Navvye/Olympiad-Problems-CSV huggingface dataset:
1985 IMO Longlists
Let $CD$ be a diameter of circle $K$. Let $AB$ be a chord that is parallel to $CD$. The line segment $AE$, with $E$ on $K$, is parallel to $CB$; $F$ is the point of intersection of line segments $AB$ and $DE$. The line segment $FG$, with $G$ on $DC$, extended is parallel to $CB$. Is $GA$ tangent to $K$ at point $A \?$
1985 IMO Longlists
Let $l$ denote the length of the smallest diagonal of all rectangles inscribed in a triangle $T$ . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of $T$ .) Determine the maximum value of $\frac{l^2}{S(T)}$ taken over all triangles ($S(T )$ denotes the area of triangle $T$ ).
1985 IMO Longlists
Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that

$$x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.$$
1985 IMO Longlists
Given that $n$ elements $a_1, a_2,\dots, a_n$ are organized into $n$ pairs $P_1, P_2, \dots, P_n$ in such a way that two pairs $P_i, P_j$ share exactly one element when $(a_i, a_j)$ is one of the pairs, prove that every element is in exactly two of the pairs.
1985 IMO Longlists
Factorise $ 5^{1985}-1$ as a product of three integers, each greater than $ 5^{100}$.
1985 IMO Longlists
Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?
1985 IMO Longlists
The sphere inscribed in tetrahedron $ABCD$ touches the sides $ABD$ and $DBC$ at points $K$ and $M$, respectively. Prove that $\angle AKB = \angle DMC$.
1985 IMO Longlists
Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:



(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and



(b) $\lim_{x\to \infty} f(x) = 0$.
1985 IMO Longlists
In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$
1984 IMO Longlists
The fraction $\frac{3}{10}$ can be written as the sum of two positive fractions with numerator $1$ as follows: $\frac{3}{10} =\frac{1}{5}+\frac{1}{10}$ and also $\frac{3}{10}=\frac{1}{4}+\frac{1}{20}$. There are the only two ways in which this can be done. In how many ways can $\frac{3}{1984}$ be written as the sum of two positive fractions with numerator $1$?

Is there a positive integer $n,$ not divisible by $3$, such that $\frac{3}{n}$ can be written as the sum of two positive fractions with numerator $1$ in exactly $1984$ ways?
1984 IMO Longlists
Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.
1984 IMO Longlists
The opposite sides of the reentrant hexagon $AFBDCE$ intersect at the points $K,L,M$ (as shown in the figure). It is given that $AL = AM = a, BM = BK = b$, $CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f$.

//cdn.artofproblemsolving.com/images/fad76261a0d1316c87c556c7b67ffd6d94dfec56.png

$(a)$ Given length $a$ and the three angles $\alpha, \beta$ and $\gamma$ at the vertices $A, B,$ and $C,$ respectively, satisfying the condition $\alpha+\beta+\gamma<180^{\circ}$, show that all the angles and sides of the hexagon are thereby uniquely determined.

$(b)$ Prove that

$$\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}$$

Easier version of $(b)$. Prove that

$$(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)$$
1984 IMO Longlists
Given a triangle $ABC$, three equilateral triangles $AEB, BFC$, and $CGA$ are constructed in the exterior of $ABC$. Prove that:

$(a) CE = AF = BG$;

$(b) CE, AF$, and $BG$ have a common point.



I could not find a separate topic for this question and I need one.  of course.
1984 IMO Longlists
For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. If $m \ge 3$, prove that

$$\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]$$
1984 IMO Longlists
Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$
1984 IMO Longlists
Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.
1984 IMO Longlists
In the plane of a given triangle $A_1A_2A_3$ determine (with proof) a straight line $l$ such that the sum of the distances from $A_1, A_2$, and $A_3$ to $l$ is the least possible.
1984 IMO Longlists
The circle inscribed in the triangle $A_1A_2A_3$ is tangent to its sides $A_1A_2, A_2A_3, A_3A_1$ at points $T_1, T_2, T_3$, respectively. Denote by $M_1, M_2, M_3$ the midpoints of the segments $A_2A_3, A_3A_1, A_1A_2$, respectively. Prove that the perpendiculars through the points $M_1, M_2, M_3$ to the lines $T_2T_3, T_3T_1, T_1T_2$ meet at one point.
1984 IMO Longlists
Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.
1984 IMO Longlists
Consider all the sums of the form

$$\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5$$

where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?
1984 IMO Longlists
Let $c$ be the inscribed circle of the triangle $ABC$, $d$ a line tangent to $c$ which does not pass through the vertices of triangle $ABC$. Prove the existence of points $A_1,B_1, C_1$, respectively, on the lines $BC,CA,AB$ satisfying the following two properties:

$(i)$ Lines $AA_1,BB_1$, and $CC_1$ are parallel.

