MathNet 1989 — Part 01

Let $A_{1}$, $A_{2}$, $A_{3}$ be three points in the plane, and for convenience, let $A_{4}=A_{1}$, $A_{5}=A_{2}$. For $n=1,2$, and $3$, suppose that $B_{n}$ is the midpoint of $A_{n} A_{n+1}$, and suppose that $C_{n}$ is the midpoint of $A_{n} B_{n}$. Suppose that $A_{n} C_{n+1}$ and $B_{n} A_{n+2}$ meet at $D_{n}$, and that $A_{n} B_{n+1}$ and $C_{n} A_{n+2}$ meet at $E_{n}$. Calculate the ratio of the area of triangle $D_{1} D_{2} D_{3}$ to the area of triangle $E_{1} E_{2} E_{3}$.
Answer: $\frac{25}{49}$.

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive real numbers, and let
$$
S = x_{1} + x_{2} + \cdots + x_{n} .
$$
Prove that
$$
\left(1 + x_{1}\right)\left(1 + x_{2}\right) \cdots \left(1 + x_{n}\right) \leq 1 + S + \frac{S^{2}}{2!} + \frac{S^{3}}{3!} + \cdots + \frac{S^{n}}{n!} .
$$

Prove that the equation
$$
6\left(6 a^{2}+3 b^{2}+c^{2}\right)=5 n^{2}
$$
has no solutions in integers except $a=b=c=n=0$.

Let $S$ be a set consisting of $m$ pairs $(a, b)$ of positive integers with the property that $1 \leq a \lt b \leq n$. Show that there are at least
$$
4m \frac{\left(m-\frac{n^{2}}{4}\right)}{3n}
$$
triples $(a, b, c)$ such that $(a, b), (a, c)$, and $(b, c)$ belong to $S$.

Determine all functions $f$ from the reals to the reals for which
(1) $f(x)$ is strictly increasing,
(2) $f(x)+g(x)=2x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$.
(Note: $f$ and $g$ are said to be composition inverses if $f(g(x))=x$ and $g(f(x))=x$ for all real $x$.)

$n$ is a positive integer such that $\frac{n(n+1)}{3}$ is a square. Show that $n$ is a multiple of $3$, and $n+1$ and $\frac{n}{3}$ are squares.

The triangle with vertices $(0, 0)$, $(0, 1)$, $(2, 0)$ is repeatedly reflected in the three lines $AB$, $BC$, $CA$ where $A$ is $(0, 0)$, $B$ is $(3, 0)$, $C$ is $(0, 3)$. Show that one of the images has vertices $(24, 36)$, $(24, 37)$ and $(26, 36)$.

The circumcenter of a tetrahedron lies inside the tetrahedron. Show that at least one of its edges is at least as long as the edge of a regular tetrahedron with the same circumsphere.
(1989)

Let $Z$ be the set of integers. $f: Z \rightarrow Z$ is defined by $f(n) = n - 10$ for $n \gt 100$ and $f(n) = f(f(n+11))$ for $n = 100$. Find the set of possible values of $f$.

$A$ and $B$ play a game. Each has $10$ tokens numbered from $1$ to $10$. The board is two rows of squares. The first row is numbered $1$ to $1492$ and the second row is numbered $1$ to $1989$. On the $n$th turn, $A$ places his token number $n$ on any empty square in either row and $B$ places his token on any empty square in the other row. $B$ wins if the order of the tokens is the same in the two rows, otherwise $A$ wins. Which player has a winning strategy? Suppose each player has $k$ tokens, numbered from $1$ to $k$. Who has the winning strategy? What if both rows are all the integers? Or both all the rationals?

Define $\{a_n\}_{n=1}$ as follows: $a_1 = 1989^{1989}$; $a_n$, $n \gt 1$, is the sum of the digits of $a_{n-1}$. What is the value of $a_5$?

Given the numbers $1, 2, 2^{2}, \ldots, 2^{n-1}$. For a specific permutation $\sigma = X_{1}, X_{2}, \ldots, X_{n}$ of these numbers we define $S_{1}(\sigma) = X_{1}$, $S_{2}(\sigma) = X_{1} + X_{2}$, $S_{3}(\sigma) = X_{1} + X_{2} + X_{3}$, $\ldots$ and $Q(\sigma) = S_{1}(\sigma) S_{2}(\sigma) \cdots S_{n}(\sigma)$. Evaluate $\sum 1 / Q(\sigma)$ where the sum is taken over all possible permutations.

There are $5$ monkeys and $5$ ladders and at the top of each ladder there is a banana. A number of ropes connect the ladders, each rope connects two ladders. No two ropes are attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. The monkeys climb up the ladders but each time they encounter a rope they climb along it to the ladder at the other end of the rope and then continue climbing upwards. Show that, no matter how many ropes there are, each monkey gets a banana.

Let $ABC$ be a right angled triangle of area $1$. Let $A'B'C'$ be the points obtained by reflecting $A$, $B$, $C$ respectively, in their opposite sides. Find the area of $\triangle A'B'C'$.

The integers $1, 2, \ldots, n$ are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?

Prove that given seven real numbers, you can choose two, say $a$ and $b$, so that $$
\sqrt{3}|a-b|\lt |1+a b|
$$ give an example of six real numbers that do not meet this property.

Let $\mathcal{D}$ be the set of complex numbers that can be written in the form $a+b \sqrt{-13}$, with $a, b$ integers. The number $14=14+0 \sqrt{-13}$ can be written as a product of two elements of $\mathcal{D}$: $14=2 \cdot 7$. Express 14 as a product of two elements of $\mathcal{D}$ in all possible ways.

Demonstrate that the number $1989$ and all its integer powers $1989^{n}$ can be written as the sum of two squares of positive integers, and at least two different ways.

Prove that $$
\frac{1}{10 \sqrt{2}} \lt \frac{1 \cdot 3 \cdot 5 \cdots 99}{2 \cdot 4 \cdot 6 \cdots 100} \lt \frac{1}{10}
$$

The program of a subject consists of $n$ questions; The exam consists of developing one of those questions, chosen at random. A student only knows a question, but can repeat the $n$ exam times. Express, as a function of $n$, the probability $p_{n}$ for the student to pass the exam. Does $p_{n}$ grow or decrease by increasing $n$? Calculate $$
\lim_{n \rightarrow \infty} p_{n}
$$ What is the highest of the lower bounds of the probabilities $p_{n}$?