Navvye/Olympiad-Problems-CSV huggingface dataset:
1974 IMO Longlists
Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$,

$$n^n\le g(2n)\le (2n)^{2n}$$
1974 IMO Longlists
Let $C_1$ and $C_2$ be circles in the same plane, $P_1$ and $P_2$ arbitrary points on $C_1$ and $C_2$ respectively, and $Q$ the midpoint of segment $P_1P_2.$ Find the locus of points $Q$ as $P_1$ and $P_2$ go through all possible positions.

Alternative version. Let $C_1, C_2, C_3$ be three circles in the same plane. Find the locus of the centroid of triangle $P_1P_2P_3$ as $P_1, P_2,$ and $P_3$ go through all possible positions on $C_1, C_2$, and $C_3$ respectively.
1974 IMO Longlists
Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$. We define the following elementary operation on $P$:

Choose $i,j\in\{1,2,\ldots ,k\}$, such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$. Then take $b$ elements from $M_i$ and place them into $M_j$, i.e., $M_j$ becomes the union of itself and a $b$-element subset of $M_i$, while the same subset is subtracted from $M_i$ (if $a=b$, $M_i$ is thus removed from the partition).

Let a finite set $M$ be given. Prove that the property “for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$” is equivalent to “the number of elements of $M$ is a power of $2$.”
1974 IMO Longlists
Let $A,B,C,D$ be points in space. If for every point $M$ on the segment $AB$ the sum

$$S_{AMC}+S_{CMD}+S_{DMB}$$

Is constant show that the points $A,B,C,D$ lie in the same plane.



Note.Note. $S_X$ denotes the area of triangle $X.$
1974 IMO Longlists
Let $y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}$ where $\alpha \neq 0, y > 0, x_i > 0$ are real numbers, and let $\lambda \neq  \alpha$ be a real number. Prove that $y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) > 0,$ and $y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) < 0.$
1974 IMO Longlists
Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.
1974 IMO Longlists
Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by

$$a_0 = a, \quad  a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.$$

Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that

$$0 < n(A - a_n) < A^3.$$
1974 IMO Longlists
Consider infinite diagrams

[asy]

import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; 

label("$n_{00} \ n_{01} \ n_{02} \ldots$", (1.14,1.38), SE*lsf); label("$n_{10} \ n_{11} \ n_{12} \ldots$", (1.2,1.8), SE*lsf); label("$n_{20} \ n_{21} \ n_{22} \ldots$", (1.2,2.2), SE*lsf); label("$\vdots \quad \vdots \qquad \vdots $", (1.32,2.72), SE*lsf);

draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);

[/asy]

where all but a finite number of the integers $n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,$ are equal to $0$. Three elements of a diagram are called adjacent if there are integers $i$ and $j$ with $i \geq 0$ and $j \geq 0$ such that the three elements are



(i) $n_{ij}, n_{i,j+1}, n_{i,j+2},$ or



(ii) $n_{ij}, n_{i+1,j}, n_{i+2,j} ,$ or



(iii) $n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.$



An elementary operation on a diagram is an operation by which three adjacent elements $n_{ij}$ are changed into $n_{ij}'$ in such a way that $|n_{ij}-n_{ij}'|=1.$ Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?
1974 IMO Longlists
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property:

If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$
1974 IMO Longlists
Consider the binomial coefficients $\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)$. Determine all positive integers $n$ for which $\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.
1974 IMO Longlists
Let $a, b$, and $c$ denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side $a$ and then propelled toward side $b$ with direction defined by the angle $\theta$. For what values of $\theta$ will the ball strike the sides $b, c, a$ in that order?
1974 IMO Longlists
The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:



(a) rectangles with vertices on the lattice and sides parallel to the coordinate axes;



(b) squares with vertices on the lattice and sides parallel to the coordinate axes;



(c) squares in total, with vertices on the lattice.
1974 IMO Longlists
Let $n$ be a positive integer, $n \geq 2$, and consider the polynomial equation

$$x^n - x^{n-2} - x + 2 = 0.$$

For each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$
1974 IMO Longlists
Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that

$$A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.$$
1974 IMO Longlists
An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$
1974 IMO Longlists
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
1974 IMO Longlists
Outside an arbitrary triangle $ABC$, triangles $ADB$ and $BCE$ are constructed such that $\angle ADB=\angle BEC=90^{\circ}$ and $\angle DAB=\angle EBC=30^{\circ}$. On the segment $AC$ the point $F$ with $AF=3FC$ is chosen. Prove that $\angle DFE=90^{\circ}$ and $\angle FDE=30^{\circ}$.
1974 IMO Longlists
Given two points $A,B$ outside of a given plane $P,$ find the positions of points $M$ in the plane $P$ for which the ratio $\frac{MA}{MB}$ takes a minimum or maximum.
1974 IMO Longlists
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
1974 IMO Longlists
Let $m$ and $n$ be natural numbers with $m>n$. Prove that

$$2(m-n)^2(m^2-n^2+1)\ge 2m^2-2mn+1$$
1974 IMO Longlists
There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.
1974 IMO Longlists
A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that:

(a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves.

