MathNet 1970 — Part 01

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either $1$ pound or $2$ pounds, and there is at least one of each weight. Prove that there are $2$ balls having different weights and different colours.

Find all number triples $(x, y, z)$ such that when any one of these numbers is added to the product of the other two, the result is $2$.

Let $f(n)$ be the sum of the first $n$ terms of the sequence
$$
0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, \ldots
$$

a) Give a formula for $f(n)$.

b) Prove that $f(s+t)-f(s-t)=s t$ where $s$ and $t$ are positive integers and $s\gt t$.

Consider all line segments of length $4$ with one end-point on the line $y = x$ and the other end-point on the line $y = 2x$. Find the equation of the locus of the midpoints of these line segments.

Given three non-collinear points $A$, $B$, $C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ to the circle are parallel.

A quadrilateral has one vertex on each side of a square of side-length $1$. Show that the lengths $a, b, c$ and $d$ of the sides of the quadrilateral satisfy the inequalities
$$
2 \leq a^{2}+b^{2}+c^{2}+d^{2} \leq 4
$$

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by $3$.

a) Find all positive integers with initial digit $6$ such that the integer formed by deleting this $6$ is $1/25$ of the original integer.


b) Show that there is no integer such that deletion of the first digit produces a result which is $1/35$ of the original integer.

Given the polynomial
$$
f(x) = x^{n} + a_{1} x^{n-1} + a_{2} x^{n-2} + \cdots + a_{n-1} x + a_{n}
$$
with integral coefficients $a_{1}, a_{2}, \ldots, a_{n}$, and given also that there exist four distinct integers $a, b, c$ and $d$ such that
$$
f(a) = f(b) = f(c) = f(d) = 5
$$
show that there is no integer $k$ such that $f(k) = 8$.

Show that given $200$ integers you can always choose $100$ with sum a multiple of $100$.

$ABC$ is an acute-angled triangle. The angle bisector $AD$, the median $BM$ and the altitude $CH$ are concurrent. Prove that angle $A$ is more than $45$ degrees.

The vertices of a regular $n$-gon are colored (each vertex has only one color). Each color is applied to at least three vertices. The vertices of any given color form a regular polygon. Show that there are two colors which are applied to the same number of vertices.

Given any positive integer $n$, show that we can find infinitely many integers $m$ such that $m$ has no zeros (when written as a decimal number) and the sum of the digits of $m$ and $mn$ is the same.

If the numbers from $11111$ to $99999$ are arranged in an arbitrary order show that the resulting $444445$ digit number is not a power of $2$.

Given a circle, diameter $AB$ and a point $C$ on $AB$, show how to construct two points $X$ and $Y$ on the circle such that (1) $Y$ is the reflection of $X$ in the line $AB$, (2) $YC$ is perpendicular to $XA$.

Two congruent rectangles of area $A$ intersect in eight points. Show that the area of the intersection is more than $A/2$.

$S$ is the set of all positive integers with $n$ decimal digits or less and with an even digit sum. $T$ is the set of all positive integers with $n$ decimal digits or less and an odd digit sum. Show that the sum of the $k$th powers of the members of $S$ equals the sum for $T$ if $1 \leq k \lt n$.

Given 5 segments such that any 3 can be used to form a triangle. Show that at least one of the triangles is acute-angled.

The product of three positive numbers is $1$, their sum is greater than the sum of their inverses. Prove that just one of the numbers is greater than $1$.

A room is an equilateral triangle side $100$ meters. It is subdivided into $100$ rooms, all equilateral triangles with side $10$ meters. Each interior wall between two rooms has a door. If you start inside one of the rooms and can only pass through each door once, show that you cannot visit more than $91$ rooms.

Suppose now the large triangle has side $k$ and is divided into $k^2$ small triangles by lines parallel to its sides. A chain is a sequence of triangles, such that a triangle can only be included once and consecutive triangles have a common side. What is the largest possible number of triangles in a chain?