1. Show that among any four points contained in a unit circle, there exist two whose distance is at most√
2.
4. Let C be a circle touching the edges of an angle ∠XOY , and let C1 be the circle touching the same edges and passing through the center of C. Let A be the second endpoint of the diameter of C1
passing through the center of C, and let B be the intersection of this diameter with C. Prove that the circle centered at A passing through B touches the edges of ∠XOY .
5. Find all integers x, y, z satisfying x2+ y2+ z2− 2xyz = 0.
6. Find the smallest integer k such that there exist two sequences{ai}, {bi} (i = 1, . . . , k) such that of An, and find the sum of all products of two distinct elements of An.
8. In an acute triangle ABC with AB6= AC, let V be the intersection of the angle bisector of A with BC, and let D be the foot of the perpendicular from A to BC. If E and F are the intersections of the circumcircle of AV D with CA and AB, respectively, show that the lines AD, BE, CF concur.
9. A word is a sequence of 8 digits, each equal to 0 or 1. Let x and y be two words differing in exactly three places. Show that the number of words differing from each of x and y in at least five places is 188.
10. Find all pairs of functions f, g : R→ R such that (a) if x < y, then f (x) < f (y);
(b) for all x, y∈ R, f(xy) = g(y)f(x) + f(y).
11. Let a1, . . . , an be positive numbers, and define A = a1+· · · + an
n G = (a1· · · an)1/n
H = n
a−11 +· · · + a−1n
.
(a) If n is even, show that HA ≤ −1 + GAn .
(b) If n is odd, show that HA ≤ −n−2n +2(nn−1) AGn
.
12. Let p1, . . . , pr be distinct primes, and let n1, . . . , nr be arbitrary natural numbers. Prove that the number of pairs (x, y) of integers satisfying x3+ y3= pn11· · · pnrr is at most 2r−1.
3.16 Poland
1. The positive integers x1, . . . , x7 satisfy the conditions
x6= 144, xn+3= xn+2(xn+1+ xn) n = 1, 2, 3, 4.
Compute x7.
2. Solve the following system of equations in real numbers x, y, z:
3(x2+ y2+ z2) = 1
x2y2+ y2z2+ z2x2 = xyz(x + y + z)3.
3. In a tetrahedron ABCD, the medians of the faces ABD, ACD, BCD from D make equal angles with the corresponding edges AB, AC, BC.
Prove that each of these faces has area less than the sum of the areas of the other two faces.
4. The sequence a1, a2, . . . is defined by
a1= 0, an= abn/2c+ (−1)n(n+1)/2 n > 1.
For every integer k≥ 0, find the number of n such that 2k≤ n < 2k+1 and an= 0.
5. Given a convex pentagon ABCDE with DC = DE and ∠BCD =
∠DEA = π/2, let F be the point on segment AB such that AF/BF = AE/BC. Show that
∠F CE = ∠F DE and ∠F EC = ∠BDC.
6. Consider n points (n≥ 2) on a unit circle. Show that at most n2/3 of the segments with endpoints among the n chosen points have length greater than√
2.
3.17 Romania
1. In the plane are given a line ∆ and three circles tangent to ∆ and externally tangent to each other. Prove that the triangle formed by the centers of the circles is obtuse, and find all possible measures of the obtuse angle.
2. Determine all sets A of nine positive integers such that for any n≥ 500, there exists a subset B of A, the sum of whose elements is n.
3. Let n≥ 4 be an integer and M a set of n points in the plane, no three collinear and not all lying on a circle. Find all functions f : M → R such that for any circle C containing at least three points of M ,
X
P∈M∩C
f (P ) = 0.
4. Let ABC be a triangle, D a point on side BC and ω the circum-circle of ABC. Show that the circum-circles tangent to ω, AD, BD and to ω, AD, DC, respectively, are tangent to each other if and only if
∠BAD = ∠CAD.
5. Let V A1· · · An be a pyramid with n≥ 4. A plane Π intersects the edges V A1, . . . , V An at B1, . . . , Bn, respectively. Suppose that the polygons A1· · · Anand B1· · · Bnare similar. Prove that Π is parallel to the base of the pyramid.
6. Let A be the set of positive integers representable in the form a2+2b2 for integers a, b with b6= 0. Show that if p2∈ A for a prime p, then p∈ A.
7. Let p≥ 5 be a prime and choose k ∈ {0, . . . , p − 1}. Find the max-imum length of an arithmetic progression, none of whose elements contain the digit k when written in base p.
8. Let p, q, r be distinct prime numbers and let A be the set A ={paqbrc: 0≤ a, b, c ≤ 5}.
Find the smallest integer n such that any n-element subset of A contains two distinct elements x, y such that x divides y.
9. Let ABCDEF be a convex hexagon. Let P, Q, R be the intersections of the lines AB and EF , EF and CD, CD and AB, respectively.
Let S, T, U be the intersections of the lines BC and DE, DE and F A, F A and BC, respectively. Show that if AB/P R = CD/RQ = EF/QP , then BC/U S = DE/ST = F A/T U .
