Navvye/Olympiad-Problems-CSV huggingface dataset:
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then

$$\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N $$



Proposed by Greece.
1987 IMO Shortlist
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:



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



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



Proposed by Netherlands.
1987 IMO Shortlist
(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.
1987 IMO Shortlist
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?



Proposed by Hungary.



RemarkI'm not sure I'm posting this in a right Forum.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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
1987 IMO Shortlist
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.
1987 IMO Shortlist
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 $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. (IMO Problem 3)



Proposed by Germany, FR
1987 IMO Shortlist
Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.(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) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$



(b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $



Proposed by Germany, FR
1987 IMO Shortlist
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
1987 IMO Shortlist
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
1987 IMO Shortlist
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
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1987 IMO Shortlist
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.
1986 IMO Shortlist
Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ?



The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$
1986 IMO Shortlist
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
1986 IMO Shortlist
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
1986 IMO Shortlist
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.
1986 IMO Shortlist
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:

$$\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.$$

Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
1986 IMO Shortlist
From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that :

(i) each pair of consecutive teams on the list have one member in common and

(ii) the chain of teams on the list are in rank order.



Alternative formulation.

Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.
1986 IMO Shortlist
Three persons $A,B,C$, are playing the following game:



A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$.



For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)
1986 IMO Shortlist
Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$



Simplified version.



Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$
1986 IMO Shortlist
A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$
1986 IMO Shortlist
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.
1986 IMO Shortlist
Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$



(a) Prove that $ABCD$ and $A''B''C''D''$ are similar.



(b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.
1986 IMO Shortlist
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.
1986 IMO Shortlist
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
1986 IMO Shortlist
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.
1986 IMO Shortlist
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that

$$r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}$$

where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
1985 IMO Shortlist
Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.
1985 IMO Shortlist
A polyhedron has $12$ faces and is such that:



(i) all faces are isosceles triangles,

(ii) all edges have length either $x$ or $y$,

(iii) at each vertex either $3$ or $6$ edges meet, and

(iv) all dihedral angles are equal.



Find the ratio $x/y.$
1985 IMO Shortlist
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: $$ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}).  $$
1985 IMO Shortlist
Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that:



(i) $i$ and $n - i$ always receive the same color, and



(ii) for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$



Prove that all numbers in $N$ must receive the same color.
1985 IMO Shortlist
Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$
1985 IMO Shortlist
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that

$$x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots$$
1985 IMO Shortlist
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
1985 IMO Shortlist
Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.
1985 IMO Shortlist
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$
1985 IMO Shortlist
Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.
1985 IMO Shortlist
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+  a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
1985 IMO Shortlist
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by

$$P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)$$

for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
1985 IMO Shortlist
Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove:



(a) If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls.



(b) If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.
1985 IMO Shortlist
A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.
1985 IMO Shortlist
Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that

$$K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? $$
1985 IMO Shortlist
If possible, construct an equilateral triangle whose three vertices are on three given circles.
1985 IMO Shortlist
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by

$$f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)$$Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
1985 IMO Shortlist
Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that

$$\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1$$
1985 IMO Shortlist
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?
1985 IMO Shortlist
A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$
1985 IMO Shortlist
The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that



(a) $\angle BAM = \angle CAX$, and



(b) $\frac{AM}{AX} = \cos\angle BAC.$
1984 IMO Shortlist
Find all solutions of the following system of $n$ equations in $n$ variables:

$$\begin{array}{c}\  x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}$$

where $a$ is a given number.
1984 IMO Shortlist
Prove:



(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$



(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
1984 IMO Shortlist
Find all positive integers $n$ such that

$$n=d_6^2+d_7^2-1,$$

where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$
1984 IMO Shortlist
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:

$$f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad  (n \geq 2).$$

Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$  such that $f_kf_{k+1}= f_r.$
1984 IMO Shortlist
(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms









is divisible by four.



(b) Solve the analogous problem for
1984 IMO Shortlist
Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations

$$\sqrt{y-a}+\sqrt{z-a}=1,$$ $$\sqrt{z-b}+\sqrt{x-b}=1,$$ $$\sqrt{x-c}+\sqrt{y-c}=1$$

has exactly one solution $(x, y, z)$ in real numbers.
1984 IMO Shortlist
Prove that the product of five consecutive positive integers cannot be the square of an integer.
1984 IMO Shortlist
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying

$$(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.$$
1984 IMO Shortlist
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
1984 IMO Shortlist
Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.



(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$



(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$
1984 IMO Shortlist
In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$
1984 IMO Shortlist
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$
1984 IMO Shortlist
The harmonic table is a triangular array:



$1$



$\frac 12 \qquad \frac 12$



$\frac 13 \qquad \frac 16 \qquad \frac 13$



$\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$



Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.
1984 IMO Shortlist
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that

$$\log_a b < \log_{a+1} (b + 1).$$
1983 IMO Shortlist
The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.
1983 IMO Shortlist
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is superabundant (P.Erdos, $1944$) if $\forall k \in  \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$

Prove that there exists an infinity of superabundant numbers.
1983 IMO Shortlist
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
1983 IMO Shortlist
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.