Found problems: 85335
A positive integer $n$ is [i]funny[/i] if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds:
$$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$
Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble
a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?
What is the smallest positive odd integer $n$ such that the product \[2^{1/7}2^{3/7}\cdots2^{(2n+1)/7}\] is greater than $1000$? (In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)
$\textbf{(A) }7\qquad\textbf{(B) }9\qquad\textbf{(C) }11\qquad\textbf{(D) }17\qquad \textbf{(E) }19$
In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.
For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle.
Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game.
Determine all pairs $(m,n)$ for which Renate has a winning strategy.
Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.
[i]Proposed by S. Berlov, S. Ivanov[/i]
From the two cities of Dobruisk and Bodruisk, the distance between which is $40$ km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at $23$ km/h and Bodie was traveling at $17$ km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of $40$ km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of $30$ km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)
Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre.
(V . Pogrebnyak , year 12 student , Vinnitsa,)
Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold:
$$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$
$$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$
$$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$
$$...$$
$$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$
$$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$
$$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$
[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$
[i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$
In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.
A survey was taken in Ms. Susan's class to see what grades the class received:
[center][img width=35]https://cdn.artofproblemsolving.com/attachments/5/c/e96cb42de6d5e1b100f37bbb71768d399842cb.png[/img][/center]
What percent of the class received an "A"?
$\textbf{(A) }3\%\qquad\textbf{(B) }5\%\qquad\textbf{(C) }10\%\qquad\textbf{(D) }15\%\qquad\textbf{(E) }27\%$
Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$.
[i]Gabriel Marsanu, Andrei Nedelcu[/i]
Triangle $ABC$ has right angle $A$, with $AB=AC=6$. Point $P$ is located within triangle $ABC$ such that the areas of triangles $ABP$, $BCP$, and $ACP$ are equal. Express $CP$ in simplest radical form.
Let $ S$ be the set of all points with coordinates $ (x,y,z)$, where $ x, y,$ and $ z$ are each chosen from the set $ \{ 0, 1, 2\}$. How many equilateral triangles have all their vertices in $ S$?
$ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 88$
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.
$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have
\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$
Find all pairs $(a,b,c)$ of positive integers such that $$\frac{a}{2^a}=\frac{b}{2^b}+\frac{c}{2^c}$$
On a table, we have ten thousand matches, two of which are inside a bowl. Anna and Bernd play the following game: They alternate taking turns and Anna begins. A turn consists of counting the matches in the bowl, choosing a proper divisor $d$ of this number and adding $d$ matches to the bowl. The game ends when more than $2024$ matches are in the bowl. The person who played the last turn wins. Prove that Anna can win independently of how Bernd plays.
[i](Richard Henner)[/i]
Positive reals $a,b,c$ satisfy $abc=1$. Prove that
$\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.