(b) If $2u\le v$, the rabbit can always run away from the fox.
1973 IMO Longlists
Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.
1973 IMO Longlists
Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "between these rays" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron $OABC$.
1973 IMO Longlists
Is the number

$$\sqrt[3]{\sqrt 5 + 2} + \sqrt[3]{\sqrt 5 - 2}$$

rational or irrational?
1973 IMO Longlists
A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions.



Note. For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$, such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle − a so-called Monge circle − with equation $x^2 + y^2 = a^2 + b^2$.
1973 IMO Longlists
Given a ball $K$. Find the locus of the vertices $A$ of all parallelograms $ABCD$ such that $ AC \leq BD$, and the diagonal $BD$ lies completely inside the ball $K$.
1973 IMO Longlists
Let $P_i (x_i, y_i)$ (with $i = 1, 2, 3, 4, 5$) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.
1973 IMO Longlists
Given a tetrahedron $ABCD$. Let $x = AB \cdot CD, y = AC \cdot BD$ and $z = AD\cdot BC$. Prove that there exists a triangle with the side lengths $x, y$ and $z$.
1973 IMO Longlists
Let $a$ be a non-zero real number. For each integer $n$, we define $S_n = a^n + a^{-n}$. Prove that if for some integer $k$, the sums $S_k$ and $S_{k+1}$ are integers, then the sums $S_n$ are integers for all integers $n$.
1973 IMO Longlists
Prove that $2^{147} - 1$ is divisible by $343$.
1972 IMO Longlists
Find all integer solutions of the equation

$$1 + x + x^2 + x^3 + x^4 = y^4.$$
1972 IMO Longlists
Find all real values of the parameter $a$ for which the system of equations

$$x^4 = yz - x^2 + a,$$

$$y^4 = zx - y^2 + a,$$

$$z^4 = xy - z^2 + a,$$

has at most one real solution.
1972 IMO Longlists
On a line a set of segments is given of total length less than $n$. Prove that every set of $n$ points of the line can be translated in some direction along the line for a distance smaller than $\frac{n}{2}$ so that none of the points remain on the segments.
1972 IMO Longlists
You have a triangle, $ABC$.  Draw in the internal angle trisectors.  Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$.  Prove that $DEF$ is equilateral.
1972 IMO Longlists
Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.
1972 IMO Longlists
Prove the inequality

$$(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1$$

for all natural numbers $n \ge 2.$
1972 IMO Longlists
Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.
1972 IMO Longlists
Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.
1972 IMO Longlists
A circle $k = (S, r)$ is given and a hexagon $AA'BB'CC'$ inscribed in it. The lengths of sides of the hexagon satisfy $AA' = A'B, BB' = B'C, CC' = C'A$. Prove that the area $P$ of triangle $ABC$ is not greater than the area $P'$ of triangle $A'B'C'$. When does $P = P'$ hold?
1972 IMO Longlists
Given a sphere $K$, determine the set of all points $A$ that are vertices of some parallelograms $ABCD$ that satisfy $AC \le BD$ and whose entire diagonal $BD$ is contained in $K$.
1972 IMO Longlists
$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that

$$p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,$$

where $a$ is the length of an edge of a tetrahedron.

$(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.



Note: Part $(b)$ is IMO 1972 Problem 6
1972 IMO Longlists
Consider the set $S$ of all the different odd positive integers that are not multiples of $5$ and that are less than $30m, m$ being a positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two integers one of which divides the other? Prove your result.
1972 IMO Longlists
A solid right circular cylinder with height $h$ and base-radius $r$ has a solid hemisphere of radius $r$ resting upon it. The center of the hemisphere $O$ is on the axis of the cylinder. Let $P$ be any point on the surface of the hemisphere and $Q$ the point on the base circle of the cylinder that is furthest from $P$ (measuring along the surface of the combined solid). A string is stretched over the surface from $P$ to $Q$ so as to be as short as possible. Show that if the string is not in a plane, the straight line $PO$ when produced cuts the curved surface of the cylinder.
1972 IMO Longlists
We have $p$ players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers $s_1 \le s_2 \le s_3 \le\cdots\le s_p$ is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if

$$(i)\displaystyle\sum_{i=1}^{p} s_i =\frac{1}{2}p(p-1)$$

$$\text{and}$$

$$(ii)\text{  for all  }k < p,\displaystyle\sum_{i=1}^{k} s_i \ge \frac{1}{2} k(k - 1).$$
1972 IMO Longlists
Let $S$ be a subset of the real numbers with the following

properties:

$(i)$ If $x \in S$ and $y \in S$, then $x - y \in S$;

$(ii)$ If $x \in S$ and $y \in S$, then $xy \in S$;

$(iii)$ $S$ contains an exceptional number $x'$ such that there is no number $y$ in $S$ satisfying $x'y + x' + y = 0$;

$(iv)$ If $x \in S$ and $x \neq x'$ , there is a number $y$ in $S$ such that $xy+x+y = 0$.

