International Competitions IMO Longlists 1987 - Art of Problem Solving

IMO Longlists 1987
1 Let x 1 , x 2 , · · · , x n be n integers. Let n = p + q, where p and q are positive integers. For
i = 1, 2, · · · , n, put
S i = x i + x i+1 + · · · + x i+p−1 and T i = x i+p + x i+p+1 + · · · + x i+n−1
(it is assumed that x i+n = x i for all i). Next, let m(a, b) be the number of indices i for which
S i leaves the remainder a and T i leaves the remainder b on division by 3, where a, b ∈ {0, 1, 2}.
Show that m(1, 2) and m(2, 1) leave the same remainder when divided by 3.
2 Suppose we have a pack of 2n cards, in the order 1, 2, ..., 2n. A perfect shuffle of these cards
changes the order to n + 1, 1, n + 2, 2, ..., n − 1, 2n, n ; i.e., the cards originally in the first
n positions have been moved to the places 2, 4, ..., 2n, while the remaining n cards, in their
original order, fill the odd positions 1, 3, ..., 2n − 1. Suppose we start with the cards in the
above order 1, 2, ..., 2n and then successively apply perfect shuffles. What conditions on the
number n are necessary for the cards eventually to return to their original order? Justify
your answer.
[hide=”Remark”] Remark. This problem is trivial. Alternatively, it may be required to find
the least number of shuffles after which the cards will return to the original order.
3 A town has a road network that consists entirely of one-way streets that are used for bus
routes. Along these routes, bus stops have been set up. If the one-way signs permit travel
from bus stop X to bus stop Y ≠ X, then we shall say Y can be reached from X. We shall
use the phrase Y comes after X when we wish to express that every bus stop from which
the bus stop X can be reached is a bus stop from which the bus stop Y can be reached, and
every bus stop that can be reached from Y can also be reached from X. A visitor to this
town discovers that if X and Y are any two different bus stops, then the two sentences Y can
be reached from X and Y comes after X have exactly the same meaning in this town. Let A
and B be two bus stops. Show that of the following two statements, exactly one is true: (i)
B can be reached from A; (ii) A can be reached from B.
4 Let a 1 , a 2 , a 3 , b 1 , b 2 , b 3 be positive real numbers. Prove that
(a 1 b 2 + a 2 b 1 + a 1 b 3 + a 3 b 1 + a 2 b 3 + a 3 b 2 ) 2 ≥ 4(a 1 a 2 + a 2 a 3 + a 3 a 1 )(b 1 b 2 + b 2 b 3 + b 3 b 1 )
and show that the two sides of the inequality are equal if and only if a 1
b 1
= a 2
b 2
= a 3
b 3
.
5 Let there be given three circles K 1 , K 2 , K 3 with centers O 1 , O 2 , O 3 respectively, which meet
at a common point P . Also, let K 1 ∩ K 2 = {P, A}, K 2 ∩ K 3 = {P, B}, K 3 ∩ K 1 = {P, C}.
Given an arbitrary point X on K 1 , join X to A to meet K 2 again in Y , and join X to C to
meet K 3 again in Z. (a) Show that the points Z, B, Y are collinear. (b) Show that the area
of triangle XY Z is less than or equal to 4 times the area of triangle O 1 O 2 O 3 .
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IMO Longlists 1987
6 Let f be a function that satisfies the following conditions:
(i) If x > y and f(y) − y ≥ v ≥ f(x) − x, then f(z) = v + z, for some number z between
x and y. (ii) The equation f(x) = 0 has at least one solution, and among the solutions of
this equation, there is one that is not smaller than all the other solutions; (iii) f(0) = 1. (iv)
f(1987) ≤ 1988. (v) f(x)f(y) = f(xf(y) + yf(x) − xy).
Find f(1987).
Proposed by Australia.
7 Let f : (0, +∞) → R be a function having the property that f(x) = f ( 1
x)
for all x > 0. Prove
( x+
1 )
that there exists a function u : [1, +∞) → R satisfying u = f(x) for all x > 0.
