Navvye/Olympiad-Problems-CSV huggingface dataset:
1987 IMO Longlists
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

$$S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;$$$$T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.$$

Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.



Prove that

$$S_1 + S_2 + S_3 = T_1 + T_2 + T_3.$$
1987 IMO Longlists
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.
1987 IMO Longlists
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.
1987 IMO Longlists
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 $\cot \frac {\theta}{2} = \frac{3}{\theta}.$
1987 IMO Longlists
Prove that if the equation $x^4 + ax^3 + bx + c = 0$ has all its roots real, then $ab \leq 0.$
1987 IMO Longlists
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.
1987 IMO Longlists
Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then

$$x + y + z \leq xyz + 2.$$
1987 IMO Longlists
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

$$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}  \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.$$
1987 IMO Longlists
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 $n.$
1987 IMO Longlists
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.
1987 IMO Longlists
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.$
1987 IMO Longlists
Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.
1987 IMO Longlists
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 $S_{1987}.$
1987 IMO Longlists
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 $\frac{6}{(n-2)^3}$
1987 IMO Longlists
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 $R_n(f) \in A$.
1987 IMO Longlists
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 $c.$
1987 IMO Longlists
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?
1987 IMO Longlists
Find the integer solutions of the equation

$$ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] $$
1987 IMO Longlists
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 $\frac 13 \binom{2n+3}{2}.$
1987 IMO Longlists
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

$$|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]$$
1987 IMO Longlists
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 $\Delta .$
1987 IMO Longlists
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.
1987 IMO Longlists
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 $d.$
1987 IMO Longlists
Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$.



VariantsVariants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$
1987 IMO Longlists
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.
1987 IMO Longlists
Let $n$ be a natural number. Solve in integers the equation

$$x^n + y^n = (x - y)^{n+1}.$$
1987 IMO Longlists
Two moving bodies $M_1,M_2$ are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines $M_1M_2.$
1987 IMO Longlists
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.
1987 IMO Longlists
Find, with argument, the integer solutions of the equation

$$3z^2 = 2x^3 + 385x^2 + 256x - 58195.$$
1987 IMO Longlists
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 $\arg x_1 - \arg x_2 = 98^{\circ}.$
1987 IMO Longlists
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 $y.$
1987 IMO Longlists
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 $BC.$
1987 IMO Longlists
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.
1987 IMO Longlists
Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion

$$(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.$$
1987 IMO Longlists
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.
1987 IMO Longlists
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 $n \geq 2.$
1987 IMO Longlists
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

$$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.$$
1987 IMO Longlists
To every natural number $k, k \geq 2$, there corresponds a sequence $a_n(k)$ according to the following rule:

$$a_0 = k, \qquad a_n = \tau(a_{n-1})  \quad \forall n \geq 1,$$

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.
1987 IMO Longlists
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) $m \times n = 1987 \times 1989 \quad ?$
1987 IMO Longlists
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

$$f(x)f'(y)=f'(x)f(y).$$
1987 IMO Longlists
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 $n \in \mathbb N,$

$$\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987$$
1987 IMO Longlists
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.
1987 IMO Longlists
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$

$$a^k(1 - a)^n < \frac{1}{(n+1)^3}.$$
1986 IMO Longlists
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality

$$1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k$$

for all real $b \geq 0.$
1986 IMO Longlists
Let $ABCD$ be a convex quadrilateral. $DA$ and $CB$ meet at $F$ and $AB$ and $DC$ meet at $E$. The bisectors of the angles $DFC$ and $AED$ are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines $AC$ and $BD.$
1986 IMO Longlists
A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$. Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$
1986 IMO Longlists
Find the last eight digits of the binary development of $27^{1986}.$
1986 IMO Longlists
Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.
1986 IMO Longlists
In an urn there are one ball marked $1$, two balls marked $2$, and so on, up to $n$ balls marked $n$. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.
1986 IMO Longlists
In a triangle $ABC$, $\angle BAC = 100^{\circ}, AB = AC$. A point $D$ is chosen on the side $AC$ such that $\angle ABD = \angle CBD$. Prove that $AD + DB = BC.$
1986 IMO Longlists
A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\frac{P}{6^{s+2}}$. Find an explicit expression for $P$.
1986 IMO Longlists
Let $O$ be an interior point of a tetrahedron $A_1A_2A_3A_4$. Let $ S_1, S_2, S_3, S_4$ be spheres with centers $A_1,A_2,A_3,A_4$, respectively, and let $U, V$ be spheres with centers at $O$. Suppose that for $i, j = 1, 2, 3, 4, i \neq j$, the spheres $S_i$ and $S_j$ are tangent to each other at a point $B_{ij}$ lying on $A_iA_j$ . Suppose also that $U $ is tangent to all edges $A_iA_j$ and $V$ is tangent to the spheres $ S_1, S_2, S_3, S_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.
1986 IMO Longlists
Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$.

Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.
1986 IMO Longlists
Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb  N$ of cardinality $l$, an infinite set $T \subset \mathbb  N$, and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$, the sum  $t+ \sum_{y \in Y} y$ belongs to $B_k.$
1986 IMO Longlists
Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties:

$$|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;$$

$$|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 $$
1986 IMO Longlists
We call a tetrahedron right-faced if each of its faces is a right-angled triangle.



(a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra.



(b) Prove that a tetrahedron with vertices $A_1,A_2,A_3,A_4$ is right-faced if and only if there exist four distinct real numbers $c_1, c_2, c_3$, and $c_4$ such that the edges $A_jA_k$ have lengths $A_jA_k=\sqrt{|c_j-c_k|}$ for $1\leq j < k \leq 4.$
1986 IMO Longlists
Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and

$$f(x + y) - f(x) = f(x) - f(x - y)$$

for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$
1986 IMO Longlists
For any angle α with $0 < \alpha < 180^{\circ}$, we call a closed convex planar set an $\alpha$-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$. Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$-set $S \subset T .$
1986 IMO Longlists
Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$
1986 IMO Longlists
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$
1986 IMO Longlists
Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$. For any point $M$ in the plane of the triangle, not on the line $BC$, denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$) of the triangle $BCM$. Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.
1986 IMO Longlists
Two families of parallel lines are given in the plane, consisting of $15$ and $11$ lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let $V$ be the set of $165$ intersection points of the lines under consideration. Show that there exist not fewer than $1986$ distinct squares with vertices in the set $V .$
1986 IMO Longlists
In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum?

A term in a sequence is a local maximum if it is greater than all its neighbors.
1986 IMO Longlists
We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M  \star O) = (O \star  I)\star M$ holds?
1986 IMO Longlists
Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.
1986 IMO Longlists
Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.
1986 IMO Longlists
Find, with proof, all solutions of the equation $\frac 1x +\frac 2y- \frac 3z = 1$ in positive integers $x, y, z.$
1986 IMO Longlists
For each non-negative integer $n$, $F_n(x)$ is a polynomial in $x$ of degree $n$. Prove that if the identity

$$F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)$$

holds for each n, then

$$F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)$$
1986 IMO Longlists
Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$
1986 IMO Longlists
Prove that the set $\{1, 2, . . . , 1986\}$ can be partitioned into $27$ disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).
1986 IMO Longlists
Let $S$ be a $k$-element set.



(a) Find the number of mappings $f : S \to S$ such that

$$\text{(i)  } f(x) \neq x \text{ for } x \in S, \quad  \text{(ii)  } f(f(x)) = x \text{ for }x \in S.$$



(b) The same with the condition $\text{(i)}$ left out.
1986 IMO Longlists
Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i \  (1 \leq i \leq m), y_j \  (1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$'s are odd, and $\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.$
1986 IMO Longlists
Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$
1986 IMO Longlists
The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?
1986 IMO Longlists
Given $n$ real numbers $a_1 \leq a_2 \leq \cdots \leq a_n$, define

$$M_1=\frac 1n \sum_{i=1}^{n} a_i , \quad M_2=\frac{2}{n(n-1)} \sum_{1 \leq i<j \leq n} a_ia_j, \quad Q=\sqrt{M_1^2-M_2}$$

Prove that

$$a_1 \leq  M_1 - Q \leq  M_1 + Q \leq a_n$$

and that equality holds if and only if $a_1 = a_2 = \cdots = a_n.$
1986 IMO Longlists
We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \  (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that:



(i) in each column there are exactly $k$ terms equal to $0$;



(ii) for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$



For what values of $k$ is this possible?
1986 IMO Longlists
Let $P$ be a convex $1986$-gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$. Starting with an arbitrary point $Q_1$ on the boundary of $P$, define recursively a sequence of points $Q_n$ as follows:

given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$. Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$.
1986 IMO Longlists
Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows:

$C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.
1986 IMO Longlists
Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$. Let $I$ be the incenter of$ ABC.$



(a) Construct the points $P$ and $Q$ on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$, where $k$ is a given rational number.



(b) Let $R$ be the intersection point of $PQ$ and $AD$. For what value of $k$ does the equality $AR = RI$ hold?



(c) In which case do the equalities $AR = RI = ID$ hold?
1986 IMO Longlists
Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.
1986 IMO Longlists
Solve the system of equations

$$\tan x_1 +\cot x_1=3 \tan x_2,$$$$\tan x_2 +\cot x_2=3 \tan x_3,$$$$\vdots$$$$\tan x_n +\cot x_n=3 \tan x_1$$
1986 IMO Longlists
For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$-tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$. Prove that

$$S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}$$

Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$