Show that

$(a)$ $S$ has more than one number in it;

$(b)$ $x' \neq -1$ leads to a contradiction;

$(c)$ $x \in S$ and $x \neq 0$ implies $1/x \in S$.
1972 IMO Longlists
Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds:

$$\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?$$
1972 IMO Longlists
The diagonals of a convex $18$-gon are colored in $5$ different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered $1, 2,\cdots$. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly $n$ pairs of diagonals of the same color and with fixed indices $i, j$.
1972 IMO Longlists
We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$
1972 IMO Longlists
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
1972 IMO Longlists
Let $A,B,C$ be points on the sides $B_1C_1, C_1A_1,A_1B_1$ of a triangle $A_1B_1C_1$ such that $A_1A,B_1B,C_1C$ are the bisectors of angles of the triangle. We have that $AC = BC$ and $A_1C_1 \neq B_1C_1.$

$(a)$ Prove that $C_1$ lies on the circumcircle of the triangle $ABC$.

$(b)$ Suppose that $\angle BAC_1 =\frac{\pi}{6};$ find the form of triangle $ABC$.
1972 IMO Longlists
Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.
1972 IMO Longlists
If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$,

show that

$$max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},$$

where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.
1972 IMO Longlists
A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$.

$(a)$ Prove that

$$U(n + 1)+U(n -1) = 4U(n);$$

$(b)$ Prove that

$$U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].$$
1972 IMO Longlists
If $p$ is a prime number greater than $2$ and $a, b, c$ integers not divisible by $p$, prove that the equation

$$ax^2 + by^2 = pz + c$$

has an integer solution.
1972 IMO Longlists
$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$,

$$(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{  and}$$

$$(ii) 2 \le \frac{4m^2}{1+m^2} < 4.$$

$(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$
1972 IMO Longlists
A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.
1972 IMO Longlists
On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?
1972 IMO Longlists
Congruent rectangles with sides $m(cm)$ and $n(cm)$ are given ($m, n$ positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
1972 IMO Longlists
How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?
1972 IMO Longlists
Prove the inequalities

$$\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{  for  }0 \le u < v \le \frac{\pi}{2}$$
1972 IMO Longlists
The ternary expansion $x = 0.10101010\cdots$ is given. Give the binary expansion of $x$.

Alternatively, transform the binary expansion $y = 0.110110110 \cdots$ into a ternary expansion.
1972 IMO Longlists
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
1972 IMO Longlists
A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.
1972 IMO Longlists
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.
1972 IMO Longlists
Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.
1971 IMO Longlists
Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ($1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.
1971 IMO Longlists
Let $a, b, c$ be positive real numbers, $0 < a \leq b \leq c$. Prove that for any positive real numbers $x, y, z$ the following inequality holds:

$$(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.$$
1971 IMO Longlists
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$
1971 IMO Longlists
Let squares be constructed on the sides $BC,CA,AB$ of a triangle $ABC$, all to the outside of the triangle, and let $A_1,B_1, C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one analogously obtains a triangle $A_2B_2C_2$. If $S, S_1, S_2$ denote the areas of triangles$ ABC,A_1B_1C_1,A_2B_2C_2$, respectively, prove that $S = 8S_1 - 4S_2.$
1971 IMO Longlists
In a triangle $ABC$, let $H$ be its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Prove that:



(a) $|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma}$ where $\alpha, \beta, \gamma$ are angles of the triangle $ABC;$



(b) $O \equiv  H$ if and only if $ABC$ is equilateral.
1971 IMO Longlists
The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$
1971 IMO Longlists
In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?
1971 IMO Longlists
A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that

$$x_1 = \log_{x_{n-1}} x_n,  x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.$$

Prove that $\prod_{k=1}^n x_k =1.$
1971 IMO Longlists
One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation:



Martian : I have spent $1/12$ of my life on Pluton.

Human : I also have.

Venusian : Me too.

Martian : But Venusian and I have spend much more time here than you, Human.

Human : That is true. However, Venusian and I are of the same age.

Venusian : Yes, I have lived $300$ Earth years.

Martian : Venusian and I have been on Pluton for the past $13$ years.



It is known that Human and Martian together have lived $104$ Earth years. Find the ages of Martian, Venusian, and Human.*



**: Note that the numbers in the problem are not necessarily in base $10.$
1971 IMO Longlists
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.