(BUL 6) Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
$$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$$
holds, where $A_1$, $B_1$, $C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP$, $BP$, $CP$, and $DP$, respectively.
(CZS 6) (IMO 1) Prove that the sum of an odd number of unit vectors passing through the same point $O$ and lying in the same half-plane whose border passes through $O$ has length greater than or equal to $1$.
(GBR 1) Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there?
(POL 2) (IMO 2) Does there exist a finite set $M$ of points in space, not all in the same plane, such that for each two points $A,B\in M$ there exist two other points $C,D\in M$ such that lines $AB$ and $CD$ are parallel?
(ROM 1) Prove that there are exactly $\binom{k}{[k/2]}$ arrays $a_1,a_2,\ldots,a_{k+1}$ of nonnegative integers such that $a_1=0$ and $|a_i-a_{i+1}|=1$ for $i=1,2,\ldots,k$.
(ROM 2) Let $Ox,Oy,Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox,Oy,Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter?
(SWE 3) (IMO 6) Let $a_1,a_2,\ldots,a_n$ be positive numbers and $q$ a given real number, $0\lt q\lt 1$. Find $n$ real numbers $b_1,b_2,\ldots,b_n$ that satisfy:
(1) $a_k\lt b_k$ for all $k=1,2,\ldots,n$;
(2) $q\lt \frac{b_{k+1}}{b_k}\lt \frac1q$ for all $k=1,2,\ldots,n-1$;
(3) $b_1+b_2+\cdots+b_n\lt \frac{1+q}{1-q}(a_1+a_2+\cdots+a_n)$.
(SWE 4) (IMO 3) Determine the minimum of $a^2+b^2$ if $a$ and $b$ are real numbers for which the equation
$$x^4+ax^3+bx^2+ax+1=0$$
has at least one real solution.
(SWE 6) Consider the two square matrices
$$A=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&-1&-1&1&1\\1&-1&-1&-1&1\\1&1&-1&1&-1\end{bmatrix}\qquad B=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&1&-1&1&-1\\1&-1&-1&1&1\\1&-1&1&-1&1\end{bmatrix}$$
with entries $1$ and $-1$. The following operations will be called elementary:
(1) Changing signs of all numbers in one row;
(2) Changing signs of all numbers in one column;
(3) Interchanging two rows (two rows exchange their positions);
(4) Interchanging two columns.
Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.
(YUG 5) (IMO 4) A soldier has to investigate whether there are mines in an area that has the form of an equilateral triangle. The radius of his detector is equal to one-half of an altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path that the soldier has to traverse in order to check the whole region.
(CUB 2) Given $a,\theta\in\mathbb R$, $m\in\mathbb N$, and $P(x)=x^{2m}-2|a|^m x^m\cos\theta+a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.
(POL 1) (IMO 5) Let $\mathcal F$ be a nonempty set of functions $f:\mathbb R\to\mathbb R$ of the form $f(x)=ax+b$, where $a$ and $b$ are real numbers and $a\ne0$. Suppose that $\mathcal F$ satisfies the following conditions:
(1) If $f,g\in\mathcal F$, then $g\circ f\in\mathcal F$, where $(g\circ f)(x)=g[f(x)]$.
(2) If $f\in\mathcal F$ and $f(x)=ax+b$, then the inverse $f^{-1}$ of $f$ belongs to $\mathcal F$ ($f^{-1}(x)=(x-b)/a$).
(3) None of the functions $f(x)=x+c$, for $c\ne0$, belong to $\mathcal F$.
Prove that there exists $x_0\in\mathbb R$ such that $f(x_0)=x_0$ for all $f\in\mathcal F$.