Navvye/Olympiad-Problems-CSV huggingface dataset:
1979 IMO Longlists
A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$, where $A_1$ lies in the plane $BCD$, $B_1$ in the plane $ACD$, etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$.
1979 IMO Longlists
Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x), g_2(x),\cdots, g_n(x)$ such that

$$f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2$$
1979 IMO Longlists
A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of $n$ days of walking from the camp, under the following conditions:

$(i)$ Each member of the expedition can pick up at most $3$ liters of water.

$(ii)$ Each member must drink one liter of water every day spent in the desert.

$(iii)$ All the members must return to the camp.

How much water do they need (at least) in order to do that?
1979 IMO Longlists
A polynomial $P(x)$ has degree at most $2k$, where $k = 0, 1,2,\cdots$. Given that for an integer $i$, the inequality $-k \le i \le k$ implies $|P(i)| \le 1$, prove that for all real numbers $x$, with $-k \le x \le k$, the following inequality holds:

$$|P(x)| < (2k + 1)\dbinom{2k}{k}$$
1979 IMO Longlists
Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.
1979 IMO Longlists
Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$
1979 IMO Longlists
Let $a, b, c$ denote the lengths of the sides $BC,CA,AB$, respectively, of a triangle $ABC$. If $P$ is any point on the circumference of the circle inscribed in the triangle, show that $aPA^2+bPB^2+cPC^2$ is constant.
1979 IMO Longlists
For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.
1979 IMO Longlists
In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.
1979 IMO Longlists
Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying:

$(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$;

$(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$, $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$

Moreover, for every prime $P$:

$(iii) f_1(P) = 2P,$

$(iv) f_2(P) < 2P.$

Prove that $|f_2(n)| < f_1(n)$ for all $n$.
1979 IMO Longlists
Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions:

$S_1$ is tangent to $CA$ and $AB$,

$S_2$ is tangent to $S_1, AB$, and $BC,$

$S_3$ is tangent to $S_2, BC$, and $CA,$

..............................

$S_7$ is tangent to $S_6, CA$ and $AB.$

Prove that the circles $S_1$ and $S_7$ coincide.
1979 IMO Longlists
Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$.
1979 IMO Longlists
An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$,

$$\frac{1}{N}\sum_{n_j\le N} n_j \le c,$$

for every $N >0$.

Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such

that

$$\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}$$

and

$$m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).$$
1979 IMO Longlists
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
1979 IMO Longlists
Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.
1979 IMO Longlists
Let $M$ be a set and $A,B,C$ given subsets of $M$. Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$. Describe all such sets.
1979 IMO Longlists
Prove that there exists a $k_0\in\mathbb{N}$ such that for every $k\in\mathbb{N},k>k_0$, there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in $k$ regions. Find $k_0$.
1979 IMO Longlists
Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$.
1979 IMO Longlists
There are two non-decreasing sequences $\{a_i\}$ and $\{b_i\}$ of $n$ real numbers each, such that $a_i\le a_{i+1}$ for each $1\le i\le n-1$, and $b_i\le b_{i+1}$ for each $1\le i\le n-1$, and $\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}$ where $m\le n$ with equality for $m=n$. For a convex function $f$ defined on the real numbers, prove that $\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}$.
1979 IMO Longlists
$T$ is a given triangle with vertices $P_1,P_2,P_3$. Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules:

$(\text{i})$ If $V$ = $P_i$, then $n(V) = i$.

$(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$, then $n(V) = i$ or $j$.

$(\text{iii})$ If $V$ lies inside the triangle $T$, then $n(V)$ is any of the numbers $1,2,3$.

Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$.
1979 IMO Longlists
Let the sequence $\{a_i\}$ of $n$ positive reals denote the lengths of the sides of an arbitrary $n$-gon. Let $s=\sum_{i=1}^{n}{a_i}$. Prove that $2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}$.
1979 IMO Longlists
From point $P$ on arc $BC$ of the circumcircle about triangle $ABC$, $PX$ is constructed perpendicular to $BC$, $PY$ is perpendicular to $AC$, and $PZ$ perpendicular to $AB$ (all extended if necessary). Prove that $\frac{BC}{PX}=\frac{AC}{PY}+\frac{AB}{PZ}$.
1979 IMO Longlists
Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.
1979 IMO Longlists
There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$. A side of triangle $T_k$ is equal to $\frac{1}{k}$, $k = 1,2, . . . ,1979$. At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don’t intersect (points of contact are allowed)?
1979 IMO Longlists
Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.
1979 IMO Longlists
In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?
1979 IMO Longlists
Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.
1979 IMO Longlists
Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space.

$(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$.

$(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.
1979 IMO Longlists
Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$. Find the sides of the triangle.
1979 IMO Longlists
By $h(n)$, where $n$ is an integer greater than $1$, let us denote the greatest prime divisor of the number $n$. Are there infinitely many numbers $n$ for which $h(n) < h(n+1)< h(n+2)$ holds?
1979 IMO Longlists
Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.
1979 IMO Longlists
Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties:

$(\text{i})$ $K$ contains three points $A,B,C$, that are the vertices of an acute-angled triangle

$(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$.

Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$.
1979 IMO Longlists
Let $\Pi$ be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped $P_0$  can be decomposed into parallelepipeds $P_1,P_2, . . . ,P_N\in \Pi$, prove that $P_0\in \Pi$.
1978 IMO Longlists
If

$$f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},$$

prove that

$$a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}$$
1978 IMO Longlists
Find all numbers $\alpha$ for which the equation

$$x^2 - 2x[x] + x -\alpha = 0$$

has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)
1978 IMO Longlists
Prove that for any triangle $ABC$ there exists a point P in the plane of the triangle and three points $A' , B'$ , and $C'$ on the lines $BC,

AC$, and $AB$ respectively such that

$$AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,$$

where $M = max\{AB,AC,BC\}$.
1978 IMO Longlists
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.
1978 IMO Longlists
For two given triangles $A_1A_2A_3$ and $B_1B_2B_3$ with areas $\Delta_A$ and $\Delta_B$, respectively, $A_iA_k \ge B_iB_k, i, k = 1, 2, 3$. Prove that $\Delta_A \ge \Delta_B$ if the triangle $A_1A_2A_3$ is not obtuse-angled.
1978 IMO Longlists
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
1978 IMO Longlists
Find all natural numbers $n < 1978$ with the following property: If $m$ is a natural number, $1 < m < n$, and $(m, n) = 1$ (i.e., $m$ and $n$ are relatively prime), then $m$ is a prime number.
1978 IMO Longlists
The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?
1978 IMO Longlists
The satellites $A$ and $B$ circle the Earth in the equatorial plane at altitude $h$. They are separated by distance $2r$, where $r$ is the radius of the Earth. For which $h$ can they be seen in mutually perpendicular directions from some point on the equator?
1978 IMO Longlists
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:

$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.

$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.

$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.

Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
1978 IMO Longlists
Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.
1978 IMO Longlists
Given a natural number $n$, prove that the number $M(n)$ of points with integer coordinates inside the circle $(O(0, 0),\sqrt{n})$ satisfies

$$\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1$$
1978 IMO Longlists
Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$.



Alternative version: A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.
1978 IMO Longlists
A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.
1978 IMO Longlists
Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$.



German IMO Selection Test 1979, problem 2
1978 IMO Longlists
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.
1978 IMO Longlists
Determine the sixth number after the decimal point in the number $(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}$
1978 IMO Longlists
Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy

$$c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{  for any  }x \neq 0, y \neq 0$$

Prove that:

$(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$;

$(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$).

Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.
1978 IMO Longlists
Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.
1978 IMO Longlists
Let the polynomials

$$P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,$$

$$Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,$$

be given satisfying the identity $P(x)^2 = (x^2 - 1)Q(x)^2 + 1$. Prove the identity

$$P'(x) = nQ(x).$$
1978 IMO Longlists
Let $\mathcal{C}$ be the circumcircle of the square with vertices $(0, 0), (0, 1978), (1978, 0), (1978, 1978)$ in the Cartesian plane. Prove that $\mathcal{C}$ contains no other point for which both coordinates are integers.
1978 IMO Longlists
A sequence $(a_n)^{\infty}_0$ of real numbers is called convex if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.
1978 IMO Longlists
A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$.

