Question bank Math Olympiad Vault 1972

Prove that the equation $x^{3} + 11^{3} = y^{3}$ has no solution in positive integers $x$ and $y$.

Describe a construction of a quadrilateral $ABCD$ given:
(i) the lengths of all four sides;
(ii) that $AB$ and $CD$ are parallel;
(iii) that $BC$ and $DA$ do not intersect.

What is the maximum number of terms in a geometric progression with common ratio greater than $1$ whose entries all come from the set of integers between $100$ and $1000$ inclusive?

During a certain election campaign, $p$ different kinds of promises are made by the various political parties ($p\gt 0$). While several parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.

Let $a$ and $b$ be distinct real numbers. Prove that there exist integers $m$ and $n$ such that $a m + b n \lt 0$, $b m + a n \gt 0$.

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_{i} a_{j}$ ($i\lt j$), i.e.,
$$
M = a_{1}(a_{2} + a_{3} + \cdots + a_{n}) + a_{2}(a_{3} + a_{4} + \cdots + a_{n}) + \cdots + a_{n-1} a_{n}
$$
Prove that the square of at least one of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ does not exceed $2M / n(n-1)$.

Four distinct lines $L_{1}, L_{2}, L_{3}, L_{4}$ are given in the plane: $L_{1}$ and $L_{2}$ are respectively parallel to $L_{3}$ and $L_{4}$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.

A $7$-gon is inscribed in a circle. The center of the circle lies inside the $7$-gon. $A$, $B$, $C$ are adjacent vertices of the $7$-gon. Show that the sum of the angles at $A$, $B$, $C$ is less than $450$ degrees.

$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. Prove that the line joining the centroids of $ABO$ and $CDO$ is perpendicular to the line joining the orthocenters of $BCO$ and $ADO$.

$P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A$, $B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on some circle center $X$.

A sequence of finite sets of positive integers is defined as follows. $S_0 = \{m\}$, where $m \gt 1$. Then given $S_n$ you derive $S_{n + 1}$ by taking $k^2$ and $k + 1$ for each element $k$ of $S_n$. For example, if $S_0 = \{5\}$, then $S_2 = \{7, 26, 36, 625\}$. Show that $S_n$ always has $2^n$ distinct elements.

Two players play the following game. At each turn the first player chooses a decimal digit, then the second player substitutes it for one of the stars in the subtraction $|**** - ****|$. The first player tries to end up with the largest possible result, the second player tries to end up with the smallest possible result. Show that the first player can always play so that the result is at least $4000$ and that the second player can always play so that the result is at most $4000$.

a, $m$, $n$ are positive integers and $a \gt 1$. Show that if $a^{m} + 1$ divides $a^{n} + 1$, then $m$ divides $n$.

The positive integer $b$ is relatively prime to $a$, show that if $a^{m} + b^{m}$ divides $a^{n} + b^{n}$ then $m$ divides $n$.

Is it possible to place the digits $0$, $1$, $2$ into unit squares of $100 \times 100$ cross-lined paper such that every $3 \times 4$ (and every $4 \times 3$) rectangle contains three $0$s, four $1$s and five $2$s?

Prove that a collection of squares with total area $1$ can always be arranged inside a square of area $2$ without overlapping.

$x_1$, $x_2$, ..., $x_n$ are positive reals with sum $1$. Let $s$ be the largest of $\dfrac{x_1}{1 + x_1}$, $\dfrac{x_2}{1 + x_1 + x_2}$, ..., $\dfrac{x_n}{1 + x_1 + ... + x_n}$. What is the smallest possible value of $s$? What are the corresponding $x_i$?

Find the largest integer $n$ such that $4^{27} + 4^{1000} + 4^{n}$ is a square.

9 lines each divide a square into two quadrilaterals with areas $\frac{2}{5}$ and $\frac{3}{5}$ that of the square. Show that 3 of the lines meet in a point.

$ABCD$ is a rectangle. $M$ is the midpoint of $AD$ and $N$ is the midpoint of $BC$. $P$ is a point on the ray $CD$ on the opposite side of $D$ to $C$. The ray $PM$ intersects $AC$ at $Q$. Show that $MN$ bisects the angle $PNQ$.