Found problems: 85335
Let $f_1(n)$ be the number of divisors that $n$ has, and define $f_k(n)=f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such that $f_k(2013^{2013})=2$.
The points $A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}$ are on the sides $AB$, $BC$ and $AC$ of an acute triangle $ABC$ such that $AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB$, $BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC$ and $CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC$. Let $k_{A}, k_{B}$ and $k_{C}$ be the circumcircles of the triangles $AA_{1}C_{2}$, $BB_{1}A_{2}$ and $CC_{1}B_{2}$ respectively. Furthermore, let $a_{B}$ and $a_{C}$ be the tangents to $k_{A}$ at $A_{1}$ and $C_{2}$, $b_{C}$ and $b_{A}$ the tangents to $k_{B}$ at $B_{1}$ and $A_{2}$ and $c_{A}$ and $c_{B}$ the tangents to $k_{C}$ at $C_{1}$ and $B_{2}$. Show that the perpendicular lines from the intersection points of $a_{B}$ and $b_{A}$, $b_{C}$ and $c_{B}$, $c_{A}$ and $a_{C}$ to $AB$, $BC$ and $CA$ respectively are concurrent.
In acute triangle $ABC$,$AC<BC$,$M$ is midpoint of $AB$ and $\Omega$ is it's circumcircle.Let $C^\prime$ be antipode of $C$ in $\Omega$. $AC^\prime$ and $BC^\prime$ intersect with $CM$ at $K,L$,respectively.The perpendicular drawn from $K$ to $AC^\prime$ and perpendicular drawn from $L$ to $BC^\prime$ intersect with $AB$ and each other and form a triangle $\Delta$.Prove that circumcircles of $\Delta$ and $\Omega$ are tangent.(M.Kungozhin)
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.
Goldfish are sold at $ 15$ cents each. The rectangular coordinate graph showing the cost of $ 1$ to $ 12$ goldfish is:
$ \textbf{(A)}\ \text{a straight line segment} \qquad \\
\textbf{(B)}\ \text{a set of horizontal parallel line segments}\qquad \\
\textbf{(C)}\ \text{a set of vertical parallel line segments}\qquad \\
\textbf{(D)}\ \text{a finite set of distinct points}\qquad
\textbf{(E)}\ \text{a straight line}$
For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$.
[i]Proposed by Harry Kim[/i]
Prove that if $x,y$ and $z$ are pairwise relatively prime positive integers, and if $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$, then $x+y, x-z, y-z$ are perfect squares of integers.
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient?
In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$.
[i](tatari/nightmare)[/i]
Let $ABC$ be a triangle. Let $D$ and $E$ be respectively points on the segments $AB$ and $AC$, and such that $DE||BC$. Let $M$ be the midpoint of $BC$. Let $P$ be a point such that $DB=DP$, $EC=EP$ and such that the open segments (segments excluding the endpoints) $AP$ and $BC$ intersect. Suppose $\angle BPD=\angle CME$. Show that $\angle CPE=\angle BMD$
How many positive integers $n$ are there such that $\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}$ is an integer?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the above}
$
Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?
Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$
Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$
Find all pairs$ (a, b)$ of positive real numbers with the following properties:
(i) For all positive real numbers $x, y, z,w$ holds $x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}$ .
(ii) There is a quadruple $(x, y, z,w)$ of positive real numbers such that in equality (i) applies.
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
Consider rows of the form: $[x], [2x], [3x], ...$
Proof that, if $N \in N$ does not occur in the sequence $([n x])$, then there is an $n \in N$ with $n - 1 < \frac{N}{x}< n -\frac{1}{x}$
Prove that, for $x, y \notin Q$: $\frac{1}{x}+\frac{1}{y} = 1$, then each $N \in N$ term is either of $([nx])$ or of $([ny])$.
Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$
Let be a natural number $ n\ge 3 $ with the property that $ 1+3n $ is a perfect square. Show that there are three natural numbers $ a,b,c, $ such that the number
$$ 1+\frac{3n+3}{a^2+b^2+c^2} $$
is a perfect square.
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior.
Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$, for each drawn segment $s$.
Estimate to determine which of the following numbers is closest to $\frac{401}{.205}$.
$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$
In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$.
(Fedak Ivan)
Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.
We consider the set
\begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \nu \geq 2 \end{align*}
and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to
\begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*}
We also consider the $\nu-$tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$th roots of $-1$, where
\begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*}
Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$, we obtain the element
\begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*}
Determine
[list=i]
[*] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) $ have measures less than or equal to $1$
[*] for $\nu =4$, the minimal value of $\kappa \in \mathbb{N}$, for which
\begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}
A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$.
[i]Proposed by Levi Iszler[/i]