Found problems: 85335
A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.
[i]Proposed by Australia[/i]
In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?
Find all real numbers $a,b,c$ and $d$ such that:
$a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$
Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are.
[i]Proposed by Sutanay Bhattacharya[/i]
Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.
A game about passing barriers rules that in the $n$th barrier, you need to throw a dice for $n$ times. If the sum of points you get is larger than $2^n$, then you can pass this barrier.
[b](a)[/b] How many barriers can you pass at most?
[b](b)[/b] Find the probablity of passing the first three barriers.
Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]
3. Let us call a bi-squared card $2 \times 1$ regular, if two positive integers are written on it and the number in the upper square is less than the number in the lower square. It is allowed at each move to change both numbers in the following manner: either add the same integer (possibly negative) to both numbers, or multiply each number by the same positive integer, or divide each number by the same positive integer. The card must remain regular after any changes made. What minimal number of moves is sufficient to get any regular card from any other regular card?
Alexey Glebov
circle $O_1$ is tangent to $AC$, $BC$(side of triangle $ABC$) at point $D, E$.
circle $O_2$ include $O_1$, is tangent to $BC$, $AB$(side of triangle $ABC$) at point $E, F$
The tangent of $O_2$ at $P(DE \cap O_2, P \neq E)$ meets $AB$ at $Q$.
A line passing through $O_1$(center of $O_1$) and parallel to $BO_2$($O_2$ is also center of $O_2$) meets $BC$ at $G$, $EQ \cap AC=K, KG \cap EF=L$, $EO_2$ meets circle $O_2$ at $N(\neq E)$, $LO_2 \cap FN=M$.
IF $N$ is a middle point of $FM$, prove that $BG=2EG$
Altitudes $BB_1$ and $CC_1$ of acute triangle $ABC$ intersect at $H$, and $\angle A = 60^{o}$, $AB < AC$. The median $AM$ intersects the circumcircle of $ABC$ at point $K$; $L$ is the midpoint of the arc $BC$ of the circumcircle that does not contain point $A$; lines $B_1C_1$ and $BC$ intersect at point $E$. Prove that $\angle EHL = \angle ABK$.
Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that:
\[ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. \]
Where $S_{XYZ}$ is the surface of the triangle $XYZ$.
Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.
Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation
$$x^3 + ax^2 + bx + c = 0$$
has three real roots creating an arithmetic progression.
Two points $A$ and $B$ are given. Find the locus of points $C$ such that triangle $ABC$ can be covered by a circle with radius $1$.
(Arseny Akopyan)
Find all triples of positive integers $(x,y,z)$ that satisfy the equation
$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$
Given real numbers $a$, $b$, $c$ satisfy inequality $\left| \frac{a^2+b^2-c^2}{ab} \right|<2$. Prove that they also satisfy equalities $\left| \frac{b^2+c^2-a^2}{bc} \right|<2$ and $\left| \frac{c^2+a^2-b^2}{ca} \right| <2$.
Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]
Prove that $$\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}}
\leq\frac{3\sqrt{3}}{2}$$
for any positive real numbers $x, y,z$ such that $x+y+z = xyz.$
[url=https://artofproblemsolving.com/community/c7h236610p10925499]2008 VTRMC #1[/url]
[url=http://www.math.vt.edu/people/plinnell/Vtregional/solutions.pdf]here[/url]
The $1 \times 1$ cells located around the perimeter of a $3 \times 3$ square are filled with the numbers $1,
2, \ldots, 8$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $8$, and in the upper left is the number $6$ (see the figure below).
[img]https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png[/img]
How many different ways to fill the remaining cells are there under these conditions?
[i]Proposed by Mariia Rozhkova[/i]
In the multiplication problem below, $ A$, $ B$, $ C$ and $ D$ are different digits. What is $ A + B$?
\begin{tabular}{cccc}
& A & B & A\\
$\times$ & & C & D\\
\hline
C & D & C & D\\
\end{tabular}
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 9$
Find the integer $n$ such that $6!\times7!=n!$.
[i]2017 CCA Math Bonanza Individual Round #1[/i]
Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.
Decompose a $5$-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.
If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $