Navvye/Olympiad-Problems-CSV huggingface dataset:
1967 IMO Longlists
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that



$$(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)$$



is divisible by the product $c_1c_2\ldots c_n$.
1967 IMO Longlists
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$
1967 IMO Longlists
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
1967 IMO Longlists
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
1967 IMO Longlists
Without using tables, find the exact value of the product:

$$P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).$$
1967 IMO Longlists
Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that

$$PA^2 + PB^2 \geq 2r^2.$$
1967 IMO Longlists
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality

$$af^2 + bfg +cg^2 \geq 0$$

holds if and only if the following conditions are fulfilled:

$$a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.$$
1967 IMO Longlists
Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).
1967 IMO Longlists
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
1967 IMO Longlists
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
1967 IMO Longlists
Find values of the parameter $u$ for which the expression

$$y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}$$

does not depend on $x.$
1967 IMO Longlists
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
1967 IMO Longlists
Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.
1967 IMO Longlists
An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.
1967 IMO Longlists
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
1967 IMO Longlists
In what case does the system of equations



$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$



have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
1967 IMO Longlists
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.
1967 IMO Longlists
Prove the identity $$\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x$$for any natural number $n$ and any angle $x.$
1967 IMO Longlists
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
1967 IMO Longlists
Prove that for arbitrary positive numbers the following inequality holds

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.$$
1967 IMO Longlists
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
1967 IMO Longlists
Show that the triangle whose angles satisfy the equality $$ \frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2 $$ is a rectangular triangle.
1967 IMO Longlists
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
1967 IMO Longlists
A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$
1967 IMO Longlists
Decompose the expression into real factors:

$$E = 1 - \sin^5(x) - \cos^5(x).$$
1967 IMO Longlists
The equation

$$x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0$$

is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
1967 IMO Longlists
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
1967 IMO Longlists
(i) Solve the equation:

$$ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.$$

(ii) Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find:



1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon.

2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.
1967 IMO Longlists
If $x,y,z$ are real numbers satisfying relations

$$x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},$$

prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.
1967 IMO Longlists
Prove the following inequality:

$$\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}

x^{n+k-1}_i,$$ where $x_i > 0,$ $k \in \mathbb{N}, n \in

\mathbb{N}.$
1967 IMO Longlists
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$
1967 IMO Longlists
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: $$ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. $$
1967 IMO Longlists
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that

$$\varphi(x,y,z) = f(x+y,z) = g(x,y+z)$$

for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that

$$\varphi(x,y,z) = h(x+y+z)$$

for all real numbers $x,y$ and $z.$
1967 IMO Longlists
A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$
1967 IMO Longlists
In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$

$$r_n \geq Cn^{\beta}, n = 1,2, \ldots$$
1967 IMO Longlists
In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct:



a) The bisector of a given angle.

b) The midpoint of a given rectilinear line segment.

c) The center of a circle through three given non-collinear

points.

d) A line through a given point parallel to a given line.
1967 IMO Longlists
Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?
1967 IMO Longlists
Find all $x$ for which, for all $n,$ $$\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.$$
1967 IMO Longlists
In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.
1967 IMO Longlists
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let

$$ c_n = \sum^8_{k=1} a^n_k$$

for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
1967 IMO Longlists
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$

$$|l(z)| \leq M \rho,$$

where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$
1967 IMO Longlists
On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?
1966 IMO Longlists
Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?
1966 IMO Longlists
Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove $$ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.$$
1966 IMO Longlists
A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.
1966 IMO Longlists
Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.
1966 IMO Longlists
Prove the inequality

$$\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1$$

for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
1966 IMO Longlists
Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$





a.) Prove that $$V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.$$



Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.)



additional question:



b.) Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$



c.) Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.



Note by Darij: I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$
1966 IMO Longlists
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
1966 IMO Longlists
We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?
1966 IMO Longlists
Find $x$ such that trigonometric

$$\frac{\sin 3x \cos (60^\circ -x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0$$

where $m$ is a fixed real number.
1966 IMO Longlists
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
1966 IMO Longlists
Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?
1966 IMO Longlists
Find digits $x, y, z$ such that the equality

$$\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}$$holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
1966 IMO Longlists
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality

$$\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2$$
1966 IMO Longlists
What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?



Posted already on the board I think...
1966 IMO Longlists
Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$
1966 IMO Longlists
We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$
1966 IMO Longlists
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.



a.) Prove that the quadrilateral $MNPQ$ is a parallelogram.



b.) What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?



(Consecutive vertices of the parallelograms are labelled in alphabetical order.
1966 IMO Longlists
Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p}, $ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist.



(Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi $''.)
1966 IMO Longlists
Construct a triangle given the radii of the excircles.
1966 IMO Longlists
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.



a.) What is the volume of this polyhedron ?



b.) Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
1966 IMO Longlists
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality

$$\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3$$

When does equality occur?
1966 IMO Longlists
Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$



Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $%

P^{\prime }.$
1966 IMO Longlists
Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.



(a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right.



(b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.
1966 IMO Longlists
There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)
1966 IMO Longlists
Prove that $$\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.$$
1966 IMO Longlists
Prove the inequality



a.) $

\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(

a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,  $



where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers.



b.) Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality



$$

a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(

a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },

$$



then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.
1966 IMO Longlists
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
1966 IMO Longlists
In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of



a.) all vertices $A$ of such triangles;



b.) all vertices $B$ of such triangles;



c.) all vertices $C$ of such triangles.
1966 IMO Longlists
A given natural number $N$ is being decomposed in a sum of some consecutive integers.



a.) Find all such decompositions for $N=500.$



b.) How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?



c.) Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$



Note by Darij: The $0$ is not considered as a natural number.
1966 IMO Longlists
Let $n$ be a positive integer, prove that :



(a) $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$



(b) $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$
1966 IMO Longlists
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?
1966 IMO Longlists
The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$



Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
1966 IMO Longlists
Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle. From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that the length of one of these tangents equals the sum of the lengths of the two other tangents.
1966 IMO Longlists
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$
1966 IMO Longlists
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
1966 IMO Longlists
Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.
1966 IMO Longlists
Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.



Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle.
1966 IMO Longlists
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
1966 IMO Longlists
Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.



a.) Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.



b.) Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.
1966 IMO Longlists
For a positive real number $p$, find all real solutions to the equation

$$\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.$$
1966 IMO Longlists
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?