8 Determine the least possible value of the natural number n such that n! ends in exactly 1987
zeros.
[hide=”Note”]Note.
positive.
Here (and generally in MathLinks) natural numbers supposed to be
9 In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y, Z} we have made a
random sequence of 28 throws. What is the probability that the sequence CUBA JULY 1987
appears in this order in the sequence already thrown?
10 In a Cartesian coordinate system, the circle C 1 has center O 1 (−2, 0) and radius 3. Denote
the point (1, 0) by A and the origin by O.Prove that there is a constant c > 0 such that for
every X that is exterior to C1,
Find the largest possible c.
OX − 1 ≥ c min{AX, AX 2 }.
11 Let S ⊂ [0, 1] be a set of 5 points with {0, 1} ⊂ S. The graph of a real function f : [0, 1] → [0, 1]
is continuous and increasing, and it is linear on every subinterval I in [0, 1] such that the
endpoints but no interior points of I are in S.
We want to compute, using a computer, the extreme values of g(x, t) = f(x+t)−f(x)
f(x)−f(x−t)
for
x − t, x + t ∈ [0, 1]. At how many points (x, t) is it necessary to compute g(x, t) with the
computer?
12 Does there exist a second-degree polynomial p(x, y) in two variables such that every nonnegative
integer n equals p(k, m) for one and only one ordered pair (k, m) of non-negative
integers?
x
2
Proposed by Finland.
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IMO Longlists 1987
13 Let A be an infinite set of positive integers such that every n ∈ A is the product of at most
1987 prime numbers. Prove that there is an infinite set B ⊂ A and a number p such that the
greatest common divisor of any two distinct numbers in B is p.
14 Given n real numbers 0 < t 1 ≤ t 2 ≤ · · · ≤ t n < 1, prove that
(
)
(1 − t 2 t 1
n)
(1 − t 2 + t 2
1 )2 (1 − t 3 + · · · + t n
2 )2 (1 − t n+1 n ) 2
< 1.
15 Let a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c 1 , c 2 , c 3 be nine strictly positive real numbers. We set
S 1 = a 1 b 2 c 3 , S 2 = a 2 b 3 c 1 , S 3 = a 3 b 1 c 2 ;
T 1 = a 1 b 3 c 2 , T 2 = a 2 b 1 c 3 , T 3 = a 3 b 2 c 1 .
Suppose that the set {S1, S2, S3, T 1, T 2, T 3} has at most two elements.
Prove that
S 1 + S 2 + S 3 = T 1 + T 2 + T 3 .
16 Let ABC be a triangle. For every point M belonging to segment BC we denote by B ′ and C ′
the orthogonal projections of M on the straight lines AC and BC. Find points M for which
the length of segment B ′ C ′ is a minimum.
17 Consider the number α obtained by writing one after another the decimal representations of
1, 1987, 1987 2 , . . . to the right the decimal point. Show that α is irrational.
18 Let ABCDEF GH be a parallelepiped with AE ‖ BF ‖ CG ‖ DH. Prove the inequality
In what cases does equality hold?
AF + AH + AC ≤ AB + AD + AE + AG.
Proposed by France.
19 How many words with n digits can be formed from the alphabet {0, 1, 2, 3, 4}, if neighboring
digits must differ by exactly one?
Proposed by Germany, FR.
20 Let x 1 , x 2 , . . . , x n be real numbers satisfying x 2 1 + x2 2 + . . . + x2 n = 1. Prove that for every
integer k ≥ 2 there are integers a 1 , a 2 , . . . , a n , not all zero, such that |a i | ≤ k − 1 for all i, and
|a 1 x 1 + a 2 x 2 + . . . + a n x n | ≤ (k−1)√ n
k n −1 . (IMO Problem 3) Proposed by Germany, FR
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IMO Longlists 1987
21 Let p n (k) be the number of permutations of the set {1, 2, 3, . . . , n} which have exactly k fixed
points. Prove that ∑ n
k=0 kp n(k) = n!.(IMO Problem 1)
Original formulation
Let S be a set of n elements. We denote the number of all permutations of S that have
exactly k fixed points by p n (k). Prove:
(a) ∑ n
k=0 kp n(k) = n! ;
(b) ∑ n
k=0 (k − 1)2 p n (k) = n!