10. Let P be the set of points in the plane and D the set of lines ithe plane. Determine whether there exists a bijective function f : P → D such that for any three collinear points A, B, C, the lines f (A), f (B), f (C) are either parallel or concurrent.
11. Find all functions f : R→ [0, ∞) such that for all x, y ∈ R, f (x2+ y2) = f (x2− y2) + f (2xy).
12. Let n≥ 2 be an integer and P (x) = xn+ an−1xn−1+· · · + a1x + 1 be a polynomial with positive integer coefficients. Suppose that ak = an−k for k = 1, 2, . . . , n− 1. Prove that there exist infinitely many pairs x, y of positive integers such that x|P (y) and y|P (x).
13. Let P (x), Q(x) be monic irreducible polynomials over the rational numbers. Suppose P and Q have respective roots α and β such that α + β is rational. Prove that the polynomial P (x)2− Q(x)2 has a rational root.
14. Let a > 1 be an integer. Show that the set {a2+ a− 1, a3+ a2− 1, . . .}
contains an infinite subset, any two members of which are relatively prime.
15. Find the number of ways to color the vertices of a regular dodecagon in two colors so that no set of vertices of a single color form a regular polygon.
16. Let Γ be a circle and AB a line not meeting Γ. For any point P ∈ Γ, let P0 be the second intersection of the line AP with Γ and let f (P ) be the second intersection of the line BP0with Γ. Given a point P0, define the sequence Pn+1= f (Pn) for n≥ 0. Show that if a positive integer k satisfies P0 = Pk for a single choice of P0, then P0 = Pk
for all choices of P0.
3.18 Russia
1. Show that the numbers from 1 to 16 can be written in a line, but not in a circle, so that the sum of any two adjacent numbers is a perfect square.
2. On equal sides AB and BC of an equilateral triangle ABC are chosen points D and K, and on side AC are chosen points E and M , so that DA + AE = KC + CM = AB. Show that the angle between the lines DM and KE equals π/3.
3. A company has 50000 employees. For each employee, the sum of the numbers of his immediate superiors and of his immediate inferiors is 7. On Monday, each worker issues an order and gives copies of it to each of his immediate inferiors (if he has any). Each day thereafter, each worker takes all of the orders he received on the previous day and either gives copies of them to all of his immediate inferiors if he has any, or otherwise carries them out himself. It turns out that on Friday, no orders are given. Show that there are at least 97 employees who have no immediate superiors.
4. The sides of the acute triangle ABC are diagonals of the squares K1, K2, K3. Prove that the area of ABC is covered by the three squares.
5. The numbers from 1 to 37 are written in a line so that each number divides the sum of the previous numbers. If the first number is 37 and the second number is 1, what is the third number?
6. Find all paris of prime numbers p, q such that p3− q5= (p + q)2. 7. (a) In Mexico City, to restrict traffic flow, for each private car are
designated two days of the week on which that car cannot be driven on the streets of the city. A family needs to have use of at least 10 cars each day. What is the smallest number of cars they must possess, if they may choose the restricted days for each car?
(b) The law is changed to restrict each car only one day per week, but the police get to choose the days. The family bribes the police so that for each car, they will restrict one of two days chosen by the family. Now what is the smallest number of cars the family needs to have access to 10 cars each day?
8. A regular 1997-gon is divided by nonintersecting diagonals into tri-angles. Prove that at least one of the triangles is acute.
9. On a chalkboard are written the numbers from 1 to 1000. Two players take turns erasing a number from the board. The game ends when two numbers remain: the first player wins if the sum of these numbers is divisible by 3, the second player wins otherwise. Which player has a winning strategy?
10. 300 apples are given, no one of which weighs more than 3 times any other. Show that the apples may be divided into groups of 4 such that no group weighs more than 11/2 times any other group.
11. In Robotland, a finite number of (finite) sequences of digits are for-bidden. It is known that there exists an infinite decimal fraction, not containing any forbidden sequences. Show that there exists an infinite periodic decimal fraction, not containing any forbidden se-quences.
12. (a) A collection of 1997 numbers has the property that if each num-ber is subtracted from the sum of the remaining numnum-bers, the same collection of numbers is obtained. Prove that the product of the numbers is 0.
(b) A collection of 100 numbers has the same property. Prove that the product of the numbers is positive.
13. Given triangle ABC, let A1, B1, C1 be the midpoints of the broken lines CAB, ABC, BCA, respectively. Let lA, lB, lC be the respective lines through A1, B1, C1 parallel to the angle bisectors of A, B, C.
Show that lA, lB, lC are concurrent.
14. The MK-97 calculator can perform the following three operations on numbers in its memory:
(a) Determine whether two chosen numbers are equal.
(b) Add two chosen numbers together.
(c) For chosen numbers a and b, find the real roots of x2+ ax + b, or announce that no real roots exist.
