Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.$$ (tatari/nightmare)

Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$. (tatari/nightmare)

Let triangle $ABC$ be a triangle right at $B$. The inscribed circle is tangent to sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $CF$ intersect the circle at the point $P$. If $\angle APB=90^{\circ}$, find the value of $\dfrac{CP+CD}{PF}$. (tatari/nightmare)

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. (tatari/nightmare)

The set $X$ of integers is called good If for each pair $a,b\in X$ , only one of $a+b,\mid a-b\mid$ is a member of $X$ ($a,b$ may be equal). Find the total number of sets with $2008$ as member. (tatari/nightmare)

Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$where $f$ appears $2009$ times. (tatari/nightmare)

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)+f(b,c,d)+f(c,d,e)+f(d,e,a)+f(e,a,b)=a+b+c+d+e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)+f(x_2,x_3,x_4)+\ldots +f(x_{n-1},x_n,x_1)+f(x_n,x_1,x_2)=x_1+x_2+\ldots+x_n\]

A positive rational number $x$ is called banzai if the following conditions are met: $\bullet$ $x=\frac{p}{q}&gt;1$ where $p,q$ are comprime natural numbers $\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$Find the total number of banzai numbers. Note:$\left\{\,x\right\}$ means fractional part of $x$ (tatari/nightmare)

$ABCD$ is a convex quadrilateral, and the point $K$ is a point on side $AB$, where $\angle KDA=\angle BCD$, let $L$ be a point on the diagonal $AC$, where $KL$ is parallel to $BC$. Prove that $$\angle KDB=\angle LDC.$$(tatari/nightmare)

In a circle, two non-intersecting chords $AB,CD$ are drawn.On the chord $AB$,a point $E$ (different from $A$,$B$) is taken Consider the arc $AB$ that does not contain the points $C,D$. With a compass and a straighthedge, find all possible point $F$ on that arc such that $\dfrac{PE}{EQ}=\dfrac{1}{2}$, where $P$ and $Q$ are the points in which the chord $AB$ meets the segment $FC$ and $FD$. (tatari/nightmare)