Proposed by Germany, FR
22 Find, with proof, the point P in the interior of an acute-angled triangle ABC for which
BL 2 + CM 2 + AN 2 is a minimum, where L, M, N are the feet of the perpendiculars from P
to BC, CA, AB respectively.
Proposed by United Kingdom.
23 A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper
and lower edges are two horizontal circles. Two points are selected on the upper smaller circle
and four points on the lower larger circle. Each of these six points has three of the others that
are its nearest neighbors at a distance d from it. By distance is meant the shortest distance
measured over the curved survace of the lampshade. Prove that the area of the lampshade is
d 2 (2θ + √ 3) where cot θ 2 = 3 θ .
24 Prove that if the equation x 4 + ax 3 + bx + c = 0 has all its roots real, then ab ≤ 0.
25 Numbers d(n, m), with m, n integers, 0 ≤ m ≤ n, are defined by d(n, 0) = d(n, n) = 0 for all
n ≥ 0 and
md(n, m) = md(n − 1, m) + (2n − m)d(n − 1, m − 1) for all 0 < m < n.
Prove that all the d(n, m) are integers.
26 Prove that if x, y, z are real numbers such that x 2 + y 2 + z 2 = 2, then
x + y + z ≤ xyz + 2.
27 Find, with proof, the smallest real number C with the following property: For every infinite
sequence {x i } of positive real numbers such that x 1 + x 2 + · · · + x n ≤ x n+1 for n = 1, 2, 3, · · · ,
we have
√
x1 + √ x 2 + · · · + √ x n ≤ C √ x 1 + x 2 + · · · + x n ∀n ∈ N.
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IMO Longlists 1987
28 In a chess tournament there are n ≥ 5 players, and they have already played
(each pair have played each other at most once).
[ ]
n 2
4
+ 2 games
(a) Prove that there are five players a, b, c, d, e for which the pairs ab, ac, bc, ad, ae, de have
already played.
[ ]
(b) Is the statement also valid for the n 2
4
+ 1 games played?
Make the proof by induction over n.
29 Is it possible to put 1987 points in the Euclidean plane such that the distance between each
pair of points is irrational and each three points determine a non-degenerate triangle with
rational area? (IMO Problem 5)
Proposed by Germany, DR
30 Consider the regular 1987-gon A 1 A 2 ...A 1987 with center O. Show that the sum of vectors
belonging to any proper subset of M = {OA j |j = 1, 2, ..., 1987} is nonzero.
31 Construct a triangle ABC given its side a = BC, its circumradius R (2R ≥ a), and the
difference 1 k = 1 c − 1 b
, where c = AB and b = AC.
32 Solve the equation 28 x = 19 y + 87 z , where x, y, z are integers.
33 Show that if a, b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then
a n
b + c +
bn
c + a +
( ) cn 2 n−2
a + b ≥ S n−1 ∀n ∈ N
3
Proposed by Greece.
34 (a) Let gcd(m, k) = 1. Prove that there exist integers a 1 , a 2 , ..., a m and b 1 , b 2 , ..., b k such that
each product a i b j (i = 1, 2, · · · , m; j = 1, 2, · · · , k) gives a different residue when divided by
mk.
(b) Let gcd(m, k) > 1. Prove that for any integers a 1 , a 2 , ..., a m and b 1 , b 2 , ..., b k there must
be two products a i b j and a s b t ((i, j) ≠ (s, t)) that give the same residue when divided by mk.
Proposed by Hungary.
