Let $ABCD$ be a tetrahedron with $BC=a$, $CA=b$, $AB=c$, $DA=a_1$, $DB=b_1$, $DC=c_1$. Prove that there is a unique point $P$ satisfying
$$PA^2+a_1^2+b^2+c^2=PB^2+b_1^2+c^2+a^2=PC^2+c_1^2+a^2+b^2=PD^2+a_1^2+b_1^2+c_1^2.$$
For this point prove $PA^2+PB^2+PC^2+PD^2\ge4R^2$, where $R$ is the circumradius of the tetrahedron. Find a necessary and sufficient condition on its edge lengths for equality.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
A country has $25$ cities. Determine the smallest integer $k$ for which two-way direct air routes can be established satisfying both conditions:
1. Every city has direct routes to exactly $k$ other cities.
2. Whenever two cities have no direct route between them, there is at least one city having direct routes to both.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Find the greatest real number $\alpha$ for which there exists an infinite sequence of natural numbers $(a_n)_{n\ge1}$ satisfying both:
1. $a_n>1997^n$ for every positive integer $n$.
2. For every $n\ge2$, $u_n\ge a_n^\alpha$, where $u_n$ is the greatest common divisor of all numbers $a_i+a_k$ with positive integers $i,k$ satisfying $i+k=n$.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $f:\mathbb N\to\mathbb Z$ satisfy $f(0)=2$, $f(1)=503$ and
$$f(n+2)=503f(n+1)-1996f(n)\qquad(n\in\mathbb N).$$
For any positive integer $k$, choose integers $s_1,\ldots,s_k$ with $s_i\ge k$ for every $i$. For each $s_i$, choose any prime divisor $p(s_i)$ of $f(2^{s_i})$. Prove that for every positive integer $t\le k$,
$$2^t\mid\sum_{i=1}^k p(s_i)\quad\Longleftrightarrow\quad2^t\mid k.$$
Here $\mathbb N$ includes $0$; the $s_i$ need not be distinct.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Determine all pairs of real numbers $a,b$ such that for every positive integer $n$ and every real solution $x_n$ of
$$4n^2x=\log_2(2n^2x+1),$$
one has
$$a^{x_n}+b^{x_n}\ge2+3x_n.$$
The powers and the logarithm must be defined as real numbers.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $n,k,p$ be positive integers with $k\ge2$ and $k(p+1)\le n$. There are $n$ distinct fixed points on a circle. Color every point either blue or red so that exactly $k$ points are blue and every clockwise arc between consecutive blue points contains at least $p$ red points. How many different colorings are there? Two colorings are different if at least one fixed point has different colors in them.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $S$ be a fixed circle of radius $R$, with center $O$. A fixed point $P$ lies inside it, with $OP=d\lt R$. A variable quadrilateral $ABCD$ has its vertices on $S$, and its diagonals $AC,BD$ intersect at $P$ and are perpendicular. Determine the largest and smallest possible perimeter of $ABCD$ in terms of $R,d$.
Let $n\gt 1$ be an integer not divisible by $1997$. Put
$$a_m=m+\frac{mn}{1997}\quad(m=1,2,\ldots,1996),$$
$$b_m=m+\frac{1997m}{n}\quad(m=1,2,\ldots,n-1).$$
Arrange all the numbers $a_i,b_j$ in increasing order. Prove that the difference between any two consecutive terms is less than $2$.
How many positive-integer-valued functions $f(n)$ defined on the positive integers satisfy $f(1)=1$ and
$$f(n)f(n+2)=f(n+1)^2+1997$$
for every positive integer $n$?
Let $k=\sqrt[3]{3}$. Find a polynomial $p(x)$ with rational coefficients and of the smallest possible degree such that $p(k+k^2)=3+k$. Does there exist a polynomial $q(x)$ with integer coefficients satisfying $q(k+k^2)=3+k$?
There are $75$ points in a unit cube, no three of which are collinear. Prove that three of them can be chosen to form a triangle of area at most $7/72$.