ILL 1987

Let $x_1, x_2,\cdots, x_n$ be $n$ integers. Let $n = p + q$ , where $p$ and $q$ are positive integers. For $i = 1, 2, \cdots, n$ , put $$S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}$$ (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 $a, b \in \{0, 1, 2\}$ . Show that $m(1, 2)$ and $m(2, 1)$ leave the same remainder when divided by $3.$

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 $Y \neq X$ , 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 $B.$

Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that $$(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)$$ and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$

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 $K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}$ . 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 $O_1O_2O_3.$

Let f be a function that satisfies the following conditions:

$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, 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; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$.

Find $f(1987)$.

Proposed by Australia.

Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$

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?

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 $C1$ , $$OX- 1 \geq c \min\{AX,AX^2\}.$$ Find the largest possible $c.$

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$ . The graph of a real function $f : [0, 1] \to [0, 1]$ 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 $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$ . At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $n$ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers?

Proposed by Finland.

Let $A$ be an infinite set of positive integers such that every $n \in A$ is the product of at most $1987$ prime numbers. Prove that there is an infinite set $B \subset A$ and a number $p$ such that the greatest common divisor of any two distinct numbers in $B$ is $p.$

Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$ , prove that $$(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.$$

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 $$S_1 = a_1b_2c_3, S_2 = a_2b_3c_1, S_3 = a_3b_1c_2;$$$$T_1 = a_1b_3c_2, T_2 = a_2b_1c_3, T_3 = a_3b_2c_1.$$ Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.

Prove that $$S_1 + S_2 + S_3 = T_1 + T_2 + T_3.$$

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.

Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality $$AF + AH + AC \leq AB + AD + AE + AG.$$

In what cases does equality hold?

Proposed by France.

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.

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$ . Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$ , not all zero, such that $|a_i|\le k-1$ for all $i$ , and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$ .

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.

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 $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$

Prove that if the equation $x^4 + ax^3 + bx + c = 0$ has all its roots real, then $ab \leq 0.$

Numbers $d(n,m)$ , with $m, n$ integers, $0 \leq m \leq n$ , are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and $$md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.$$ Prove that all the $d(n,m)$ are integers.

Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$ , then $$x + y + z \leq xyz + 2.$$

Find, with proof, the smallest real number $C$ with the following property: For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$ , we have $$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \forall n \in \mathbb N.$$

In a chess tournament there are $n \geq 5$ players, and they have already played $\left[ \frac{n^2}{4} \right] +2$ games (each pair have played each other at most once).(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 $\left[ \frac{n^2}{4} \right] +1$ games played?

Make the proof by induction over $n.$

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

Consider the regular $1987$ -gon $A_1A_2 . . . A_{1987}$ with center $O$ . Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.

Construct a triangle $ABC$ given its side $a = BC$ , its circumradius $R \ (2R \geq a)$ , and the difference $\frac{1}{k} = \frac{1}{c}-\frac{1}{b}$ , where $c = AB$ and $b = AC.$

Solve the equation $28^x = 19^y +87^z$ , where $x, y, z$ are integers.

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then $$\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \forall n \in \mathbb N$$

Proposed by Greece.

(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 $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) 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 $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$

Proposed by Hungary.

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 $\frac{1}{2^n}$ . Determine the probability that the stone will stop exactly on square $S_{1987}.$

Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$ . 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 $\frac{6}{(n-2)^3}$

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.

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions:(i) if $f \in A$ and $\deg( f ) \leq 1$ , then $f(x) = x - 1$ ;(ii) if $f \in A$ and $\deg( f ) \geq 2$ , then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$ ;(iii) for every $g, h \in A$ , 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 $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$ , and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$ .

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 $\lambda$ . Given $b = AC$ and $a = BC$ , find the length of side $c.$

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?