35 Does there exist a set M in usual Euclidean space such that for every plane λ the intersection
M ∩ λ is finite and nonempty ?
Proposed by Hungary.
[hide=”Remark”]I’m not sure I’m posting this in a right Forum.
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IMO Longlists 1987
36 A game consists in pushing a flat stone along a sequence of squares S 0 , S 1 , S 2 , ... that are
arranged in linear order. The stone is initially placed on square S 0 . When the stone stops on
a square S k it is pushed again in the same direction and so on until it reaches S 1987 or goes
beyond it; then the game stops. Each time the stone is pushed, the probability that it will
advance exactly n squares is 1
2
. Determine the probability that the stone will stop exactly
n
on square S 1987 .
37 Five distinct numbers are drawn successively and at random from the set {1, · · · , n}. Show
that the probability of a draw in which the first three numbers as well as all five numbers can
be arranged to form an arithmetic progression is greater than
6
(n−2) 3
38 Let S 1 and S 2 be two spheres with distinct radii that touch externally. The spheres lie inside
a cone C, and each sphere touches the cone in a full circle. Inside the cone there are n
additional solid spheres arranged in a ring in such a way that each solid sphere touches the
cone C, both of the spheres S 1 and S 2 externally, as well as the two neighboring solid spheres.
What are the possible values of n?
Proposed by Iceland.
39 Let A be a set of polynomials with real coefficients and let them satisfy the following conditions:
(i) if f ∈ A and deg(f) ≤ 1, then f(x) = x − 1;
(ii) if f ∈ A and deg deg(f) ≥ 2, then either there exists g ∈ A such that f(x) = x 2+deg(g) +
xg(x) − 1 or there exist g, h ∈ A such that f(x) = x 1+deg(g) g(x) + h(x);
(iii) for every f, g ∈ A, both x 2+deg(g) + xg(x) − 1 and x 1+deg(g) g(x) + h(x) belong to A.
Let R n (f) be the remainder of the Euclidean division of the polynomial f(x) by x n . Prove
that for all f ∈ A and for all natural numbers n ≥ 1 we have
R n (f)(1) ≤ 0 and R n (f)(1) = 0 =⇒ R n (f) ∈ A.
40 The perpendicular line issued from the center of the circumcircle to the bisector of angle C
in a triangle ABC divides the segment of the bisector inside ABC into two segments with
ratio of lengths . Given b = AC and a = BC, find the length of side c.
41 Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments
between pairs of these points. What is the minimum number of segments that can be colored
red in such a way that among any four points, three of them are connected by segments that
form a red triangle?
42 Find the integer solutions of the equation
[√ ] [
2m = n(2 + √ ]
2)
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IMO Longlists 1987
43 Let 2n + 3 points be given in the plane in such a way that no three lie on a line and no four
lie on a circle. Prove that the number of circles that pass through three of these points and
contain exactly n interior points is not less than 1 ( 2n+3
)
3 2 .
44 Let θ 1 , θ 2 , · · · , θ n be n real numbers such that sin θ 1 + sin θ 2 + · · · , + sin θ n = 0. Prove that
[ ] n
2
| sin θ 1 + 2 sin θ 2 + · · · , +n sin θ n | ≤
4
45 Let us consider a variable polygon with 2n sides (n ∈ N) in a fixed circle such that 2n − 1 of
its sides pass through 2n − 1 fixed points lying on a straight line ∆. Prove that the last side
also passes through a fixed point lying on ∆.
46 Given five real numbers u 0 , u 1 , u 2 , u 3 , u 4 , prove that it is always possible to find five real
numbers v0, v 1 , v 2 , v 3 , v 4 that satisfy the following conditions:
(i) u i − v i ∈ N, 0 ≤ i ≤ 4
(ii) ∑ 0≤i

IMO Longlists 1987
51 The function F is a one-to-one transformation of the plane into itself that maps rectangles
into rectangles (rectangles are closed; continuity is not assumed). Prove that F maps squares
into squares.