The results of each operation are accumulated in memory. Initially the memory contains a single number x. How can one determine, using the MK-97, whether x is equal to 1?
15. The circles S1 and S2 intersect at M and N . Show that if vertices A and C of a rectangle ABCD lie on S1 while vertices B and D lie on S2, then the intersection of the diagonals of the rectangle lies on the line M N .
16. For natural numbers m, n, show that 2n− 1 divides (2m− 1)2if and only if n divides m(2m− 1).
17. Can three faces of a cube of side length 4 be covered with 16 1× 3 rectangles?
18. The vertices of triangle ABC lie inside a square K. Show that if the triangle is rotated 180◦ about its centroids, at least one vertex remains inside the square.
19. Let S(N ) denote the sum of the digits of the natural number N . Show that there exist infinitely many natural numbers n such that S(3n)≥ S(3n+1).
20. The members of Congress form various overlapping factions such that given any two (not necessarily distinct) factions A and B, the complement of A∪B is also a faction. Show that for any two factions A and B, A∪ B is also a faction.
21. Show that if 1 < a < b < c, then
loga(logab) + logb(logbc) + logc(logca) > 0.
22. Do there exist pyramids, one with a triangular base and one with a convex n-sided base (n ≥ 4), such that the solid angles of the triangular pyramid are congruent to four of the solid angles of the n-sided pyramid?
23. For which α does there exist a nonconstant function f : R→ R such that
f (α(x + y)) = f (x) + f (y)?
24. Let P (x) be a quadratic polynomial with nonnegative coefficients.
Show that for any real numbers x and y, we have the inequality P (xy)2≤ P (x2)P (y2).
25. Given a convex polygon M invariant under a 90◦rotation, show that there exist two circles, the ratio of whose radii is√
2, one containing M and the other contained in M .
26. (a) The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage’s head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat.
The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case?
(b) The king decides to use three colors of hats: white, black and red. Now what is the smallest number of sages guaranteed to survive?
27. The lateral sides of a box with base a×b and height c (where a, b, c are natural numbers) are completely covered without overlap by rectan-gles whose edges are parallel to the edges of the box, each containing an odd number of unit squares. Prove that if c is odd, then the num-ber of rectangles covering lateral edges of the box is even.
28. Do there exist real numbers b and c such that each of the equations x2+ bx + c = 0 and 2x2+ (b + 1)x + c + 1 = 0 have two integer roots?
29. A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name.
30. The incircle of triangle ABC touches sides AB, BC, CA at M, N, K, respectively. The line through A parallel to N K meets M N at D.
The line through A parallel to M N meets N K at E. Show that the line DE bisects sides AB and AC of triangle ABC.
31. The numbers from 1 to 100 are arranged in a 10× 10 table so that no two adjacent numbers sum to S. Find the smallest value of S for which this is possible.
32. Find all integer solutions of the equation (x2− y2)2= 1 + 16y.
33. An n× n square grid (n ≥ 3) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells.
34. Two circles intersect at A and B. A line through A meets the first circle again at C and the second circle again at D. Let M and N be the midpoints of the arcs BC and BD not containing A, and let K be the midpoint of the segment CD. Show that ∠M KN = π/2.
(You may assume that C and D lie on opposite sides of A.) 35. A polygon can be divided into 100 rectangles, but not into 99. Prove
that it cannot be divided into 100 triangles.
36. Do there exist two quadratic trinomials ax2+ bx + c and (a + 1)x2+ (b + 1)x + (c + 1) with integer coefficients, both of which have two integer roots?
37. A circle centered at O and inscribed in triangle ABC meets sides AC, AB, BC at K, M, N , respectively. The median BB1 of the tri-angle meets M N at D. Show that O, D, K are collinear.
38. Find all triples m, n, l of natural numbers such that
m + n = gcd(m, n)2, m + l = gcd(m, l)2, n + l = gcd(n, l)2. 39. On an infinite (in both directions) strip of squares, indexed by the
natural numbers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types:
(a) Remove one stone from each of the squares n− 1 and n and place one stone on square n + 1.
(b) Remove two stones from square n and place one stone on each of the squares n + 1, n− 2.
Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves.
40. An n× n × n cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished.
Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color.
41. Of the quadratic trinomials x2+ px + q where p, q are integers and 1≤ p, q ≤ 1997, which are there more of: those having integer roots or those not having real roots?
42. We are given a polygon, a line l and a point P on l in general position:
all lines containing a side of the polygon meet l at distinct points differing from P . We mark each vertex of the polygon whose sides both meet the line l at points differing from P . Show that P lies inside the polygon if and only if for each choice of l there are an odd number of marked vertices.
43. A sphere inscribed in a tetrahedron touches one face at the inter-section of its angle bisectors, a second face at the interinter-section of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular.
44. In an m× n rectangular grid, where m and n are odd integers, 1 × 2 dominoes are initially placed so as to exactly cover all but one of the 1× 1 squares at one corner of the grid. It is permitted to slide a domino towards the empty square, thus exposing another square.
Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle.