Found problems: 85335
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ \text{Infinitely many}
\qquad\textbf{e)}\ \text{None of above}
$
Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was
$\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$
For any natural $n$, $f(n)$ is the number of labeled digraphs with $n$ vertices such that for any vertex the number if in-edges is equal to the number of out-edges and the total of (in+out) edges is even. Let $g(n)$ be the odd-analogous of $f(n)$. Find $g(n)-f(n)$ with proof .
[hide=original formulation] Dado $n$ natural, seja $f(n)$ o número de grafos rotulados direcionados com $n$ vértices de modo que em cada vértice o número de arestas que chegam é igual ao número de
arestas que saem e o número de arestas total do grafo é par . Defina $g(n)$ analogamente trocando "par" por "ímpar" na definição acima. Calcule $f(n) - g (n)$.
(Observação: Um grafo rotulado direcionado é um par $G = (V, E)$ onde $V = \{1, 2, …, n\}$ e $E$ é um subconjunto de $V^2 -\{(i, i); 0 < i < n + 1\}$).[/hide]
Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.
Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base $10$) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base $10$) number $\underline{b} \, \underline{c} \, \underline{a}$ is also a multiple of $x$.
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}
$N$ lines are drawn on the plane that divide it into a certain number for finite and endless parts. For which number of straight lines $n$ can there be more finite than infinite among the resulting level parts?
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares.
[i](V. Dolnikov)[/i]
Find all pairs $(a, b)$ of positive integers such that $a \le b$ and
$$ \gcd(x, a) \gcd(x, b) = \gcd(x, 20) \gcd(x, 22) $$
holds for every positive integer $x$.
Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player $A$ guarantees to take $k$ balls, find the maximum possible value of $k$.
Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?
Let $x\neq y$. Two sequences $x,a_1,a_2,a_3,y$ and $b_1,x,b_2,b_3,y,b_4$ are arithmetic sequence. Then $\frac{b_4-b_3}{a_2-a_1}=$________.
How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?
We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element.
a) If in $A$, $1+1=0$, prove that $A$ has property $(P)$ if and only if $A$ is a field.
b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$.
[i]Dan Schwarz[/i]
Show that there exists a degree $58$ monic polynomial
$$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$
such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.
The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
[i]Proposed by Athanasios Kontogeorgis, Greece[/i]
The edges of a graph $G$ are coloured in two colours. Such that for each colour all the connected components of this graph formed by edges of this colour contains at most $n>1$ vertices. Prove there exists a proper colouring for the vertices of this graph with $n$ colours.
Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .