Navvye/Olympiad-Problems-CSV huggingface dataset:
1986 IMO Longlists
Find the least integer $n$ with the following property:

For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points.
1986 IMO Longlists
Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ ,  $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$
1986 IMO Longlists
Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \  (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.
1986 IMO Longlists
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ in the points $A',B', C'$, respectively; the excircle in the angle $A$ touches the lines containing these sides in $A_1,B_1, C_1$, and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$. Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $ (A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $ (A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.
1986 IMO Longlists
Prove the inequality

$$(-a+b+c)^2(a-b+c)^2(a+b-c)^2 \geq (-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)$$

for all real numbers $a, b, c.$
1986 IMO Longlists
Given a positive integer $n$, find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$, where $X$ and $C$ are sets of cardinality $n$ and $p$, respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$. ($\mathbb P(X)$ is the family of all subsets of $X$.)
1986 IMO Longlists
Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number.
1986 IMO Longlists
Let $AA',BB', CC'$ be the bisectors of the angles of a triangle $ABC \ (A' \in BC, B' \in CA, C' \in AB)$. Prove that each of the lines $A'B', B'C', C'A'$ intersects the incircle in two points.
1986 IMO Longlists
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
1986 IMO Longlists
Let $A_1A_2A_3A_4$ be a quadrilateral inscribed in a circle $C$. Show that there is a point $M$ on $C$ such that $MA_1 -MA_2 +MA_3 -MA_4 = 0.$
1986 IMO Longlists
One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
1986 IMO Longlists
Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.
1986 IMO Longlists
Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.
1986 IMO Longlists
A one-person game with two possible outcomes is played as follows:

After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$
1986 IMO Longlists
Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$
1986 IMO Longlists
The incenter of a triangle is the midpoint of the line segment of length $4$ joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.
1986 IMO Longlists
Find all integers $x,y,z$ such that

$$x^3+y^3+z^3=x+y+z=8$$
1986 IMO Longlists
If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality

$$\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.$$
1986 IMO Longlists
Let $AA_1,BB_1, CC_1$ be the altitudes in an acute-angled triangle $ABC$, $K$ and $M$ are points on the line segments $A_1C_1$ and $B_1C_1$ respectively. Prove that if the angles $MAK$ and $CAA_1$ are equal, then the angle $C_1KM$ is bisected by $AK.$
1986 IMO Longlists
Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$
1985 IMO Longlists
We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$
1985 IMO Longlists
A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?
1985 IMO Longlists
Let $x, y$, and $z$ be real numbers satisfying $x + y + z = xyz.$ Prove that

$$x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.$$
1985 IMO Longlists
On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars.



(a) What is the probability that the pedestrian crosses the street uninjured?



(b) Can he improve this probability by crossing the road in a direction other than perpendicular?
1985 IMO Longlists
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$
1985 IMO Longlists
Let $K $ be a convex set in the $xy$-plane, symmetric with respect to the origin and having area greater than $4 $. Prove that there exists a point $(m, n) \neq (0, 0)$ in $K$ such that $m$ and $n$ are integers.
1985 IMO Longlists
Let $a$ and $ b$ be integers and $n$ a positive integer. Prove that

$$\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}$$

is an integer.
1985 IMO Longlists
Find the maximum value of

$$\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n$$

subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$
1985 IMO Longlists
Find the average of the quantity

$$(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2$$

taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$
1985 IMO Longlists
Let $k$ be a positive integer. Define $u_0 = 0, u_1 = 1$, and $u_n=ku_{n-1}-u_{n-2} , n \geq 2.$ Show that for each integer $n$, the number $u_1^3 + u_2^3 +\cdots+ u_n^3 $ is a multiple of $u_1 + u_2 +\cdots+ u_n.$
1985 IMO Longlists
Superchess is played on on a $12 \times 12$ board, and it uses superknights, which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a superknight to visit every other cell of a superchessboard exactly once and return to its starting cell ?
1985 IMO Longlists
Set

$$A_n=\sum_{k=1}^n \frac{k^6}{2^k}.$$

Find $\lim_{n\to\infty} A_n.$
1985 IMO Longlists
The circles $(R, r)$ and $(P, \rho)$, where $r > \rho$, touch externally at $A$. Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$. The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$. If $|BC| = 6|DE|$, prove that:

(a) the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and



(b) $|AB| = 2|AC|.$
1985 IMO Longlists
Solve the system of simultaneous equations

$$\sqrt x - \frac 1y - 2w + 3z = 1,$$$$x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,$$$$x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,$$$$x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.$$
1985 IMO Longlists
Let $T$ be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points $(x, y, z)$ and $(u, v,w)$ are called neighbors if $|x - u| + |y - v| + |z - w| = 1$. Show that there exists a subset $S$ of $T$ such that for each $p \in T$ , there is exactly one point of $S$ among $p$ and its neighbors.
1985 IMO Longlists
Let $A$ be a set of positive integers such that for any two distinct elements $x, y\in A$ we have $|x-y| \geq \frac{xy}{25}.$ Prove that $A$ contains at most nine elements. Give an example of such a set of nine elements.
1985 IMO Longlists
Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that

$$S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.$$
1985 IMO Longlists
Let $d \geq 1$ be an integer that is not the square of an integer. Prove that for every integer $n \geq 1,$

$$(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1$$
1985 IMO Longlists
Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property:



For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$
1985 IMO Longlists
Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$).

Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy



(i) $M_0 = M_k$, and



(ii) if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.
1985 IMO Longlists
a) Let $M$ be the set of the lengths of the edges of an octahedron whose sides are congruent quadrangles. Prove that $M$ has at most three elements.



b)  Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.
1985 IMO Longlists
a) Call a four-digit number $(xyzt)_B$ in the number system with base $B$ stable if $(xyzt)_B = (dcba)_B - (abcd)_B$, where $a \leq b \leq c \leq d$ are the digits of $(xyzt)_B$ in ascending order. Determine all stable numbers in the number system with base $B.$



b) With assumptions as in a, determine the number of bases $B \leq 1985$ such that there exists a stable number with base $B.$
1985 IMO Longlists
A plane rectangular grid is given and a “rational point” is defined as a point $(x, y)$ where $x$ and $y$ are both rational numbers. Let $A,B,A',B'$ be four distinct rational points. Let $P$ be a point such that $\frac{A'B'}{AB}=\frac{B'P}{BP} = \frac{PA'}{PA}.$ In other words, the triangles $ABP, A'B'P$ are directly or oppositely similar. Prove that $P$ is in general a rational point and find the exceptional positions of $A'$ and $B'$ relative to $A$ and $B$ such that there exists a $P$ that is not a rational point.
1985 IMO Longlists
Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.
1985 IMO Longlists
A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$



Similar Problem :



A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$
1985 IMO Longlists
We call a coloring $f$ of the elements in the set $M = \{(x, y) | x = 0, 1, \dots , kn - 1; y = 0, 1, \dots , ln - 1\}$ with $n$ colors allowable if every color appears exactly $k$ and $ l$ times in each row and column and there are no rectangles with sides parallel to the coordinate axes such that all the vertices in $M$ have the same color. Prove that every allowable coloring $f$ satisfies $kl \leq n(n + 1).$
1985 IMO Longlists
Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection
1985 IMO Longlists
Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies

$$\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.$$



Remark.Remark. Can we determine all of equality cases ?
1985 IMO Longlists
Given a triangle $ABC$ and external points $X, Y$ , and $Z$ such that $\angle BAZ = \angle CAY , \angle CBX = \angle ABZ$, and $\angle ACY = \angle BCX$, prove that $AX,BY$ , and $CZ$ are concurrent.
1985 IMO Longlists
Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and

$$\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.$$

where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.
1985 IMO Longlists
Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.
1985 IMO Longlists
Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$
1985 IMO Longlists
Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?
1985 IMO Longlists
Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$; t intersects $C$ at another point $Q$. Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$. Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$
1985 IMO Longlists
Let $F$ be the correspondence associating with every point $P = (x, y)$ the point $P' = (x', y')$ such that