52 Given a nonequilateral triangle ABC, the vertices listed counterclockwise, find the locus of
the centroids of the equilateral triangles A ′ B ′ C ′ (the vertices listed counterclockwise) for
which the triples of points A, B ′ , C ′ ; A ′ , B, C ′ ; and A ′ , B ′ , C are collinear.
Proposed by Poland.
53 Prove that there exists a four-coloring of the set M = {1, 2, · · · , 1987} such that any arithmetic
progression with 10 terms in the set M is not monochromatic.
Alternative formulation
Let M = {1, 2, · · · , 1987}. Prove that there is a function f : M → {1, 2, 3, 4} that is not
constant on every set of 10 terms from M that form an arithmetic progression.
54 Let n be a natural number. Solve in integers the equation
x n + y n = (x − y) n+1 .
Proposed by Romania
55 Two moving bodies M 1 , M 2 are displaced uniformly on two coplanar straight lines. Describe
the union of all straight lines M 1 M 2 .
56 For any integer r ≥ 1, determine the smallest integer h(r) ≥ 1 such that for any partition
of the set {1, 2, · · · , h(r)} into r classes, there are integers a ≥ 0 ; 1 ≤ x ≤ y, such that
a + x, a + y, a + x + y belong to the same class.
Proposed by Romania
57 The bisectors of the angles B, C of a triangle ABC intersect the opposite sides in B ′ , C ′
respectively. Prove that the straight line B ′ C ′ intersects the inscribed circle in two different
points.
58 Find, with argument, the integer solutions of the equation
3z 2 = 2x 3 + 385x 2 + 256x − 58195.
59 It is given that a 11 , a 22 are real numbers, that x 1 , x 2 , a 12 , b 1 , b 2 are complex numbers, and
that a 11 a 22 = a 12 a 12 (Where a 12 is he conjugate of a 12 ). We consider the following system in
x 1 , x 2 :
x 1 (a 11 x 1 + a 12 x 2 ) = b 1 ,
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IMO Longlists 1987
x 2 (a 12 x 1 + a 22 x 2 ) = b 2 .
(a) Give one condition to make the system consistent.
(b) Give one condition to make arg x 1 − arg x 2 = 98 ◦ .
60 It is given that x = −2272, y = 10 3 + 10 2 c + 10b + a, and z = 1 satisfy the equation
ax + by + cz = 1, where a, b, c are positive integers with a < b < c. Find y.
61 Let P Q be a line segment of constant length λ taken on the side BC of a triangle ABC with
the order B, P, Q, C, and let the lines through P and Q parallel to the lateral sides meet AC
at P 1 and Q 1 and AB at P 2 and Q 2 respectively. Prove that the sum of the areas of the
trapezoids P QQ 1 P 1 and P QQ 2 P 2 is independent of the position of P Q on BC.
62 Let l, l ′ be two lines in 3-space and let A, B, C be three points taken on l with B as midpoint
of the √ segment AC. If a, b, c are the distances of A, B, C from l ′ , respectively, show that
a
b ≤
2 +c 2
2
, equality holding if l, l ′ are parallel.
63 Compute ∑ 2n
k=0 (−1)k a 2 k where a k are the coefficients in the expansion
(1 − √ 2x + x 2 ) n =
2n∑
k=0
a k x k .
64 Let r > 1 be a real number, and let n be the largest integer smaller than r. Consider
an arbitrary real number x with 0 ≤ x ≤
n
r−1
. By a base-r expansion of x we mean a
representation of x in the form
where the a i are integers with 0 ≤ a i < r.
x = a 1
r + a 2
r 2 + a 3
r 3 + · · ·
You may assume without proof that every number x with 0 ≤ x ≤
expansion.
Prove that if r is not an integer, then there exists a number p, 0 ≤ p ≤
infinitely many distinct base-r expansions.
n
r−1
has at least one base-r
n
r−1
, which has
65 The runs of a decimal number are its increasing or decreasing blocks of digits. Thus 024379
has three runs : 024, 43, and 379. Determine the average number of runs for a decimal number
in the set {d 1 d 2 · · · d n |d k ≠ d k+1 , k = 1, 2, · · · , n − 1}, where n ≥ 2.
