Found problems: 85335
2016 Purple Comet Problems, 4
One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the
perimeter of the rectangle.
2016 Swedish Mathematical Competition, 6
Each cell in a $13 \times 13$ grid table is painted in black or white. Each move consists of choosing a subsquare of size either $2 \times 2$ or $9 \times 9$, and painting all white cells of the choosen subsquare black, and painting all its black cells white. It is always possible to get all cells of the original square black, after a finite number of such moves ?
2018 Czech-Polish-Slovak Junior Match, 5
An acute triangle $ABC$ is given in which $AB <AC$. Point $E$ lies on the $AC$ side of the triangle, with $AB = AE$. The segment $AD$ is the diameter of the circumcircle of the triangle $ABC$, and point $S$ is the center of this arc $BC$ of this circle to which point $A$ does not belong. Point $F$ is symmetric of point $D$ wrt $S$. Prove that lines $F E$ and $AC$ are perpendicular.
2022 MOAA, 12
Triangle $ABC$ has circumcircle $\omega$ where $B'$ is the point diametrically opposite $B$ and $C'$ is the point diametrically opposite $C$. Given $B'C'$ passes through the midpoint of $AB$, if $AC' = 3$ and $BC = 7$, find $AB'^2$..
2019 Kyiv Mathematical Festival, 3
There were $2n,$ $n\ge2,$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same non-zero number of points?
Kvant 2019, M2577
Inside the acute-angled triangle $ABC$ we take $P$ and $Q$ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of $\angle BAC$ passing through $P$ and $Q$ intersect the segments $AC$ and $AB$ at the points $B_p\in AC$, $B_q\in AC$, $C_p\in AB$ and $C_q\in AB$, respectively. Let $W$ be the midpoint of the arc $BAC$ of the circle $(ABC)$. The line $WP$ intersects the circle $(ABC)$ again at $P_1$ and the line $WQ$ intersects the circle $(ABC)$ again at $Q_1$. Prove that the points $P_1$, $Q_1$, $B_p$, $B_q$, $C_p$ and $C_q$ lie on a circle.
[i]Proposed by P. Bibikov[/i]
1963 Putnam, B6
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$, define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$
2015 India IMO Training Camp, 3
There are $n\ge 2$ lamps, each with two states: $\textbf{on}$ or $\textbf{off}$. For each non-empty subset $A$ of the set of these lamps, there is a $\textit{soft-button}$ which operates on the lamps in $A$; that is, upon $\textit{operating}$ this button each of the lamps in $A$ changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most $2^{n-1}+1$ operations.
2000 Junior Balkan Team Selection Tests - Romania, 4
Two identical squares havind a side length of $ 5\text{cm} $ are each divided separately into $ 5 $ regions through intersection with some lines. Show that we can color the regions of the first square with five colors and the regions of the second with the same five colors such that the sum of the areas of the resultant regions that have the same colors at superpositioning the two squares is at least $ 5\text{cm}^2. $
1980 IMO Shortlist, 16
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
2009 Stanford Mathematics Tournament, 7
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.
2010 Bosnia And Herzegovina - Regional Olympiad, 1
Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$ $$y-\frac{x+3y}{x^2+y^2}=0$$
2018 India IMO Training Camp, 2
A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.
1995 Czech And Slovak Olympiad IIIA, 3
Five distinct points and five distinct lines are given in the plane. Prove that one can select two of the points and two of the lines so that none of the selected lines contains any of the selected points.
2024 Ukraine National Mathematical Olympiad, Problem 8
There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities.
[i]Proposed by Lior Shayn[/i]
2014 Bulgaria National Olympiad, 3
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
2005 Sharygin Geometry Olympiad, 10.1
A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.
2009 Korea National Olympiad, 1
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $ BIC, CIA, AIB$. Let $ P, Q, R$ be the midpoints of segments $ DI, EI, FI $. Prove that the circumcenter of triangle $PQR $, $M$, is the midpoint of segment $IO$.
2017 Saudi Arabia BMO TST, 4
Let $ABC$ be a triangle with $A$ is an obtuse angle. Denote $BE$ as the internal angle bisector of triangle $ABC$ with $E \in AC$ and suppose that $\angle AEB = 45^o$. The altitude $AD$ of triangle $ABC$ intersects $BE$ at $F$. Let $O_1, O_2$ be the circumcenter of triangles $FED, EDC$. Suppose that $EO_1, EO_2$ meet $BC$ at $G, H$ respectively. Prove that $\frac{GH}{GB}= \tan \frac{a}{2}$
2006 MOP Homework, 7
Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$.
Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.
2003 Cuba MO, 7
Let S(n) be the sum of the digits of the positive integer $n$. Determine
$$S(S(S(2003^{2003}))).$$
2016 AMC 10, 23
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
$\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$
2019 VJIMC, 2
A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$
a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]?
b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]?
[i]Proposed by Arturas Dubickas (Vilnius University).
[/i]
2018 Thailand Mathematical Olympiad, 5
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?
2023 Turkey MO (2nd round), 1
Prove that there exist infinitely many positive integers $k$ such that the equation
$$\frac{n^2+m^2}{m^4+n}=k$$
don't have any positive integer solution.