Question bank Math Olympiad Vault 1995

Find all real-valued functions on the positive integers such that $f(x+1019) = f(x)$ for all integer $x$, and $f(xy) = f(x)f(y)$ for all integers $x, y$.

Let $p(n)$ be the largest prime which divides $n$. Show that there are infinitely many positive integers $n$ such that $p(n) \lt p(n + 1) \lt p(n + 2)$.

A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

$X$ has $n$ elements. $\mathcal{F}$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ members which does not contain any members of $\mathcal{F}$.

Show that the $n$th root of a rational (for $n$ a positive integer) cannot be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.

$ABCD$ is a quadrilateral with a circumcircle center $O$ and an inscribed circle center $I$. The diagonals intersect at $S$. Show that if two of $O$, $I$, $S$ coincide, then it must be a square.

Prove that if both coordinates of every vertex of a convex pentagon are integers, then the area of this pentagon is not less than $\frac{5}{2}$.

The real numbers $a$, $b$ and $c$ satisfy the inequalities $|a| \geq |b+c|$, $|b| \geq |c+a|$ and $|c| \geq |a+b|$. Prove that $a+b+c=0$.

Prove that $\sin^{3} 18^{\circ} + \sin^{2} 18^{\circ} = \frac{1}{8}$.

John is older than Mary. He notices that if he switches the two digits of his age (an integer), he gets Mary's age. Moreover, the difference between the squares of their ages is the square of an integer. How old are Mary and John?

Prove that
$$
\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\cdots-\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\cdots+\frac{1995}{1996} .
$$

Find all triples $(x, y, z)$ of positive integers satisfying the system of equations
$$
\left\{\begin{array}{l}
x^{2}=2(y+z) \\
x^{6}=y^{6}+z^{6}+31\left(y^{2}+z^{2}\right)
\end{array}\right.
$$

The following construction is used for training astronauts: A circle $C_{2}$ of radius $2R$ rolls along the inside of another, fixed circle $C_{1}$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_{3}$ of radius $R$ which rolls along the inside of circle $C_{2}$ in such a way that the touching point of the circles $C_{2}$ and $C_{3}$ remains at maximum distance from the touching point of the circles $C_{1}$ and $C_{2}$ at all times (see Figure 3).

How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_{3}$ while the circle $C_{2}$ completes one full lap around the inside of circle $C_{1}$?


Figure 3

Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x, y, x+y$,
(iii) $(x+y) f(x+y)=x y f(x) f(y)$ for all non-zero $x, y, x+y$.

In the triangle $A B C$, let $l$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of the segment $A B$ parallel to $l$ meets the line $A C$ at $E$. Determine $|C E|$, if $|A C|=7$ and $|C B|=4$.

Assume we have $95$ boxes and $19$ balls distributed in these boxes in an arbitrary manner. We take six new balls at a time and place them in six of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the $95$ boxes contains an equal number of balls?

Let $a$ and $k$ be positive integers such that $a^{2}+k$ divides $(a-1) a(a+1)$. Prove that $k \geq a$.

Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.

A polygon with $2n+1$ vertices is given. Show that it is possible to label the vertices and midpoints of the sides of the polygon, using all the numbers $1, 2, \ldots, 4n+2$, so that the sums of the three numbers assigned to each side are all equal.

The positive integers $a$, $b$, $c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^{2} + b^{2} = c^{2}$. Prove that $b + c$ is a square of an integer.