MathNet 2005 — Part 01

Prove that for every irrational real number $a$, there are irrational real numbers $b$ and $b^{\prime}$ so that $a+b$ and $a b^{\prime}$ are both rational while $a b$ and $a+b^{\prime}$ are both irrational.

Let $a$, $b$ and $c$ be positive real numbers such that $a b c = 8$. Prove that
$$
\frac{a^{2}}{\sqrt{\left(1+a^{3}\right)\left(1+b^{3}\right)}} + \frac{b^{2}}{\sqrt{\left(1+b^{3}\right)\left(1+c^{3}\right)}} + \frac{c^{2}}{\sqrt{\left(1+c^{3}\right)\left(1+a^{3}\right)}} \geq \frac{4}{3}
$$

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1,1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time $0$, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended neighbors of each house which was on fire at time $t$. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a neighbor of a house indexed by $(k, \ell)$ if $|i-k|+|j-\ell|=1$.

In a triangle $A B C$, points $M$ and $N$ are on sides $A B$ and $A C$, respectively, such that $M B = B C = C N$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $A B C$, respectively. Express the ratio $M N / B C$ in terms of $R$ and $r$.

How many real numbers $x$ are solutions to the following equation?
$$
2003^{x} + 2004^{x} = 2005^{x}
$$

Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Find the maximum possible value of
$$
\frac{x y + y z + z x}{x^{2} + y^{2} + z^{2}}.
$$

If $a, b, c \gt 0$, what is the smallest possible value of $\left\lfloor\frac{a+b}{c}\right\rfloor + \left\lfloor\frac{b+c}{a}\right\rfloor + \left\lfloor\frac{c+a}{b}\right\rfloor$? (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers?

Find the sum of the $x$-coordinates of the distinct points of intersection of the plane curves given by $x^{2}=x+y+4$ and $y^{2}=y-15 x+36$.

Let $x$ be a positive real number. Find the maximum possible value of
$$
\frac{x^{2}+2-\sqrt{x^{4}+4}}{x} .
$$

Compute
$$
\sum_{n=0}^{\infty} \frac{n}{n^{4}+n^{2}+1}
$$

The number $27,000,001$ has exactly four prime factors. Find their sum.

Find the sum of the absolute values of the roots of $x^{4}-4 x^{3}-4 x^{2}+16 x-8=0$.

Let $f(x) = x^{3} + a x + b$, with $a \neq b$, and suppose the tangent lines to the graph of $f$ at $x = a$ and $x = b$ are parallel. Find $f(1)$.

A plane curve is parameterized by $x(t) = \int_{t}^{\infty} \frac{\cos u}{u} du$ and $y(t) = \int_{t}^{\infty} \frac{\sin u}{u} du$ for $1 \leq t \leq 2$. What is the length of the curve?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function with $\int_{0}^{1} f(x) f^{\prime}(x) d x=0$ and $\int_{0}^{1} f(x)^{2} f^{\prime}(x) d x=18$. What is $\int_{0}^{1} f(x)^{4} f^{\prime}(x) d x$?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a smooth function such that $f'(x)^2 = f(x) f''(x)$ for all $x$. Suppose $f(0) = 1$ and $f^{(4)}(0) = 9$. Find all possible values of $f'(0)$.

Calculate
$$
\lim_{x \rightarrow 0^{+}} \left(x^{x^{x}} - x^{x}\right).
$$

The graph of $r=2+\cos 2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.