Question bank Math Olympiad Vault 1981

Show that there are at least 3 and at most 4 powers of $2$ with $m$ digits. For which $m$ are there $4$?

For which $k$ does the system $x^2 - y^2 = 0$, $(x - k)^2 + y^2 = 1$ have exactly
(1) two, (2) three real solutions?

The centers of the faces of a cube form a regular octahedron of volume $V$. Through each vertex of the cube we may take the plane perpendicular to the long diagonal from the vertex. These planes also form a regular octahedron. Show that its volume is $27V$.

A graph has 100 points. Given any four points, there is one joined to the other three. Show that one point must be joined to all 99 other points. What is the smallest number possible of such points (that are joined to all the others)?

Given a sheet of paper and the use of a rule, compass and pencil, show how to draw a straight line that passes through two given points, if the length of the ruler and the maximum opening of the compass are both less than half the distance between the two points. You may not fold the paper.

Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3$ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $\frac{m+n}{2}$ is divisible by $\gcd(m, n)$.

For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8]=8$, $[\pi]=3$ and $[-5 / 2]=-3$. Show that the equation
$$
[x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345
$$
has no real solution.

$P(x)$ and $Q(x)$ are two polynomials that satisfy the identity $P(Q(x)) \equiv Q(P(x))$ for real numbers $x$. If the equation $P(x)=Q(x)$ has no real solution, show that the equation $P(P(x))=Q(Q(x))$ also has no real solution.

Eleven theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least one performance of every other group. At least how many days did the festival last?

Given a finite collection of lines in a plane $P$, show that it is possible to draw an arbitrarily large circle in $P$ which does not meet any of them. On the other hand, show that it is possible to arrange an infinite sequence of lines (first line, second line, third line, etc.) in $P$ so that every circle in $P$ meets at least one of the lines. (A point is not considered to be a circle.)

Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.

Find all solutions $(x, y)$ in positive integers to $x^{3} - y^{3} = xy + 61$.

Find an example of $m$ and $n$ such that $m$ is the product of $n$ consecutive positive integers and also the product of $n + 2$ consecutive positive integers. Show that we cannot have $n = 2$.

Eighteen teams are playing in a tournament. So far, each team has played exactly eight games, each with a different opponent. Show that there are three teams none of which has yet played the other.

The positive reals $x$, $y$ satisfy $x^{3} + y^{3} = x - y$. Show that $x^{2} + y^{2} \lt 1$.

The reals $a$ and $b$ are such that $a \cos x + b \cos 3x \gt 1$ has no real solutions. Show that $|b| \leq 1$.

$AB$ is a diameter of the circle $C$. $M$ and $N$ are any two points on the circle. The chord $MA'$ is perpendicular to the line $NA$ and the chord $MB'$ is perpendicular to the line $NB$. Show that $AA'$ and $BB'$ are parallel.

$ABCDEF$ is a prism. Its base $ABC$ and its top $DEF$ are congruent equilateral triangles. The side edges are $AD$, $BE$ and $CF$. Find all points on the base which are equidistant from the three lines $AE$, $BF$ and $CD$.

Is it possible to color half the cells in a rectangular array white and half black so that in each row and column more than $\frac{3}{4}$ of the cells are the same color?

An integer is initially written at each vertex of a cube. A move is to add $1$ to the numbers at two vertices connected by an edge. Is it possible to equalise the numbers by a series of moves in the following cases?

(1) The initial numbers are $0$, except for one vertex which is $1$.

(2) The initial numbers are $0$, except for two vertices which are $1$ and diagonally opposite on a face of the cube.

(3) Initially, the numbers going round the base are $1$, $2$, $3$, $4$. The corresponding vertices on the top are $6$, $7$, $4$, $5$ (with $6$ above the $1$, $7$ above the $2$ and so on).