Navvye/Olympiad-Problems-CSV huggingface dataset:
1971 IMO Longlists
Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$



Show that if there exists a right circular cone with vertex $V$, with the properties:



(1) its axis passes through $O$, and



(2) its curved surface passes through $A,B,C$ and $D,$ then

$$OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.$$



Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties (1) and (2).
1971 IMO Longlists
Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that

$$(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.$$
1971 IMO Longlists
In a triangle $P_1P_2P_3$ let $P_iQ_i$ be the altitude from $P_i$ for $i = 1, 2,3$ ($Q_i$ being the foot of the altitude). The circle with diameter $P_iQ_i$ meets the two corresponding sides at two points different from $P_i.$ Denote the length of the segment whose endpoints are these two points by $l_i.$ Prove that $l_1 = l_2 = l_3.$
1971 IMO Longlists
Let $M$ be the circumcenter of a triangle $ABC.$ The line through $M$ perpendicular to $CM$ meets the lines $CA$ and $CB$ at $Q$ and $P,$ respectively. Prove that

$$\frac{\overline{CP}}{\overline{CM}} \cdot \frac{\overline{CQ}}{\overline{CM}}\cdot \frac{\overline{AB}}{\overline{PQ}}= 2.$$
1971 IMO Longlists
We are given an $n \times n$ board, where $n$ is an odd number. In each cell of the board either $+1$ or $-1$ is written. Let $a_k$ and $b_k$ denote them products of numbers in the $k^{th}$ row and in the $k^{th}$ column respectively. Prove that the sum $a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n$ cannot be equal to zero.
1971 IMO Longlists
Find all integer solutions of the equation

$$x^2+y^2=(x-y)^3.$$
1971 IMO Longlists
Let $A, B,$ and $C$ denote the angles of a triangle. If $\sin^2 A + \sin^2 B + \sin^2 C = 2$, prove that the triangle is right-angled.
1971 IMO Longlists
Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.
1971 IMO Longlists
An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates $(0, 0), (p, 0), (p, q), (0, q)$ for some positive integers $p, q$. Show that there must exist two among them one of which is entirely contained in the other.
1971 IMO Longlists
A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.
1971 IMO Longlists
Prove that the system of equations

$$2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, $$

$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
1971 IMO Longlists
Two half-lines $a$ and $b$, with the common endpoint $O$, make an acute angle $\alpha$. Let $A$ on $a$ and $B$ on $b$ be points such that $OA=OB$, and let $b$ be the line through $A$ parallel to $b$. Let $\beta$ be the circle with centre $B$ and radius $BO$. We construct a sequence of half-lines $c_1,c_2,c_3,\ldots $, all lying inside the angle $\alpha$, in the following manner:

(i) $c_i$ is given arbitrarily;

(ii) for every natural number $k$, the circle $\beta$ intercepts on $c_k$ a segment that is of the same length as the segment cut on $b'$ by $a$ and $c_{k+1}$.

Prove that the angle determined by the lines $c_k$ and $b$ has a limit as $k$ tends to infinity and find that limit.
1971 IMO Longlists
A square $2n\times 2n$ grid is given. Let us consider all possible paths along grid lines, going from the centre of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its centre at the grid centre is passed through only once.

(a) Prove that the number of all such paths is equal to $4\prod_{i=2}^n(16i-9)$.

(b) Find the number of pairs of such paths that divide the grid into two congruent figures.

(c) How many quadruples of such paths are there that divide the grid into four congruent parts?
1971 IMO Longlists
Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$. Let $N(\alpha )=n$ denote the number of elements in $\alpha$. We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$.

Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$.

Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$.
1971 IMO Longlists
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
1971 IMO Longlists
Consider the set of grid points $(m,n)$ in the plane, $m,n$ integers. Let $\sigma$ be a finite subset and define

$$S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|) $$

Find the maximum of $S$, taken over the set of all such subsets $\sigma$.
1971 IMO Longlists
Let $L_i,\ i=1,2,3$, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths $l_i,\ i=1,2,3$. By $L_i^{\ast}$ we denote the segment of length $l_i$ with its midpoint on the midpoint of the corresponding side of the triangle. Let $M(L)$ be the set of points in the plane whose orthogonal projections on the sides of the triangle are in $L_1,L_2$, and $L_3$, respectively; $M(L^{\ast})$ is defined correspondingly. Prove that if $l_1\ge l_2+l_3$, we have that the area of $M(L)$ is less than or equal to the area of $M(L^{\ast})$.
1971 IMO Longlists
Show that for nonnegative real numbers $a,b$ and integers $n\ge 2$,

