MathNet 1986 — Part 01

A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

The Poincaré plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

Find all 10 digit numbers $a_0a_1\ldots a_9$ such that, for each $k$, $a_k$ is the number of times that the digit $k$ appears in the number.

A number is written in each square of a chessboard, so that each number not on the border is the mean of the 4 neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

In the diagram line segments $AB$ and $CD$ are of length $1$ while angles $ABC$ and $CBD$ are $90^{\circ}$ and $30^{\circ}$ respectively. Find $AC$.

For positive integers $n$ and $k$, define $F(n, k) = \sum_{r=1}^{n} r^{2k-1}$. Prove that $F(n, 1)$ divides $F(n, k)$.

A chord $ST$ of constant length slides around a semicircle with diameter $AB$. $M$ is the mid-point of $ST$ and $P$ is the foot of the perpendicular from $S$ to $AB$. Prove that angle $SPM$ is constant for all positions of $ST$.

Let $u_{1}, u_{2}, u_{3}, \ldots$ be a sequence of integers satisfying the recurrence relation $u_{n+2} = u_{n+1}^{2} - u_{n}$. Suppose $u_{1} = 39$ and $u_{2} = 45$. Prove that $1986$ divides infinitely many terms of the sequence.

A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_{1}$ points were awarded for first place, $p_{2}$ for second and $p_{3}$ for third, where $p_{1} \gt p_{2} \gt p_{3} \gt 0$ and $p_{1}, p_{2}, p_{3}$ are integers. The final score for $A$ was $22$, for $B$ was $9$ and for $C$ was also $9$. $B$ won the $100$ metres. What is the value of $M$ and who was second in the high jump?

Sean $A$, $B$ adjacent vertices of a $n$-regular agone $(n \geq 5)$ of the plane that has center on $O$. A triangle $XYZ$ that is consistent with $OAB$ and initially matches it, moves in the plane so that $Y$ and $Z$ describe the boundary of the Polygon, leaving $X$ inside. Find the geometric place of $X$.

We indicate by $m(a, b)$ the arithmetic mean of the positive real numbers $a$ and $b$. Given the positive real function $g$ that has the first and the second positive derivatives, we define the mean $\mu(a, b)$ relative to the function $g$ by means of $$
2 g(\mu(a, b))=g(a)+g(b)
$$ say, Reasonedly, which of the two means $m$ and $\mu$ is greater.

Find all the functions $f$ defined in the set of nonnegative real numbers and that take nonnegative real values, such that 1) $\quad f(x f(y)) f(y)=f(x+y)$ for all $x, y \geq 0$, 2) $\quad f(2)=0$, 3) $\quad f(x) \neq 0$ for $0 \leq x\lt 2$.

Let $d$ be a positive integer other than $2$, $5$ and $13$. Demonstrate that you can find different elements $a, b$ in the set $\{2, 5, 13, d\}$, so that $ab - 1$ is not a perfect square.

We have in the plane a point $P_{0}$ and a triangle $A_{1} A_{2} A_{3}$. We define $A_{s}=A_{s-3}$ for all $s \geq 4$. We build a succession of $P_{1}, P_{2}, P_{3}, \ldots$ points, so that $P_{k+1}$ is the image of $P_{k}$ by the center rotation $A_{k+1}$ and angle $120^{\circ}$ clockwise, for $k=0,1,2 \ldots$ to show that if $P_{1986}=P_{0}$, then triangle $A_{1} A_{2} A_{3}$ is equilateral.

We have a finite set of plane points, each with entire coordinates. It is asked if it is possible to color some points of the set in red and the rest in white so that every line $L$ parallel to one of the coordinate axes contains red and white dots in quantities whose difference in absolute value is 1 maximum. justify the answer.

Find the values of $n \in \mathbb{N}$ such that $5^{n}+3$ is a power of 2 of natural exponent.

Calculate $$
\prod_{k=1}^{14} \cos \left(\frac{k \pi}{15}\right)
$$

Each vertex of a pentagon is assigned an integer, so that the sum of the five integers is positive. If three consecutive vertices have assigned numbers $x, y, z$, respectively, and is $y\lt 0$, then the following operation is allowed to do: The numbers $x, y, z$ are replaced respectively by $x+y$, $-y$, $z+y$. This operation can be done repeatedly while at least one of the five numbers is negative. Determine if this process necessarily ends a finite number of steps.

A $d$ segment divides the $s$ segment if there is a natural $n$ such that $$
n d = d + d + \stackrel{n}{n}^{*} + d = s
$$
a) shows that if the segment $d$ divides the segments $s$ and $s'$ with $s \lt s'$, then divide the segment difference $s' - s$.
b) Demonstrate that no segment divides on the side $s$ and the diagonal $s'$ of a regular pentagon (reasoning on the regular pentagon whose sides are contained in the diagonals of the given pentagon, without making numerical calculations).