Navvye/Olympiad-Problems-CSV huggingface dataset:
1991 IMO Shortlist
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”
1990 IMO Shortlist
The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 = 4+5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 = 4+5 = 2+3+4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?
1990 IMO Shortlist
Given $ n$ countries with three representatives each, $ m$ committees $ A(1),A(2), \ldots, A(m)$ are called a cycle if



(i) each committee has $ n$ members, one from each country;

(ii) no two committees have the same membership;

(iii) for $ i = 1, 2, \ldots,m$, committee $ A(i)$ and committee $ A(i + 1)$ have no member in common, where $ A(m + 1)$ denotes $ A(1);$

(iv) if $ 1 < |i - j| < m - 1,$ then committees $ A(i)$ and $ A(j)$ have at least one member in common.



Is it possible to have a cycle of 1990 committees with 11 countries?
1990 IMO Shortlist
Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r = \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j + 1} - a_j \leq m,$ $ (1 \leq j \leq k - 1)$.
1990 IMO Shortlist
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
1990 IMO Shortlist
Let $ f(0) = f(1) = 0$ and



$$ f(n+2) = 4^{n+2} \cdot  f(n+1) - 16^{n+1} \cdot f(n) + n \cdot 2^{n^2}, \quad n = 0, 1, 2, \ldots$$



Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$
1990 IMO Shortlist
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n+1}(k) = f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$
1990 IMO Shortlist
The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$
1990 IMO Shortlist
A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part.





Original formulation:



A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.
1990 IMO Shortlist
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
1990 IMO Shortlist
In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that



(i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x = j$ or $ y = k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1;



(ii) each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition.



Prove that there exist at least two triangles in the partition each of which has two good sides.
1990 IMO Shortlist
Determine for which positive integers $ k$ the set $$ X = \{1990, 1990 + 1, 1990 + 2, \ldots, 1990 + k\}$$ can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$
1990 IMO Shortlist
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$



(a) Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.

(b) The same question as (a) with the extra condition that $ A = B.$
1990 IMO Shortlist
Let $ a, b \in \mathbb{N}$  with $ 1 \leq a \leq b,$ and $ M = \left[\frac {a + b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by

$$ f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\
n - b, & \text{if } n >M. \end{cases}

$$Let $ f^1(n) = f(n),$ $ f_{i + 1}(n) = f(f^i(n)),$ $ i = 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) = 0.$
1990 IMO Shortlist
Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$
1990 IMO Shortlist
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
1990 IMO Shortlist
Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that



$$ \frac{(1 + 2^p + 2^{n-p})N - 1}{2^n}$$



is an integer.
1990 IMO Shortlist
Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.
1990 IMO Shortlist
Let $ w, x, y, z$ are non-negative reals such that $ wx + xy + yz + zw = 1$.

Show that $ \frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}$.
1990 IMO Shortlist
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n = p(q_{n + 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n + k} = q_n$ for all positive $ n$.
1990 IMO Shortlist
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n - 1$ digits $ 1$ and one digit $ 7$ is prime.
1990 IMO Shortlist
Prove that on the coordinate plane it is impossible to draw a closed broken line such that



(i) the coordinates of each vertex are rational;

(ii) the length each of its edges is 1;

(iii) the line has an odd number of vertices.
1989 IMO Shortlist
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
1989 IMO Shortlist
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
1989 IMO Shortlist
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation $$ \sum^n_{k=1} \frac{x^k}{k!} + 1 = 0$$ has no rational roots.
1989 IMO Shortlist
Find the roots $ r_i \in \mathbb{R}$  of the polynomial  $$ p(x) = x^n + n \cdot x^{n-1} + a_2 \cdot x^{n-2} + \ldots + a_n$$ satisfying $$ \sum^{16}_{k=1} r^{16}_k = n.$$
1989 IMO Shortlist
For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality



$$ 16Q^3 \geq 27 r^4 P,$$



where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.
1989 IMO Shortlist
Show that any two points lying inside a regular $ n-$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 - \frac{2}{n} \right) \cdot \pi.$
1989 IMO Shortlist
Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions:



(i) The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$

(ii) The interiors of any two different rectangles $ R_i$ are disjoint.

(iii) Each rectangle $ R_i$ has at least one side of integral length.



Prove that $ R$ has at least one side of integral length.



Variant: Same problem but with rectangular parallelepipeds having at least one integral side.
1989 IMO Shortlist
$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such



$$ \left(1 + 4 \cdot \sqrt[3]{2} - 4 \cdot \sqrt[3]{4} \right)^n = a_n + b_n \cdot \sqrt[3]{2} + c_n \cdot \sqrt[3]{4}.$$



Prove that $ c_n = 0$ implies $ n = 0.$
1989 IMO Shortlist
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 = 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that

$$ f(z) + f(\omega z + a) = g(z),z\in \mathbb{C}

$$
1989 IMO Shortlist
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d = 2^n.$ Show that $ n|a_n.$
1989 IMO Shortlist
There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.
1989 IMO Shortlist
A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.
1989 IMO Shortlist
Let $ a, b, c, d,m, n \in \mathbb{Z}^+$ such that $$ a^2+b^2+c^2+d^2 = 1989,$$

$$ a+b+c+d = m^2,$$ and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values  of $m$ and $ n.$
1989 IMO Shortlist
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:



(i) $ a_0 = a_n = 0,$

(ii) for $ 1 \leq k \leq n - 1,$ $$ a_k = c + \sum^{n-1}_{i=k} a_{i-k} \cdot \left(a_i + a_{i+1} \right)$$



Prove that $ c \leq \frac{1}{4n}.$
1989 IMO Shortlist
Given seven points in the plane, some of them are connected by segments such that:



(i) among any three of the given points, two are connected by a segment;

(ii) the number of segments is minimal.



How many segments does a figure satisfying (i) and (ii) have? Give an example of such a figure.
1989 IMO Shortlist
Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i = 1, 2, \ldots, n.$
1989 IMO Shortlist
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
1989 IMO Shortlist
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
1989 IMO Shortlist
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
1989 IMO Shortlist
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if $$ x^2 - ay^2 - bz^2 + abw^2 = 0$$ has a nontrivial solution in integers, then so does $$ x^2 - ay^2 - bz^2 = 0.$$
1989 IMO Shortlist
Let $ n \in \mathbb{Z}^+$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which



$$ \sum^n_{i=0} x_i = a \text{ and } \sum^n_{i=0} x^2_i = b,$$



where $ x_0, x_1, \ldots , x_n$ are real variables.
1989 IMO Shortlist
Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n - 1.$
1989 IMO Shortlist
Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that $$ \sigma(OA_iA_j) \geq 1 \quad \forall i, j = 1, 2, 3, 4, i \neq j.$$  where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$  Prove that there exists at least  one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that $$ \sigma(OA_iA_j) \geq \sqrt{2}.$$
1989 IMO Shortlist
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$  Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
1989 IMO Shortlist
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^+$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation



$$ \sum^3_{k=1} \frac{a_k}{x_k} = 1.$$



where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that $$ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 + ln(2 a_1)).$$
1989 IMO Shortlist
The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$
1988 IMO Shortlist
An integer sequence is defined by $${ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.$$ Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
1988 IMO Shortlist
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial

$$ u_n(x) = (x^2 + x + 1)^n.

$$
1988 IMO Shortlist
The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$
1988 IMO Shortlist
An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$
1988 IMO Shortlist
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
1988 IMO Shortlist
Let $ a$ be the greatest positive root of the equation $ x^3 - 3 \cdot x^2 + 1 = 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$
1988 IMO Shortlist
Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$
1988 IMO Shortlist
Let $ N = \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F = \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i = 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F = \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?
1988 IMO Shortlist
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe  (assuming the "right combination" is not known)?
1988 IMO Shortlist
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that

$$ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.

$$
1988 IMO Shortlist
For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?
1988 IMO Shortlist
Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.
1988 IMO Shortlist
In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.
1988 IMO Shortlist
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) + f(m)) = m + n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
1988 IMO Shortlist
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
1988 IMO Shortlist
Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$
1988 IMO Shortlist
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation

$$ \sum^p_{i = 1} x^2_i - \frac {4}{4 \cdot p + 1} \left( \sum^p_{i = 1} x_i \right)^2 = 1

$$

OR



Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying

$$ \sum^p_{i = 1} x^2_i - \frac {4}{4 \cdot p + 1} \left( \sum^p_{i = 1} x_i \right)^2 = 1

$$
1988 IMO Shortlist
Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$

$$ a(PA)^2 + b(PB)^2 + c(PC)^2 = a(QA)^2 + b(QB)^2 + c(QC)^2 + (a + b + c)(QP)^2,

$$

where $ a = BC, b = CA$ and $ c = AB.$
1988 IMO Shortlist
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:

$$ a_k - 2  a_{k + 1} + a_{k + 2} \geq 0

$$

and $ \sum^k_{j = 1} a_j \leq 1$ for all $ k = 1,2, \ldots$. Prove that:

$$ 0 \leq a_{k} - a_{k + 1} < \frac {2}{k^2}

$$

for all $ k = 1,2, \ldots$.
1988 IMO Shortlist
A positive integer is called a double number if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.
1988 IMO Shortlist
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A + v^2\cdot\tan B + w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
1988 IMO Shortlist
The sequence $ \{a_n\}$ of integers is defined by

$$ a_1 = 2, a_2 = 7

$$

and

$$ - \frac {1}{2} < a_{n + 1} - \frac {a^2_n}{a_{n - 1}} \leq \frac {}{}, n \geq 2.

$$

Prove that $ a_n$ is odd for all $ n > 1.$
1988 IMO Shortlist
A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$
1988 IMO Shortlist
A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that

$$ BM^{2} = X \cot \left( \frac {B}{2}\right)

$$

where X is the area of triangle $ ABC.$
1988 IMO Shortlist
Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.