$(a)$ Prove that there exists a constant $C >0$ such that

$$\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)$$

for all concave positive sequences $(a_n)^N_0$

$(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best

possible.
1978 IMO Longlists
The integers $1$ through $1000$ are located on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e.,$1, 16, 31, \cdots$ ) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?
1978 IMO Longlists
Simplify

$$\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},$$

where $a, b, c$ are positive real numbers.
1978 IMO Longlists
Given a circle, construct a chord that is trisected by two given noncollinear radii.
1978 IMO Longlists
$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.
1978 IMO Longlists
If $C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)$, prove the identity

$$C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}$$

and then evaluate the sum

$$S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.$$
1978 IMO Longlists
$A,B,C,D,E$ are points on a circle $O$ with radius equal to $r$. Chords $AB$ and $DE$ are parallel to each other and have length equal to $x$. Diagonals $AC,AD,BE, CE$ are drawn. If segment $XY$ on $O$ meets $AC$ at $X$ and $EC$ at $Y$ , prove that lines $BX$ and $DY$ meet at $Z$ on the circle.
1978 IMO Longlists
If $p$ is a prime greater than $3$, show that at least one of the numbers

$$\frac{3}{p^2} , \frac{4}{p^2} , \cdots, \frac{p-2}{p^2}$$

is expressible in the form $\frac{1}{x} + \frac{1}{y}$, where $x$ and $y$ are positive integers.
1978 IMO Longlists
In  $ABC$ with $\angle C = 60^{\circ}$, prove that

$$\frac{c}{a} + \frac{c}{b} \ge2.$$
1978 IMO Longlists
If $r > s >0$ and $a > b > c$, prove that

$$a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.$$
1978 IMO Longlists
Given the expression

$$P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],$$

prove:

$(a) P_n(x)$ satisfies the identity

$$P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.$$

$(b) P_n(x)$ is a polynomial in $x$ of degree $n.$
1978 IMO Longlists
Let $A,B,C,D$ be four arbitrary distinct points in space.

$(a)$ Prove that using the segments $AB +CD, AC +BD$ and $AD +BC$, it is always possible to construct a triangle $T$ that is non-degenerate and has no obtuse angle.

$(b)$ What should these four points satisfy in order for the triangle $T$ to be right-angled?
1978 IMO Longlists
A variable tetrahedron $ABCD$ has the following properties:

Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?
1978 IMO Longlists
Find the relations among the angles of the triangle $ABC$ whose altitude $AH$ and median $AM$ satisfy $\angle BAH =\angle CAM$.
1978 IMO Longlists
Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$
1977 IMO Longlists
A pentagon $ABCDE$ inscribed in a circle for which $BC<CD$ and $AB<DE$ is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, prove that $BS>CS$.
1977 IMO Longlists
In a company of $n$ persons, each person has no more than $d$ acquaintances, and in that company there exists a group of $k$ persons, $k\ge d$, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than $\left[ \frac{n^2}{4}\right]$.
1977 IMO Longlists
We are given $n$ points in space. Some pairs of these points are connected by line segments so that the number of segments equals $[n^2/4],$ and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle.
1977 IMO Longlists
Let $x_1, x_2, \ldots , x_n  \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi,  \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$
1977 IMO Longlists
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that

$$ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,$$

then either all these numbers are nonnegative or all these numbers are nonpositive.
1977 IMO Longlists
A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.
1977 IMO Longlists
Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.
1977 IMO Longlists
Let $n$ and $z$ be integers greater than $1$ and $(n,z)=1$. Prove:

(a) At least one of the numbers $z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,$ is divisible by $n$.

(b) If $(z-1,n)=1$, then at least one of the numbers $z_i$ is divisible by $n$.
1977 IMO Longlists
Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \  k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$
1977 IMO Longlists
Let $n$ be an integer greater than $1$. In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$. Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$.
1977 IMO Longlists
A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Moreover, for two balls $K_1$ and $K_2$ tangent to $K$ and tangent to each other there exist two other balls tangent to $K_1,K_2$ and also to $K$. How many balls are tangent to $K$? For a given $r$ determine $R$.
1977 IMO Longlists
Given an isosceles triangle $ABC$ with a right angle at $C,$ construct the center $M$ and radius $r$ of a circle cutting on segments $AB, BC, CA$ the segments $DE, FG,$ and $HK,$ respectively, such that $\angle DME + \angle FMG + \angle HMK = 180^\circ$ and $DE : FG : HK = AB : BC : CA.$
1977 IMO Longlists
Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^n$ are a sequence $a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_i$ is even, then $a_{i-1}$ is odd, and if $a_i$ is odd, then $a_{i-1}$ is even).
1977 IMO Longlists
Given that $x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,$ prove that:

$$ \frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}$$
1977 IMO Longlists
Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation

$$f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).$$