MathNet 1973 — Part 01

For every positive integer $n$, let
$$
h(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}
$$
For example, $h(1) = 1$, $h(2) = 1 + \frac{1}{2}$, $h(3) = 1 + \frac{1}{2} + \frac{1}{3}$. Prove that for $n = 2, 3, 4, \ldots$
$$
n + h(1) + h(2) + h(3) + \cdots + h(n-1) = n h(n)
$$

i. Solve the simultaneous inequalities, $x \lt \frac{1}{4x}$ and $x \lt 0$; i.e., find a single inequality equivalent to the two given simultaneous inequalities.

ii. What is the greatest integer which satisfies both inequalities $4x + 13 \lt 0$ and $x^2 + 3x \gt 16$?

iii. Give a rational number between $\frac{11}{24}$ and $\frac{6}{13}$.

iv. Express $100000$ as a product of two integers neither of which is an integral multiple of $10$.

v. Without the use of logarithm tables evaluate
$$
\frac{1}{\log_{2} 36} + \frac{1}{\log_{3} 36}
$$

Observe that
$$
\frac{1}{1} = \frac{1}{2} + \frac{1}{2} ; \quad \frac{1}{2} = \frac{1}{3} + \frac{1}{6} ; \quad \frac{1}{3} = \frac{1}{4} + \frac{1}{12} ; \quad \frac{1}{4} = \frac{1}{5} + \frac{1}{20}
$$
State a general law suggested by these examples, and prove it.

Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that
$$
\frac{1}{n} = \frac{1}{i(i+1)} + \frac{1}{(i+1)(i+2)} + \frac{1}{(i+2)(i+3)} + \cdots + \frac{1}{j(j+1)}
$$

Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.

Find all the real numbers which satisfy the equation $|x+3| - |x-1| = x+1$. (Note: $|a| = a$ if $a \geq 0$; $|a| = -a$ if $a \lt 0$.)

If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).

Players numbered $1$ to $1024$ play in a knock-out tournament. There are no draws, the winner of a match goes through to the next round and the loser is knocked-out, so that there are $512$ matches in the first round, $256$ in the second and so on. If $m$ plays $n$ and $m \lt n - 2$ then $m$ always wins. What is the largest possible number for the winner?

Prove that a 9 digit decimal number whose digits are all different, which does not end with 5 and or contain a 0, cannot be a square.

$ABC$ is an acute-angled triangle. $D$ is the reflection of $A$ in $BC$, $E$ is the reflection of $B$ in $AC$, and $F$ is the reflection of $C$ in $AB$. Show that the circumcircles of $DBC$, $ECA$, $FAB$ meet at a point and that the lines $AD$, $BE$, $CF$ meet at a point.

A triangle has area $1$, and sides $a \geq b \geq c$. Prove that $b^2 \geq 2$

Given $n \gt 4$ points, show that you can place an arrow between each pair of points, so that given any point you can reach any other point by travelling along either one or two arrows in the direction of the arrow.

Define $p(x) = a x^2 + b x + c$. If $p(x) = x$ has no real roots, prove that $p(p(x)) = 0$ has no real roots.

$OA$ and $OB$ are tangent to a circle at $A$ and $B$. The line parallel to $OB$ through $A$ meets the circle again at $C$. The line $OC$ meets the circle again at $E$. The ray $AE$ meets the line $OB$ at $K$. Prove that $K$ is the midpoint of $OB$.

At time $1$, $n$ unit squares of an infinite sheet of paper ruled in squares are painted black, the rest remain white. At time $k + 1$, the color of each square is changed to the color held at time $k$ by a majority of the following three squares: the square itself, its northern neighbour and its eastern neighbour. Prove that all the squares are white at time $n + 1$.

You are given 14 coins. It is known that genuine coins all have the same weight and that fake coins all have the same weight, but weigh less than genuine coins. You suspect that 7 particular coins are genuine and the other 7 fake. Given a balance, how can you prove this in three weighings (assuming that you turn out to be correct)?

A convex $n$-gon has no two sides parallel. Given a point $P$ inside the $n$-gon, show that there are at most $n$ lines through $P$ which bisect the area of the $n$-gon.

$p(x) = ax^2 + bx + c$ is a real quadratic such that $|p(x)| \leq 1$ for all $|x| \leq 1$. Prove that $|cx^2 + bx + a| \leq 2$ for $|x| \leq 1$.

A king moves on an $8 \times 8$ chessboard. He can move one square at a time, diagonally or orthogonally (so away from the borders he can move to any of eight squares). He makes a complete circuit of the board, starting and finishing on the same square and visiting every other square just once. His trajectory is drawn by joining the center of the squares he moves to and from for each move. The trajectory does not intersect itself. Show that he makes at least 28 moves parallel to the sides of the board (the others being diagonal) and that a circuit is possible with exactly 28 moves parallel to the sides of the board. If the board has side length 8, what is the maximum and minimum possible length for such a trajectory.

Given 4 points which do not lie in a plane, how many parallelepipeds have all 4 points as vertices?

a, b, c, d, e are positive reals. Show that $$(a + b + c + d + e)^2 \geq 4(ab + bc + cd + de + ea)$$