MathNet 1976 — Part 01

Let $P(x, y)$ be a polynomial in two variables $x, y$ such that $P(x, y) = P(y, x)$ for every $x, y$ (for example, the polynomial $x^{2} - 2 x y + y^{2}$ satisfies this condition). Given that $(x - y)$ is a factor of $P(x, y)$, show that $(x - y)^{2}$ is a factor of $P(x, y)$.

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students from grade eight participated in the chess tournament? Is the solution unique?

If $A, B, C, D$ are four points in space, such that
$$
\angle A B C=\angle B C D=\angle C D A=\angle D A B=\pi / 2
$$
prove that $A, B, C, D$ lie in a plane.

Let $AB$ be a diameter of a circle, $C$ be any fixed point between $A$ and $B$ on this diameter, and $Q$ be a variable point on the circumference of the circle. Let $P$ be the point on the line determined by $Q$ and $C$ for which $\frac{AC}{CB} = \frac{QC}{CP}$. Describe, with proof, the locus of the point $P$.

Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.

Suppose
$$
n(n+1) a_{n+1} = n(n-1) a_n - (n-2) a_{n-1}
$$
for every positive integer $n \geq 1$.
Given that $a_0 = 1$, $a_1 = 2$, find
$$
\frac{a_0}{a_1} + \frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{50}}{a_{51}}.
$$

$S$ is a sphere with unit radius. $P$ is a plane through the center. For any point $x$ on the sphere, $f(x)$ is the perpendicular distance from $x$ to $P$. Show that if $x$, $y$, $z$ are the ends of three mutually perpendicular radii, then $f(x)^2 + f(y)^2 + f(z)^2 = 1$ (*).

Now let $g(x)$ be any function on the points of $S$ taking non-negative real values and satisfying (*). Regard the intersection of $P$ and $S$ as the equator, the poles as the points with $f(x) = 1$ and lines of longitude as semicircles through both poles.

(1) If $x$ and $y$ have the same longitude and both lie on the same side of the equator with $x$ closer to the pole, show that $g(x) \gt g(y)$.

(2) Show that for any $x$, $y$ on the same side of the equator with $x$ closer to the pole than $y$ we have $g(x) \gt g(y)$.

(3) Show that if $x$ and $y$ are the same distance from the pole then $g(x) = g(y)$.

(4) Show that $g(x) = f(x)$ for all $x$.

You are given a regular $n$-gon. Each vertex is marked $+1$ or $-1$. A move consists of changing the sign of all the vertices which form a regular $k$-gon for some $1 \lt k \leq n$. [A regular $2$-gon means two vertices which have the center of the $n$-gon as their midpoint.] For example, if we label the vertices of a regular $6$-gon $1, 2, 3, 4, 5, 6$, then you can change the sign of $\{1, 4\}$, $\{2, 5\}$, $\{3, 6\}$, $\{1, 3, 5\}$, $\{2, 4, 6\}$ or $\{1, 2, 3, 4, 5, 6\}$. Show that for (1) $n = 15$, (2) $n = 30$, (3) any $n \gt 2$, we can find some initial marking which cannot be changed to all $+1$ by any series of moves. Let $f(n)$ be the largest number of markings, so that no one can be obtained from any other by any series of moves. Show that $f(200) = 2^{80}$.

$n$ real numbers are written around a circle. One of the numbers is $1$ and the sum of the numbers is $0$. Show that there are two adjacent numbers whose difference is at least $n/4$. Show that there is a number which differs from the arithmetic mean of its two neighbours by at least $8/n^2$. Improve this result to some $k/n^2$ with $k \gt 8$. Show that for $n = 30$, we can take $k = 1800/113$. Give an example of $30$ numbers such that no number differs from the arithmetic mean of its two neighbours by more than $2/113$.

Initially, there is one beetle on each square in the set $S$. Suddenly each beetle flies to a new square, subject to the following conditions:

(1) the new square may be the same as the old or different;

(2) more than one beetle may choose the same new square;

(3) if two beetles are initially in squares with a common vertex, then after the flight they are either in the same square or in squares with a common vertex.

Suppose $S$ is the set of all squares in the middle row and column of a $99 \times 99$ chess board, is it true that there must always be a beetle whose new square shares a vertex with its old square (or is identical with it)?

What if $S$ also includes all the border squares (so $S$ is rows $1$, $50$ and $99$ and columns $1$, $50$ and $99$)?

What if $S$ is all squares of the board?

There are three straight roads. On each road a man is walking at constant speed. At time $t = 0$, the three men are not collinear. Prove that they will be collinear for $t \gt 0$ at most twice.

Can you label each vertex of a cube with a different three digit binary number so that the numbers at any two adjacent vertices differ in at least two digits?

$a_1$ and $a_2$ are positive integers less than $1000$. Define $a_n = \min\{|a_i - a_j| : 0 \lt i \lt j \lt n\}$. Show that $a_{21} = 0$.

$n$ rectangles are drawn on a rectangular sheet of paper. Each rectangle has its sides parallel to the sides of the paper. No pair of rectangles has an interior point in common. If the rectangles were removed show that the rest of the sheet would be in at most $n + 1$ parts.

The circles $C_1$, $C_2$, $C_3$ with equal radius all pass through the point $X$. $C_i$ and $C_j$ also intersect at the point $Y_{ij}$. Show that $\angle X O_1 Y_{12} + \angle X O_2 Y_{23} + \angle X O_3 Y_{31} = 180^\circ$ where $O_i$ is the center of circle $C_i$.

1000 numbers are written in line 1, then further lines are constructed as follows. If the number $m$ occurs in line $n$, then we write under it in line $n + 1$, each time it occurs, the number of times that $m$ occurs in line $n$. Show that lines 11 and 12 are identical. Show that we can choose numbers in line 1, so that lines 10 and 11 are not identical.

Call a triangle big if each side is longer than $1$. Show that we can draw $100$ big triangles inside an equilateral triangle with side length $5$ so that all the triangles are disjoint. Show that you can draw $100$ big triangles with every vertex inside or on an equilateral triangle with side $3$, so that they cover the equilateral triangle, and any two big triangles either (1) are disjoint, or (2) have as intersection a common vertex, or (3) have as intersection a common side.

$S$ is a set of $1976$ points which form a regular $1976$-gon. $T$ is the set of all points which are the midpoint of at least one pair of points in $S$. What is the greatest number of points of $T$ which lie on a single circle?

50 watches, all keeping perfect time, lie on a table. Show that there is a moment when the sum of the distances from the center of the table to the center of each dial equals the sum of the distances from the center of the table to the tip of each minute hand.