$(ii)$ Lines $AA_1,BB_1$, and $CC_1$ meet $d$ respectively at points $A' ,B'$, and $C'$ such that

$$\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}$$
1984 IMO Longlists
Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by

$$F(M) = BM +CM-\sqrt{3}AM$$
1984 IMO Longlists
$(1)$ Start with $a$ white balls and $b$ black balls.

$(2)$ Draw one ball at random.

$(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$.

Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$
1984 IMO Longlists
A $2\times 2\times 12$ box fixed in space is to be filled with twenty-four $1 \times 1 \times 2$ bricks. In how many ways can this be done?
1984 IMO Longlists
A cylindrical container has height $6 cm$ and radius $4 cm$. It rests on a circular hoop, also of radius $4 cm$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points.

The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder’s vertical ends.
1984 IMO Longlists
The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and

$$f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)$$

for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.
1984 IMO Longlists
A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$

$$t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)$$

Prove that all $t_{n,m}$ are integers.
1984 IMO Longlists
Let $S_n = \{1, \cdots, n\}$ and let $f$ be a function that maps every subset of $S_n$ into a positive real number and satisfies the following condition: For all $A \subseteq S_n$ and $x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)$. Prove that for all $A,B \subseteq S_n$ the following inequality holds:

$$f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)$$
1984 IMO Longlists
Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.
1984 IMO Longlists
Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that

$$\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha$$
1984 IMO Longlists
One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
1984 IMO Longlists
Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions

$$\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}$$

where $\pi$ is the ratio between a circle and its diameter.
1984 IMO Longlists
The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.
1984 IMO Longlists
$(MOR 1)$ Denote by $[x]$ the greatest integer not exceeding $x$. For all real $k > 1$, define two sequences:

$$a_n(k) = [nk]\text{ and  } b_n(k) =\left[\frac{nk}{k - 1}\right]$$

If $A(k) = \{a_n(k) : n \in\mathbb{N}\}$ and $B(k) = \{b_n(k) : n \in \mathbb{N}\}$, prove that $A(k)$ and $B(k)$ form a partition of $\mathbb{N}$ if and only if $k$ is irrational.
1984 IMO Longlists
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that

$$f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.$$
1984 IMO Longlists
Let $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?
1984 IMO Longlists
Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)
1984 IMO Longlists
Triangle $ABC$ is given for which $BC = AC + \frac{1}{2}AB$. The point $P$ divides $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2\angle CPA$.
1984 IMO Longlists
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.



It was proposed by Poland. Have fun!  :lol:
1984 IMO Longlists
Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.
1984 IMO Longlists
Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.
1984 IMO Longlists
Let $ABC$ be a triangle with interior angle bisectors $AA_1, BB_1, CC_1$ and incenter $I$. If $\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC]$, where $\sigma[ABC]$ denotes the area of $ABC$, show that $ABC$ is isosceles.
1984 IMO Longlists
Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set

$$S_k = x_1^k+ x^k_2+\cdots+ x^k_n$$

for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$
1984 IMO Longlists
Two cyclists leave simultaneously a point $P$ in a circular runway with constant velocities $v_1, v_2 (v_1 > v_2)$ and in the same sense. A pedestrian leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1+v_2}{12}$ . If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist $91$ seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him $187$ seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.
1984 IMO Longlists
Construct a scalene triangle such that

$$a(\tan B - \tan C) = b(\tan A - \tan C)$$
1984 IMO Longlists
Find a sequence of natural numbers $a_i$ such that $a_i = \displaystyle\sum_{r=1}^{i+4} d_r$, where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$.
1984 IMO Longlists
Let $P$ be a convex planar polygon with equal angles. Let $l_1,\cdots, l_n$ be its sides. Show that a necessary and sufficient condition for $P$ to be regular is that the sum of the ratios $\frac{l_i}{l_{i+1}} (i = 1,\cdots, n; l_{n+1}= l_1)$ equals the number of sides.
1984 IMO Longlists
Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers.

$(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number.

$(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.
1984 IMO Longlists
Let $a, b, c$ be nonnegative integers such that $a \le b \le c, 2b \neq a + c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d, e$, and $f$ such that $d \le e \le f, f \neq c$, and such that $a^2+b^2+c^2 = d^2 + e^2 + f^2$?
1984 IMO Longlists
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$
1984 IMO Longlists
Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.
1984 IMO Longlists
Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.
1984 IMO Longlists
A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.
1984 IMO Longlists
From a point $P$ exterior to a circle $K$, two rays are drawn intersecting $K$ in the respective pairs of points $A, A'$ and $B,B' $. For any other pair of points $C, C'$ on $K$, let $D$ be the point of intersection of the circumcircles of triangles $PAC$ and $PB'C'$ other than point $P$. Similarly, let $D'$ be the point of intersection of the circumcircles of triangles $PA'C'$ and $PBC$ other than point $P$. Prove that the points $P, D$, and $D'$ are collinear.
1984 IMO Longlists
For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set

$$a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{  for  }i = 1,\cdots,m\text{  and  }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{  for  }j = 1, . . . , n\longrightarrow(1)$$

By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.
1984 IMO Longlists
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
1984 IMO Longlists
Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that:

$$d_6^{2} +d_7^{2} - 1=n$$
1984 IMO Longlists
With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that

$$R_m>\frac{5}{6}R$$
1984 IMO Longlists
In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.
1983 IMO Longlists
Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities 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 Longlists
(a) Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes.



(b) If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.
1983 IMO Longlists
Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational.



(a) Prove that the union of segments with vertices from $\mathbb Q^2$  is the entire set $\mathbb R^2$.



(b) Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?
1983 IMO Longlists
Find all numbers $x \in \mathbb Z$ for which the number

$$x^4 + x^3 + x^2 + x + 1$$

is a perfect square.
1983 IMO Longlists
Which of the numbers $1, 2, \ldots , 1983$ has the largest number of divisors?
1983 IMO Longlists
A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible given that the line $AB$ and the lake are exterior to each other.
1983 IMO Longlists
The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?
1983 IMO Longlists
Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that

$$p \equiv a_i a_j  \pmod r.$$
1983 IMO Longlists
Let $\ell$ be tangent to the circle $k$ at $B$. Let $A$ be a point on $k$ and $P$ the foot of perpendicular from $A$ to $\ell$. Let $M$ be symmetric to $P$ with respect to $AB$. Find the set of all such points $M.$
1983 IMO Longlists
Find all possible finite sequences $\{n_0, n_1, n_2, \ldots, n_k \}$ of integers such that for each $i, i$ appears in the sequence $n_i$ times $(0 \leq i \leq k).$
1983 IMO Longlists
In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times  n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which:

(i) $a_1 < a_2 < \cdots < a_n,$



(ii)  $b_1 < b_2 <\cdots < b_n,$



(iii) $a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$
1983 IMO Longlists
Let $b \geq 2$ be a positive integer.



(a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits.



(b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits.



(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.



(d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.
1983 IMO Longlists
Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$



(a) Prove that the function $g : \mathbb R  \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$



(b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$
1983 IMO Longlists
Prove that there are infinitely many positive integers $n$ for which it is possible for a knight, starting at one of the squares of an $n \times n$ chessboard, to go through each of the squares exactly once.
1983 IMO Longlists
Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$
1983 IMO Longlists
Every $x, 0 \leq x \leq  1$, admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$, where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$. If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$, and

$$f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}$$

show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$
1983 IMO Longlists
How many permutations $a_1, a_2, \ldots, a_n$ of $\{1, 2, . . ., n \}$ are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange $a_i$ and $a_{i+1}$ whenever $a_i > a_{i+1}$ for $i$ running from $1$ up to $n - 1 \ ?$
1983 IMO Longlists
Let $a, b, c$ be positive integers satisfying $\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1$. Show that $2abc-ab-bc-ca$ cannot be represented as $bcx+cay +abz$ with nonnegative integers $x, y, z.$
1983 IMO Longlists
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$  and show that the tangents at $A,D$ to the circle meet on the line $BC.$
1983 IMO Longlists
Prove the existence of a unique sequence $\{u_n\} \ (n = 0, 1, 2 \ldots )$ of positive integers such that

$$u_n^2 = \sum_{r=0}^n \binom{n+r}{r} u_{n-r} \qquad \text{for all  }  n \geq 0$$
1983 IMO Longlists
Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and

$$f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all  } n \geq 1.$$

Prove that if $b + c < 1$, there is a real number $k$ such that

$$f(n) \leq  kn \qquad \text{  for all  } n \qquad (1)$$

while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq  K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$