Let $x, y$ and $z$ be consecutive integers such that \[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\] Find the maximum value of $x + y + z$.

A cone is constructed with a semicircular piece of paper, with radius 10. Find the height of the cone.

$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D, E, F$, respectively. Show that $AD$ is perpendicular to $EF$.

Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are multiples of $33$?

A point $P$ is outside of a circle and the distance to the center is $13$. A secant line from $P$ meets the circle at $Q$ and $R$ so that the exterior segment of the secant, $PQ$, is $9$ and $QR$ is $7$. Find the radius of the circle.

The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.