$$\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n$$

When does equality hold?
1971 IMO Longlists
Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.
1971 IMO Longlists
A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?
1971 IMO Longlists
The diagonals of a convex quadrilateral $ABCD$ intersect at a point $O$. Find all angles of this quadrilateral if $\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}$, and $\measuredangle OAD=30^{\circ}$.
1971 IMO Longlists
Suppose that the sides $AB$ and $DC$ of a convex quadrilateral $ABCD$ are not parallel. On the sides $BC$ and $AD$, pairs of points $(M,N)$ and $(K,L)$ are chosen such that $BM=MN=NC$ and $AK=KL=LD$. Prove that the areas of triangles $OKM$ and $OLN$ are different, where $O$ is the intersection point of $AB$ and $CD$.
1971 IMO Longlists
Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.
1971 IMO Longlists
A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied:

$(1)$ every member of the set $P$ knows every other member of the set $P$;

$(2)$ no member of the set $P$ knows any other member of the set $P$.
1971 IMO Longlists
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying

$$|\lambda |+|\mu |+|\nu |\le \sqrt{2} $$
1970 IMO Longlists
Prove that $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le \frac{a+b+c}{2}$, where $a,b,c\in\mathbb{R}^{+}$.
1970 IMO Longlists
Prove that the two last digits of $9^{9^{9}}$ and $9^{9^{9^{9}}}$ are the same in decimal representation.
1970 IMO Longlists
Prove that $(a!\cdot b!) | (a+b)!$ $\forall a,b\in\mathbb{N}$.
1970 IMO Longlists
Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$.

$$x^2 + xy = a^2 + ab$$ $$y^2 + xy = a^2 - ab$$
1970 IMO Longlists
Prove that $\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1$ for $2 \le n \in \mathbb{N}$.
1970 IMO Longlists
There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.
1970 IMO Longlists
Let $ABCD$ be an arbitrary quadrilateral. Squares with centers $M_1, M_2, M_3, M_4$ are constructed on $AB,BC,CD,DA$ respectively, all outwards or all inwards. Prove that $M_1 M_3=M_2 M_4$ and $M_1 M_3\perp M_2 M_4$.
1970 IMO Longlists
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that

$$\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.$$
1970 IMO Longlists
For even $n$, prove that $\sum_{i=1}^{n}{\left((-1)^{i+1}\cdot\frac{1}{i}\right)}=2\sum_{i=1}^{n/2}{\frac{1}{n+2i}}$.
1970 IMO Longlists
In $\triangle ABC$, prove that $1< \sum_{cyc}{\cos A}\le \frac{3}{2}$.
1970 IMO Longlists
Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.
1970 IMO Longlists
Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$.

$(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other.

$(b)$ Generalize the result to every prime number.
1970 IMO Longlists
Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:

$(1)$ $A,B,C$ are assigned $1,2,3$ respectively

$(2)$ Points on $AB$ are assigned $1$ or $2$

$(3)$ Points on $BC$ are assigned $2$ or $3$

$(4)$ Points on $CA$ are assigned $3$ or $1$

Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.
1970 IMO Longlists
Let $\alpha + \beta +\gamma = \pi$. Prove that $\sum_{cyc}{\sin 2\alpha} = 2\cdot \left(\sum_{cyc}{\sin \alpha}\right)\cdot\left(\sum_{cyc}{\cos \alpha}\right)- 2\sum_{cyc}{\sin \alpha}$.
1970 IMO Longlists
Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$.

Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.
1970 IMO Longlists
Show that the equation $\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$ has no real roots.
1970 IMO Longlists
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: $$ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). $$ When do we have equality?
1970 IMO Longlists
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
1970 IMO Longlists
Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$, where $\alpha _i\le \frac{1}{i(i+1)}$. Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$.
1970 IMO Longlists
Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that

$$ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}$$

($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).
1970 IMO Longlists
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
1970 IMO Longlists
Let $E$ be a finite set, $P_E$ the family of its subsets, and $f$ a mapping from $P_E$ to the set of non-negative reals, such that for any two disjoint subsets $A,B$ of $E$, $f(A\cup B)=f(A)+f(B)$. Prove that there exists a subset $F$ of $E$ such that if with each $A \subset E$, we associate a subset $A'$ consisting of elements of $A$ that are not in $F$, then $f(A)=f(A')$ and $f(A)$ is zero if and only if $A$ is a subset of $F$.
1970 IMO Longlists
Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.
1970 IMO Longlists
A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.
1970 IMO Longlists
Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.
1970 IMO Longlists
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.
1970 IMO Longlists
A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled:

$(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$.

$(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$.

$(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x =  v$, and an element $y\in G$ such that $y\circ u = v$.



Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.
1970 IMO Longlists
Prove that the equation $4^x +6^x =9^x$ has no rational solutions.
1970 IMO Longlists
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that

$$1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .$$
1970 IMO Longlists
Prove that for any triangle with sides $a, b, c$ and area $P$ the following inequality holds:

$$P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.$$

Find all triangles for which equality holds.
1970 IMO Longlists
Let there be given an acute angle $\angle AOB = 3\alpha$, where $\overline{OA}= \overline{OB}$. The point $A$ is the center of a circle with radius $\overline{OA}$. A line $s$ parallel to $OA$ passes through $B$. Inside the given angle a variable line $t$ is drawn through $O$. It meets the circle in $O$ and $C$ and the given line $s$ in $D$, where $\angle AOC = x$. Starting from an arbitrarily chosen position $t_0$ of $t$, the series $t_0, t_1, t_2, \ldots$ is determined by defining $\overline{BD_{i+1}}=\overline{OC_i}$ for each $i$ (in which $C_i$ and $D_i$ denote the positions of $C$ and $D$, corresponding to $t_i$). Making use of the graphical representations of $BD$ and $OC$ as functions of $x$, determine the behavior of $t_i$ for $i\to \infty$.
1970 IMO Longlists
The vertices of a given square are clockwise lettered $A,B,C,D$. On the side $AB$ is situated a point $E$ such that $AE = AB/3$. Starting from an arbitrarily chosen point $P_0$ on segment $AE$ and going clockwise around the perimeter of the square, a series of points $P_0, P_1, P_2, \ldots$ is marked on the perimeter such that $P_iP_{i+1} = AB/3$ for each $i$. It will be clear that when $P_0$ is chosen in $A$ or in $E$, then some $P_i$ will coincide with $P_0$. Does this possibly also happen if $P_0$ is chosen otherwise?
1970 IMO Longlists
In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
1970 IMO Longlists
Find for every value of $n$ a set of numbers $p$ for which the following statement is true: Any convex $n$-gon can be divided into $p$ isosceles triangles.
1970 IMO Longlists
Let $x, y, z$ be non-negative real numbers satisfying

$$x^2 + y^2 + z^2 = 5 \quad \text{ and } \quad yz + zx + xy = 2.$$

Which values can the greatest of the numbers $x^2 -yz, y^2 - xz$ and $z^2 - xy$ have?
1970 IMO Longlists
Solve the set of simultaneous equations

\begin{align*}

v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\
u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\
u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\
u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\
u^2+ v^2+ w^2+ x^2 &= 6- 2y.

\end{align*}
1970 IMO Longlists
Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$.
1970 IMO Longlists
$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.
1970 IMO Longlists
Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.
1970 IMO Longlists
Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.
1970 IMO Longlists
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
1970 IMO Longlists
Prove that the equation

$$x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0$$

has one root $x_1 = \tan \frac{\pi}{36}$, and find the other roots.
1970 IMO Longlists
If $a, b, c$ are side lengths of a triangle, prove that

$$(a + b)(b + c)(c + a) \geq  8(a + b - c)(b + c - a)(c + a - b).$$
1970 IMO Longlists
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.



(a) Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.



(b) Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
1970 IMO Longlists
Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$.
1970 IMO Longlists
Given a polynomial

$$P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2  +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),$$where $a, b, c \neq 0$, prove that $P(x)$ is divisible by

$$Q(x) = abx^2 + (a^2 + b^2)x + ab$$and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.
1970 IMO Longlists
Let a polynomial $p(x)$ with integer coefficients take the value $5$ for five different integer values of $x.$ Prove that $p(x)$ does not take the value $8$ for any integer $x.$
1970 IMO Longlists
For $n \in \mathbb N$, let $f(n)$ be the number of positive integers $k \leq n$ that do not contain the digit $9$. Does there exist a positive real number $p$ such that $\frac{f(n)}{n} \geq p$ for all positive integers $n$?
1970 IMO Longlists
The area of a triangle is $S$ and the sum of the lengths of its sides is $L$. Prove that $36S \leq L^2\sqrt 3$ and give a necessary and sufficient condition for equality.
1970 IMO Longlists
Let $p$ be a prime number. A rational number $x$, with $0 < x < 1$, is written in lowest terms. The rational number obtained from $x$ by adding $p$ to both the numerator and the denominator differs from $x$ by $1/p^2$. Determine all rational numbers $x$ with this property.
1970 IMO Longlists
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.



a.) Prove that $0\le b_n<2$.



b.) Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
1970 IMO Longlists
A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.
1970 IMO Longlists
Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$
1970 IMO Longlists
A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha <  \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
1970 IMO Longlists
A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L >\sqrt{2a^2+h^2}$, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).