Found problems: 85335
Find all positive integers $n$ such that \[2n = \varphi(n)^{\frac{2}{3}}(\varphi(n)^{\frac{2}{3}}+1)\]
Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$. Construct an $n \times 3$ table where the entries of the $k$-th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.
[list=a]
[*] Prove that at least three entries remain at the end of the procedure.
[*] Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain.
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Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.
On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?
In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$.
(a) Show that the lines $AD, BC$ and $PQ$ go through the same point.
(b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.
Let $P(x)$ be a monic polynomial of degree $3$. Suppose that $P(x)$ has remainder $R(x)$ when it is divided by $(x - 1)(x - 4)$ and $2R(x)$ when it is divided by $(x - 2)(x - 3)$. Given that $P(0) = 5$, find $P(5)$.
Call a positive real number special if it has a decimal representation that consists entirely of digits $ 0$ and $ 7$. For example, $ \frac{700}{99} \equal{} 7.\overline{07} \equal{} 7.070707\cdots$ and $ 77.007$ are special numbers. What is the smallest $ n$ such that $ 1$ can be written as a sum of $ n$ special numbers?
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad\\
\textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.}$
Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following:
$$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$
For how many positive integers $n$ is \[\left( 1999+\frac{1}{2}\right)^{n}+\left(2000+\frac{1}{2}\right)^{n}\] an integer?
Let $x$ and $y$ be real numbers such that
\[
2 < \frac{x - y}{x + y} < 5.
\]
If $\frac{x}{y}$ is an integer, what is its value?
There are $n$ coins in a row, $n\geq 2$. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$. No move is allowed if all $n$ coins are tails.
Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $ \lfloor\frac {2^m}{3}\rfloor $ moves
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions:
(i) $f(x)$ is an integer if and only if $x$ is an integer;
(ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.
Let $ABC$ be a triangle with $BC = a, AC = b$ and $AB = c$. A point $P$ inside the triangle has the property that for any line passing through $P$ and intersects the lines $AB$ and $AC$ in the distinct points $E$ and $F$ we have the relation $\frac{1}{AE} +\frac{1}{AF} =\frac{a + b + c}{bc}$. Prove that the point $P$ is the center of the circle inscribed in the triangle $ABC$.
Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:
(a) $P(2017)=2016$ and
(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.
A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^2=4ax$ is always $90^\circ$. The locus of $P$ is
$\textbf{(A)}~\text{a parabola}$
$\textbf{(B)}~\text{a circle}$
$\textbf{(C)}~\text{an ellipse}$
$\textbf{(D)}~\text{a straight line}$
Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$\gcd(f(x),y)f(xy)=f(x)f(y)$$ for all positive integers $x, y$.
[b]a)[/b] Let be a sequence $ \left( x_n \right)_{n\ge 1} $ defined by the recursion $ x_{n+1}=\frac{1+x_n}{1-x_n} , $ with $ x_1=2006. $ Calculate $ \lim_{n\to\infty } \frac{x_1+x_2+\cdots +x_n}{n} . $
[b]b)[/b] Prove that if a convergent sequence $ \left( s_n \right)_{n\ge 1} $ verifies $ a_{2^n} =na_n , $ for any natural numbers $ n, $ then $ a_n=0, $ for any natural numbers $ n. $
[i]Cornel Stoicescu[/i]
Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$.
a) Prove there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,15R$.
b) Does there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,2R$?
Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$
Let $f(x)=x^3-x^2$. For a given value of $x$, the graph of $f(x)$, together with the graph of the line $c+x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.
In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.
In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?