66 At a party attended by n married couples, each person talks to everyone else at the party
except his or her spouse. The conversations involve sets of persons or cliques C 1 , C 2 , · · · , C k
with the following property: no couple are members of the same clique, but for every other
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IMO Longlists 1987
pair of persons there is exactly one clique to which both members belong. Prove that if n ≥ 4,
then k ≥ 2n.
Proposed by USA.
67 If a, b, c, d are real numbers such that a 2 +b 2 +c 2 +d 2 ≤ 1, find the maximum of the expression
(a + b) 4 + (a + c) 4 + (a + d) 4 + (b + c) 4 + (b + d) 4 + (c + d) 4 .
68 Let α, β, γ be positive real numbers such that α + β + γ < π, α + β > γ,β + γ > α, γ + α > β.
Prove that with the segments of lengths sin α, sin β, sin γ we can construct a triangle and that
its area is not greater than
A = 1 (sin 2α + sin 2β + sin 2γ) .
8
Proposed
by Soviet Union
69 Let n ≥ 2 be an integer. Prove that if k 2 + k + n is prime for all integers k such that
0 ≤ k ≤ √ n
3 , then k2 +k +n is prime for all integers k such that 0 ≤ k ≤ n−2.(IMO Problem
6)
Original Formulation
Let f(x) = x 2 + x + p, p ∈ N. Prove that if the numbers f(0), f(1), · · · , f(
√
p
3
) are primes, then all the numbers f(0), f(1), · · · , f(p − 2) are primes.
Proposed by Soviet Union.
70 In an acute-angled triangle ABC the interior bisector of angle A meets BC at L and meets the
circumcircle of ABC again at N. From L perpendiculars are drawn to AB and AC, with feet
K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal
areas.(IMO Problem 2)
Proposed by Soviet Union.
71 To every natural number k, k ≥ 2, there corresponds a sequence a n (k) according to the following
rule:
a 0 = k, a n = τ(a n−1 ) ∀n ≥ 1,
in which τ(a) is the number of different divisors of a. Find all k for which the sequence a n (k) does
not contain the square of an integer.
72 Is it possible to cover a rectangle of dimensions m × n with bricks that have the trimino angular
shape (an arrangement of three unit squares forming the letter L) if:
(a) m × n = 1985 × 1987;
(b) m × n = 1987 × 1989 ?
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IMO Longlists 1987
73 Let f(x) be a periodic function of period T > 0 defined over R. Its first derivative is continuous on
R. Prove that there exist x, y ∈ [0, T ) such that x ≠ y and
f(x)f ′ (y) = f ′ (x)f(y).
74 Does there exist a function f : N → N, such that f(f(n)) = n + 1987 for every natural number n?
(IMO Problem 4)
Proposed by Vietnam.
75 Let a k be positive numbers such that a 1 ≥ 1 and a k+1 − a k ≥ 1 (k = 1, 2, ...). Prove that for every
n ∈ N,
1987
∑
k=1
1
a k+1
1987 √ a k
< 1987
76 Given two sequences of positive numbers {a k } and {b k } (k ∈ N) such that:
(i) a k < b k ,
(ii) cos a k x + cos b k x ≥ − 1 k
prove the existence of lim k→∞
a k
b k
for all k ∈ N and x ∈ R,
and find this limit.
77 Find the least positive integer k such that for any a ∈ [0, 1] and any positive integer n,
a k (1 − a) n <
1
(n + 1) 3 .
78 Prove that for every natural number k (k ≥ 2) there exists an irrational number r such that for
every natural number m,
[r m ] ≡ −1 (mod k).
Remark. An easier variant: Find r as a root of a polynomial of second degree with integer coefficients.
Proposed by Yugoslavia.
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