Find the integer solutions of the equation $$\left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right]$$

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 $\frac 13 \binom{2n+3}{2}.$

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$ . Prove that $$|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]$$

Let us consider a variable polygon with $2n$ sides ( $n \in N$ ) in a fixed circle such that $2n - 1$ of its sides pass through $2n - 1$ fixed points lying on a straight line $\Delta$ . Prove that the last side also passes through a fixed point lying on $\Delta .$

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:

$(i)$ $u_i-v_i \in \mathbb N, 0 \leq i \leq 4$

$(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$

Proposed by Netherlands.

Through a point $P$ within a triangle $ABC$ the lines $l, m$ , and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$ , $m$ intersects $AC$ in $R$ , and $n$ intersects $AB$ in $S$ , then the points $Q, R$ , and $S$ are collinear.

Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:

$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.

$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .

Proposed by Poland.

In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil$ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $\lceil d \rceil$ is the least integer not less than $d.$

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.

Attachments:

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.

Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.

Alternative formulation

Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.

Proposed by Romania

Let $n$ be a natural number. Solve in integers the equation $$x^n + y^n = (x - y)^{n+1}.$$

Attachments:

Two moving bodies $M_1,M_2$ are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines $M_1M_2.$ Attachments:

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.

Proposed by Romania

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.

Attachments:

Find, with argument, the integer solutions of the equation $$3z^2 = 2x^3 + 385x^2 + 256x - 58195.$$

Attachments:

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 $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$ ). We consider the following system in $x_1, x_2$ : $$\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,$$$$\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.$$**(a)Give one condition to make the system consistent.(b)**Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$ Attachments:

It is given that $x = -2272$ , $y = 10^3+10^2c+10b+a$ , and $z = 1$ satisfy the equation $ax + by + cz = 1$ , where $a, b, c$ are positive integers with $a < b < c$ . Find $y.$ Attachments:

Let $PQ$ be a line segment of constant length $\lambda$ 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 $PQQ_1P_1$ and $PQQ_2P_2$ is independent of the position of $PQ$ on $BC.$ Attachments:

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 $b \leq \sqrt{ \frac{a^2+c^2}{2}}$ , equality holding if $l, l'$ are parallel.

Attachments:

Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion $$(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.$$

Attachments:

Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$ . Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a base- $r$ expansion of $x$ we mean a representation of $x$ in the form $$x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots$$ where the $a_i$ are integers with $0 \leq a_i < r.$ You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one base- $r$ expansion.

Prove that if $r$ is not an integer, then there exists a number $p$ , $0 \leq p \leq \frac{n}{r-1}$ , which has infinitely many distinct base- $r$ expansions.

Attachments:

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 $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$ , where $n \geq 2.$ Attachments:

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 $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$.

Proposed by USA.

If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$ , find the maximum of the expression $$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.$$

Attachments:

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$, $\beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma$ we can construct a triangle and that its area is not greater than $$A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).$$

Proposed by Soviet Union

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$. (IMO Problem 6)

Original Formulation

Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.

Proposed by Soviet Union.

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.

To every natural number $k, k \geq 2$ , there corresponds a sequence $a_n(k)$ according to the following rule: $$a_0 = k, a_n = \tau(a_{n-1}) \forall n \geq 1,$$ in which $\tau(a)$ 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.

Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$ ) if:(a) $m \times n = 1985 \times 1987;$ (b) $m \times n = 1987 \times 1989 ?$ Attachments:

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$ . Its first derivative is continuous on $\mathbb R$ . Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and $$f(x)f'(y)=f'(x)f(y).$$

Attachments:

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? (IMO Problem 4)

Proposed by Vietnam.

Let $a_k$ be positive numbers such that $a_1 \geq 1$ and $a_{k+1} -a_k \geq 1 \ (k = 1, 2, . . . )$ . Prove that for every $n \in \mathbb N,$ $$\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987$$

Attachments:

Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:(i) $a_k < b_k,$ (ii) $\cos a_kx + \cos b_kx \geq -\frac 1k$ for all $k \in \mathbb N$ and $x \in \mathbb R,$ prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit.

Attachments:

Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ $$a^k(1 - a)^n < \frac{1}{(n+1)^3}.$$

Attachments:

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, $$[r^m] \equiv -1 \pmod k .$$

Remark. An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.

Proposed by Yugoslavia.