$$ x'= ax + b,\qquad  y'= ay + 2b.  \qquad (1)$$

Show that if $a \neq 1$, all lines $PP'$ are concurrent. Find the equation of the set of points corresponding to $P = (1, 1)$ for $b = a^2$. Show that the composition of two mappings of type $(1)$ is of the same type.
1985 IMO Longlists
In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, $A, B$, and $C$. Person $A$ says, “If $C$ is a knight, $B$ is a knave.” Person $C$ says, “$A$ and I are different; one is a knight and the other is a knave.” Who are the knights, and who are the knaves ?
1985 IMO Longlists
From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$
1985 IMO Longlists
Let $f_1 = (a_1, a_2, \dots , a_n) , n > 2$, be a sequence of integers. From $f_1$ one constructs a sequence $f_k$ of sequences as follows: if $f_k = (c_1, c_2, \dots, cn)$, then $f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1)$, where $(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}})$ is a permutation of $(c_1, c_2, \dots, c_n)$. Give a necessary and sufficient condition for $f_1$ under which it is possible for $f_k$ to be a constant sequence $(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n$, for some $k.$
1985 IMO Longlists
In the triangle $ABC$, let $B_1$ be on $AC, E$ on $AB, G$ on $BC$, and let $EG$ be parallel to $AC$. Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$. Let $r, r_1$, and $r_2$ be the inradii of the triangles $ABC, ABB_1$, and $BFG$, respectively. Prove that $r = r_1 + r_2.$
1985 IMO Longlists
Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$
1985 IMO Longlists
The points $A,B,C$ are in this order on line $D$, and $AB = 4BC$. Let $M$ be a variable point on the perpendicular to $D$ through $C$. Let $MT_1$ and $MT_2$ be tangents to the circle with center $A$ and radius $AB$. Determine the locus of the orthocenter of the triangle $MT_1T_2.$
1985 IMO Longlists
Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$
1985 IMO Longlists
a) The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$





b) Using the same assumptions in a), prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$
1985 IMO Longlists
Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1 + 2 + \cdots + k = (k + 1) + (k + 2)+\cdots + N.$
1985 IMO Longlists
The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.
1985 IMO Longlists
Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$
1985 IMO Longlists
A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed $\frac{\pi}{3}$ cm per second (relative to the disk). Describe the path of the insect



(a) relative to the disk;



(b) relative to the wall.
1985 IMO Longlists
Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?
1985 IMO Longlists
Define the functions $f, F : \mathbb N \to \mathbb N$, by

$$f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in  \mathbb N|f^k(n) > 0 \},$$

where $f^k = f \circ \cdots  \circ f$ is $f$ iterated $n$ times. Prove that $F(k + 2) = 3F(k + 1) - F(k)$ for all $k \in  \mathbb N.$
1985 IMO Longlists
Let $k \geq 2$ and $n_1, n_2, . . . , n_k \geq  1$ natural numbers having the property $n_2 | 2^{n_1} - 1, n_3 | 2^{n_2} -1 , \cdots, n_k | 2^{n_k-1}-1$, and $n_1 | 2^{n_k} - 1$. Show that $n_1 = n_2 = \cdots = n_k = 1.$
1985 IMO Longlists
Show that the sequence $\{a_n\}_{n\geq1}$ defined by $a_n = [n \sqrt 2]$ contains an infinite number of integer powers of $2$. ($[x]$ is the integer part of $x$.)
1985 IMO Longlists
Let $A$ and $B$ be two finite disjoint sets of points in the plane such that no three distinct

points in $A \cup B$ are collinear. Assume that at least one of the sets $A, B$ contains at least five points. Show that there exists a triangle all of whose vertices are contained in $A$ or in $B$ that does not contain in its interior any point from the other set.
1985 IMO Longlists
Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in  \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.
1985 IMO Longlists
For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.
1985 IMO Longlists
Construct a triangle $ABC$ given the side $AB$ and the distance $OH$ from the circumcenter $O$ to the orthocenter $H$, assuming that $OH$ and $AB$ are parallel.
1985 IMO Longlists
Let $A_1A_2,B_1B_2, C_1C_2$ be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines $B_2C_1, C_2A_1,A_2B_1$, the segments $B_2C_1, C_2A_1,A_2B_1$ are proportional to the sides in which they are contained.
1985 IMO Longlists
Find all triples of positive integers $x, y, z$ satisfying

$$\frac{1}{x} +\frac{1}{y} + \frac{1}{z} = \frac{4}{5} .$$
1985 IMO Longlists
Let $ABCD$ be a rectangle, $AB = a, BC = b$. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$. Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find:



(a) how $L $ varies,



(b) a necessary and sufficient condition for $L$ to be a constant, and



(c) the value of this constant.
1985 IMO Longlists
Are there integers $m$ and $n$ such that

$$5m^2 - 6mn + 7n^2 = 1985 \ ?$$
1985 IMO Longlists
Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$
1985 IMO Longlists
Let $a, b$, and $c$ be real numbers such that

$$\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.$$

Prove that

$$\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.$$
1985 IMO Longlists
Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$
1985 IMO Longlists
Given the side $a$ and the corresponding altitude $h_a$ of a triangle $ABC$, find a relation between $a$ and $h_a$ such that it is possible to construct, with straightedge and compass, triangle $ABC$ such that the altitudes of $ABC$ form a right triangle admitting $h_a$ as hypotenuse.
1985 IMO Longlists
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.
1985 IMO Longlists
Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$