t (t = 0, 1, , n; kany integer)

1sin 2x +

1sin 4x + · · · +

1sin 2nx = cot x − cot 2

63 Let ABC be a triangle, and let P , Q, R be three points in the interiors of the sides BC, CA,

AB of this triangle Prove that the area of at least one of the three triangles AQR, BRP ,

CP Q is less than or equal to one quarter of the area of triangle ABC

Alternative formulation: Let ABC be a triangle, and let P , Q, R be three points on thesegments BC, CA, AB, respectively Prove that

min {|AQR| , |BRP | , |CP Q|} ≤ 14 · |ABC|,

where the abbreviation |P1P2P3| denotes the (non-directed) area of an arbitrary triangle

P1P2P3

1 Prove that all numbers of the sequence

3 Prove the trigonometric inequality cos x < 1 − x22 +x164, when x ∈ 0,π2

4 Suppose medians ma and mb of a triangle are orthogonal Prove that:

a.) Using medians of that triangle it is possible to construct a rectangular triangle

b.) The following inequality:

5(a2+ b2− c2) ≥ 8ab,

is valid, where a, b and c are side length of the given triangle

5 Solve the system of equations:

Democratic Republic Of Germany

1 Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius r,there exist one (or more) with maximum area If so, determine their shape and area

2 Which fractions p

q, where p, q are positive integers < 100, is closest to

√2? Find all digitsafter the point in decimal representation of that fraction which coincide with digits in decimalrepresentation of √2 (without using any table)

Great Britain

1 Let k, m, n be natural numbers such that m + k + 1 is a prime greater than n + 1 Let

cs= s(s + 1) Prove that

(cm+1− ck)(cm+2− ck) (cm+n− ck)

is divisible by the product c1c2 cn

2 If x is a positive rational number show that x can be uniquely expressed in the form x =

3 The n points P1, P2, , Pnare placed inside or on the boundary of a disk of radius 1 in such

a way that the minimum distance Dn between any two of these points has its largest possiblevalue Dn Calculate Dn for n = 2 to 7 and justify your answer

2 In the space n ≥ 3 points are given Every pair of points determines some distance Supposeall distances are different Connect every point with the nearest point Prove that it isimpossible to obtain (closed) polygonal line in such a way.

3 Without using tables, find the exact value of the product:



4 Let k1 and k2 be two circles with centers O1 and O2 and equal radius r such that O1O2= r.Let A and B be two points lying on the circle k1 and being symmetric to each other withrespect to the line O1O2 Let P be an arbitrary point on k2 Prove that

1 A0B0C0 and A1B1C1 are acute-angled triangles Describe, and prove, how to construct thetriangle ABC with the largest possible area which is circumscribed about A0B0C0 (so BCcontains B0, CA contains B0, and AB contains C0) and similar to A1B1C1

2 Let ABCD be a regular tetrahedron To an arbitrary point M on one edge, say CD, sponds the point P = P (M ) which is the intersection of two lines AH and BK, drawn from

corre-A orthogonally to BM and from B orthogonally to corre-AM What is the locus of P when Mvaries ?

3 Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

4 Find values of the parameter u for which the expression

y = tan(x − u) + tan(x) + tan(x + u)tan(x − u) tan(x) tan(x + u)does not depend on x

1 Given m + n numbers: ai, i = 1, 2, , m, bj, j = 1, 2, , n, determine the number of pairs(ai, bj) for which |i − j| ≥ k, where k is a non-negative integer

2 An urn contains balls of k different colors; there are ni balls of i − th color Balls are selected

at random from the urn, one by one, without replacement, until among the selected balls mballs of the same color appear Find the greatest number of selections

3 Determine the volume of the body obtained by cutting the ball of radius R by the trihedronwith vertex in the center of that ball, it its dihedral angles are α, β, γ

4 In what case does the system of equations

6 Prove the identity

1 Prove that a tetrahedron with just one edge length greater than 1 has volume at most 18

2 Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincideswith the center of a sphere inscribed in that tetrahedron if and only if the skew edges of thetetrahedron are equal

3 Prove that for arbitrary positive numbers the following inequality holds

4 Does there exist an integer such that its cube is equal to 3n2+ 3n + 7, where n is an integer

5 Show that a triangle whose angles A, B, C satisfy the equality

sin2A + sin2B + sin2Ccos2A + cos2B + cos2C = 2

is a rectangular triangle

6 A line l is drawn through the intersection point H of altitudes of acute-angle triangles Provethat symmetric images la, lb, lc of l with respect to the sides BC, CA, AB have one point incommon, which lies on the circumcircle of ABC

4 (i) Solve the equation:

sin3(x) + sin3 2π

3 + x

+ sin3 4π

3 + x

+3

4cos 2x = 0.

(ii) Supposing the solutions are in the form of arcs AB with one end at the point A, thebeginning of the arcs of the trigonometric circle, and P a regular polygon inscribed in thecircle with one vertex in A, find:

1) The subsets of arcs having the other end in B in one of the vertices of the regular dodecagon.2) Prove that no solution can have the end B in one of the vertices of polygon P whose number

of sides is prime or having factors other than 2 or 3

5 If x, y, z are real numbers satisfying relations

x + y + z = 1 and arctan x + arctan y + arctan z = π

4,prove that x2n+1+ y2n+1+ z2n+1= 1 holds for all positive integers n

6 Prove the following inequality:

Socialists Republic Of Czechoslovakia

1 The parallelogram ABCD has AB = a, AD = 1, ∠BAD = A, and the triangle ABD has allangles acute Prove that circles radius 1 and center A, B, C, D cover the parallelogram if andonly

5 Let n be a positive integer Find the maximal number of non-congruent triangles whose sideslengths are integers ≤ n

6 Given a segment AB of the length 1, define the set M of points in the following way: itcontains two points A, B, and also all points obtained from A, B by iterating the followingrule: With every pair of points X, Y the set M contains also the point Z of the segment XYfor which Y Z = 3XZ

2 .

