Found problems: 85335
Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that
[b]i.)[/b] no three points of $ S$ are collinear, and
[b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$
Prove that:
\[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n}
\]
Let $A$ be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest $k$ for which one can ensure that we can color elements of $A$ with $k$ colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.
Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence:
$a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$
Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$.
($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$
Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?
Prove that if the polynomial $ f(x) = ax^3 + bx^2 + cx + d $ with integer coefficients takes odd values for $ x = 0 $ and $ x = 1 $, then the equation $ f(x) = 0 $ has no integer roots.
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $
Prove that $xyz \leq 36$
Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.
Is there exist a function $f:\mathbb {N}\to \mathbb {N}$ with for $\forall m,n \in \mathbb {N}$ $$f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?$$
Let $ABCD$ be a cyclic quadrilateral such that $|AD| =|BD|$ and let $M$ be the intersection of its diagonals. Furthermore, let $N$ be the second intersection of the diagonal $AC$ with the circle passing through points $B, M$ and the center of the circle inscribed in triangle $BCM$. Prove that $AN \cdot NC = CD \cdot BN$
Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy
$11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$.
A fractional number $x$ is called [i][b]pretty[/b][/i] if it has finite expression in base$-b$ numeral system, $b$ is a positive integer in $[2;2022]$. Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$
Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$
Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.
There are $n -1$ red balls, $n$ green balls and $n + 1$ blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is $299$. What is the value of $n$?
Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black)
The figure can be rotated $90°, 180°$ or $270°$.
Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.
Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated $ 45^\circ$, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $ B$ from the line on which the bases of the original squares were placed?
[asy]unitsize(1inch);
defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((0,0)--((1/3) + 3*(1/2),0));
fill(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle, rgb(.7,.7,.7));
draw(((1/6),0)--((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6),(1/2))--cycle);
draw(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle);
draw(((1/6) + 1,0)--((1/6) + 1,(1/2))--((1/6) + (3/2),(1/2))--((1/6) + (3/2),0)--cycle);
draw((2,0)--(2 + (1/3) + (3/2),0));
draw(((2/3) + (3/2),0)--((2/3) + 2,0)--((2/3) + 2,(1/2))--((2/3) + (3/2),(1/2))--cycle);
draw(((2/3) + (5/2),0)--((2/3) + (5/2),(1/2))--((2/3) + 3,(1/2))--((2/3) + 3,0)--cycle);
label("$B$",((1/6) + (1/2),(1/2)),NW);
label("$B$",((2/3) + 2 + (1/4),(29/30)),NNE);
draw(((1/6) + (1/2),(1/2)+0.05)..(1,.8)..((2/3) + 2 + (1/4)-.05,(29/30)),EndArrow(HookHead,3));
fill(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle, rgb(.7,.7,.7));
draw(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle);[/asy]$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ \sqrt {2}\qquad \textbf{(C)}\ \frac {3}{2}\qquad \textbf{(D)}\ \sqrt {2} \plus{} \frac {1}{2}\qquad \textbf{(E)}\ 2$
Let $ABC$ be a triangle with incenter $I$. Points $E$ and $F$ are on segments $AC$ and $BC$ respectively such that, $AE=AI$ and $BF=BI$. If $EF$ is the perpendicular bisector of $CI$, then $\angle{ACB}$ in degrees can be written as $\frac{m}{n}$ where $m$ and $n$ are co-prime positive integers. Find the value of $m+n$.
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
Find all $a,b \in \mathbb{N}$ such that
$$2049^ba^{2048}-2048^ab^{2049}=1.$$
[i]Proposed by fattypiggy123 and 61plus[/i]
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men?
$\textbf{(A)}\ \frac{1}{3}\qquad
\textbf{(B)}\ \frac{3}{8}\qquad
\textbf{(C)}\ \frac{2}{5}\qquad
\textbf{(D)}\ \frac{5}{12}\qquad
\textbf{(E)}\ \frac{3}{5}$
Let $N(a,b)$ be the number of ways to cover a table $a \times b$ with domino tiles. Let $M(a,2b+1)$ be the number of ways to cover a table $a \times 2b+1$ with domino tiles, such that there are [b]no[/b] vertical tile in the central column. Prove that
$$M(2m,2n+1)=2^m \cdot N(2m,n)\cdot N(2m,n-1)$$