MathNet 1999 — Part 01

Determine all pairs $(a, b)$ of integers with the property that the numbers $a^{2}+4b$ and $b^{2}+4a$ are both perfect squares.

Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ in its exterior. Prove that the number of good circles has the same parity as $n$.

Find the smallest positive integer $n$ with the following property: There does not exist an arithmetic progression of $1999$ terms of real numbers containing exactly $n$ integers.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_{1}$ and $\Gamma_{2}$ touches $\Gamma_{1}$ at $A$ and $\Gamma_{2}$ at $B$. The tangent of $\Gamma_{1}$ at $P$ meets $\Gamma_{2}$ at $C$, which is different from $P$ and the extension of $A P$ meets $B C$ at $R$. Prove that the circumcircle of triangle $P Q R$ is tangent to $B P$ and $B R$.

Let $a_{1}, a_{2}, \cdots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_{i}+a_{j}$ for all $i, j=1,2, \cdots$. Prove that
$$
a_{1}+\frac{a_{2}}{2}+\frac{a_{3}}{3}+\cdots+\frac{a_{n}}{n} \geq a_{n}
$$
for each positive integer $n$.

Let $ABCDE$ be a regular pentagon such that the star $ACEBD$ has area $1$. Let $P$ be the point of intersection of $AC$ and $BE$ and $Q$ be the point of intersection of $BD$ and $CE$. Find the area of $APQD$.

Given a triangle $ABC$, explain how to construct with ruler and compass a triangle $A'B'C'$ of minimum area such that $C' \in AC$, $A' \in AB$, $B' \in BC$ and $\angle B'A'C' = \angle BAC$, $\angle A'C'B' = \angle ACB$.

Planet *Zork* is spherical and has many towns. For each town there is a corresponding antipodal town (i.e. symmetric in relation to the centre of the planet).
There are roads connecting pairs of towns in Zork. If there is a road connecting towns $P$ and $Q$ then there is also a road connecting towns $P'$ and $Q'$, where $P'$ is the antipode of $P$ and $Q'$ is the antipode of $Q$. Besides the roads do not cross each other and for any given two towns $P$ and $Q$ it is possible to travel from $P$ to $Q$ through some sequence of roads.
The prices of *Kriptonita* in *Urghs* (the planetary currency) in two towns connected by a road differ by no more than 100 *Urghs*. Prove that there exist two antipodal towns such that the prices of *Kriptonita* in these towns differ by no more than 100 *Urghs*.

$n$ tokens are to be placed on the squares of a $10 \times 10$ board such that no 4 tokens be the vertices of a rectangle with sides parallel to the sides of the board. Find the greatest value of $n$ for which this is possible.

There are $n$ football teams in *Tumbólia*.
A championship is to be organized in which each team plays against every other exactly once. Every match takes place on Sundays and one team must not play more than once in the same day.
Find the least integer $m$ for which it's possible to set up a championship lasting $m$ Sundays.

Prove that there is at least one nonzero digit between the 1,000,000th and the 3,000,000th decimal digits of $\sqrt{2}$.

In a triangle $A B C$ it is given that $2|A B|=|A C|+|B C|$. Prove that the incentre of $A B C$, the circumcentre of $A B C$, and the midpoints of $A C$ and $B C$ are concyclic.

Does there exist a finite sequence of integers $c_{1}, \ldots, c_{n}$ such that all the numbers $a+c_{1}, \ldots, a+c_{n}$ are primes for more than one but not infinitely many different integers $a$?

Can the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by distance $1$?

Let $a$, $b$, $c$ and $d$ be prime numbers such that $a \gt 3b \gt 6c \gt 12d$ and $a^{2} - b^{2} + c^{2} - d^{2} = 1749$. Determine all possible values of $a^{2} + b^{2} + c^{2} + d^{2}$.

What is the least number of moves it takes a knight to get from one corner of an $n \times n$ chessboard, where $n \geqslant 4$, to the diagonally opposite corner?

Determine all positive integers $n \geqslant 3$ such that the inequality
$$
a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n-1} a_{n}+a_{n} a_{1} \leqslant 0
$$
holds for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ which satisfy $a_{1}+\cdots+a_{n}=0$.

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing the last three decimal digits of $n$.

A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1,2, \ldots, 27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel to the edges of the cube). At most how many of the $27$ row sums can be odd?

Two squares on an $8 \times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?