Question bank Math Olympiad Vault 1983

Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.

Show that there are only finitely many solutions to
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1983}
$$
in positive integers.

Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.

Show that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ is not an integer for $n \gt 1$.

Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n^{1/n} \le k$ for all positive integers $n$.

An equilateral triangle $ABC$ has side $a$. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.

Find all positive integers $w, x, y$ and $z$ which satisfy $w! = x! + y! + z!$.

For each real number $r$ let $T_{r}$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^{r} x, r 2^{r} x + 2^{r} y)$. Let $F$ be the family of all such transformations i.e. $F = \{ T_{r} : r \text{ a real number} \}$. Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^{n}-n$.

The geometric mean (G.M.) of a $k$ positive numbers $a_{1}, a_{2}, \ldots, a_{k}$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of $3,4,18$ is $6$. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.

be $A$ one of the two distinct intersection points of two different circles $C_{1}$, $C_{2}$ of centers $O_{1}$, $O_{2}$, respectively. One of the common tangents to the two circles touches $C_{1}$ on $P_{1}$ y a $C_{2}$ on $P_{2}$, while the other touches $C_{1}$ on $Q_{1}$ y a $C_{2}$ on $Q_{2}$. be $M_{1}$ the midpoint of $P_{1} Q_{1}$ y $M_{2}$ the midpoint of $P_{2} Q_{2}$. show that $\widehat{O_{1} A O_{2}} = \widehat{M_{1} A M_{2}}$.

Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Demonstrate that $$
a^{2} b(a-b) + b^{2} c(b-c) + c^{2} a(c-a) \geq 0
$$ determine in which cases the equality is met.

Say if it is possible to choose $1983$ different positive integers, all less than or equal to $10^{5}$, so that three of them are not consecutive terms of an arithmetic progression. justify the answer.

Find all functions $f$ defined in the set of real numbers, which take positive real values and satisfy the conditions 1) $\quad f(x f(y))=y f(x)$ for all $x, y$ positive, 2) $\quad f(x) \rightarrow 0$ if $x \rightarrow \infty$.

Let $ABC$ be an equilateral triangle, and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$ and $CA$ (with $A$, $B$ and $C$ included). Determine if it is true that for each partition of $\mathcal{E}$ into two disjoint sets, at least one of the two sets contains the vertices of a right triangle. justify the answer.

Let $a$, $b$, $c$ be positive integers, two to two cousins together. Demonstrate that $2abc - ab - bc - ca$ is the largest integer that cannot be expressed in the form $x\,bc + y\,ca + z\,ab$, where $x$, $y$, and $z$ are non-negative integers.

Interior points $D$, $E$, $F$ are chosen on the sides $BC$, $CA$, $AB$ (not at the vertices). Let $k$ be the length of the longest side of $DEF$. Let $a$, $b$, $c$ be the lengths of the longest sides of $AFE$, $BDF$, $CDE$ respectively. Show that $k \geq \dfrac{\sqrt{3}}{2} \min(a, b, c)$. When do we have equality?

Find all real solutions $x$, $y$ to
$$
y^2 = x^3 - 3x^2 + 2x, \quad x^2 = y^3 - 3y^2 + 2y.
$$

$M$ is the midpoint of $BC$. $E$ is any point on the side $AC$ and $F$ is any point on the side $AB$. Show that area $\{MEF\} \leq$ area $\{BMF\} +$ area $CME$.