4 In a group of interpreters each one speaks one of several foreign languages, 24 of them speakJapanese, 24 Malaysian, 24 Farsi Prove that it is possible to select a subgroup in whichexactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speakFarsi

5 A linear binomial l(z) = Az + B with complex coefficients A and B is given It is known thatthe maximal value of |l(z)| on the segment −1 ≤ x ≤ 1 (y = 0) of the real line in the complexplane z = x + iy is equal to M Prove that for every z

|l(z)| ≤ M ρ,where ρ is the sum of distances from the point P = z to the points Q1 : z = 1 and Q3: z = −1

6 On the circle with center 0 and radius 1 the point A0is fixed and points A1, A2, , A999, A1000are distributed in such a way that the angle ∠A00Ak = k (in radians) Cut the circle at points

A0, A1, , A1000 How many arcs with different lengths are obtained ?

ϕ(x, y, z) = h(x + y + z)for all real numbers x, y and z.

4 A subset S of the set of integers 0 - 99 is said to have property A if it is impossible to fill acrossword-puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01,and so on) Determine the maximal number of elements in the set S with the property A

5 In the plane a point O is and a sequence of points P1, P2, P3, are given The distances

OP1, OP2, OP3, are r1, r2, r3, Let α satisfies 0 < α < 1 Suppose that for every n thedistance from the point Pnto any other point of the sequence is ≥ rαn Determine the exponent

β, as large as possible such that for some C independent of n

rn≥ Cnβ, n = 1, 2,

6 In making Euclidean constructions in geometry it is permitted to use a ruler and a pair ofcompasses In the constructions considered in this question no compasses are permitted,but the ruler is assumed to have two parallel edges, which can be used for constructing twoparallel lines through two given points whose distance is at least equal to the breadth of therule Then the distance between the parallel lines is equal to the breadth of the ruler Carrythrough the following constructions with such a ruler Construct:

1 Two ships sail on the sea with constant speeds and fixed directions It is known that at 9 : 00the distance between them was 20 miles; at 9 : 35, 15 miles; and at 9 : 55, 13 miles At whatmoment were the ships the smallest distance from each other, and what was that distance ?

2 Find all triangles whose side lengths are consecutive integers, and one of whose angles is twiceanother

3 Prove that every tetrahedron has a vertex whose three edges have the right lengths to form

7 Prove that the product of the radii of three circles exscribed to a given triangle does notexceed A = 3

√ 3

8 times the product of the side lengths of the triangle When does equalityhold?

8 Given an oriented line ∆ and a fixed point A on it, consider all trapezoids ABCD one ofwhose bases AB lies on ∆, in the positive direction Let E, F be the midpoints of AB and

CD respectively Find the loci of vertices B, C, D of trapezoids that satisfy the following:

(i) |AB| ≤ a (a fixed);

(ii) |EF | = l (l fixed);

(iii) the sum of squares of the nonparallel sides of the trapezoid is constant

[hide=”Remark”]

Remark The constants are chosen so that such trapezoids exist

9 Let ABC be an arbitrary triangle and M a point inside it Let da, db, dc be the distancesfrom M to sides BC, CA, AB; a, b, c the lengths of the sides respectively, and S the area ofthe triangle ABC Prove the inequality

abdadb+ bcdbdc+ cadcda≤ 4S

2

3 .Prove that the left-hand side attains its maximum when M is the centroid of the triangle

10 Consider two segments of length a, b (a > b) and a segment of length c =√ab

(a) For what values of a/b can these segments be sides of a triangle ?

(b) For what values of a/b is this triangle right-angled, obtuse-angled, or acute-angled ?

11 Find all solutions (x1, x2, , xn) of the equation

14 A line in the plane of a triangle ABC intersects the sides AB and AC respectively at points

X and Y such that BX = CY Find the locus of the center of the circumcircle of triangleXAY

15 Let n be a natural number Prove that

16 A polynomial p(x) = a0xk+ a1xk−1+ · · · + ak with integer coefficients is said to be divisible

by an integer m if p(x) is divisible by m for all integers x Prove that if p(x) is divisible by

m, then k!a0 is also divisible by m Also prove that if a0, k, m are non-negative integers forwhich k!a0 is divisible by m, there exists a polynomial p(x) = a0xk+ · · · + ak divisible by m

17 Given a point O and lengths x, y, z, prove that there exists an equilateral triangle ABC forwhich OA = x, OB = y, OC = z, if and only if x + y ≥ z, y + z ≥ x, z + x ≥ y (the points

20 Given n (n ≥ 3) points in space such that every three of them form a triangle with one anglegreater than or equal to 120◦, prove that these points can be denoted by A1, A2, , An insuch a way that for each i, j, k, 1 ≤ i < j < k ≤ n, angle AiAjAk is greater than or equal to

120◦

21 Let a0, a1, , ak (k ≥ 1) be positive integers Find all positive integers y such that

a0|y, (a0+ a1)|(y + a1), , (a0+ an)|(y + an)

22 Find all natural numbers n the product of whose decimal digits is n2− 10n − 22

23 Find all complex numbers m such that polynomial

x3+ y3+ z3+ mxyzcan be represented as the product of three linear trinomials

25 Given k parallel lines l1, , lk and ni points on the line li, i = 1, 2, , k, find the maximumpossible number of triangles with vertices at these points

26 Let f be a real-valued function defined for all real numbers, such that for some a > 0 we have