Found problems: 85335
Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.
Two people play a game on a graph with $2022$ points. Initially, there are no edges in the graph. They take turns and connect two non-neighbouring vertices with an edge. Whoever makes the graph connected loses. Which player has a winning strategy?
[i]ST, danny2915[/i]
suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. we know that for every $A_i,A_j\in \mathcal F$ that $A_i\supseteq A_j$ we have $3\le |A_i|-|A_j|$. prove that:
$|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$
(20 points)
Let $ABC$ be a triangle with incenter $I$, and let $D$ be the foot of the angle bisector from $A$ to $BC$. Let $\Gamma$ be the circumcircle of triangle $BIC$, and let $PQ$ be a chord of $\Gamma$ passing through $D$. Prove that $AD$ bisects $\angle PAQ$.
Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.
The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.
Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.
Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:
$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$
$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
From a square board $1000\times 1000$ four rectangles $2\times 994$ have been cut off as shown on the picture. Initially, on the marked square there is a centaur - a piece that moves to the adjacent square to the left, up, or diagonally up-right in each move. Two players alternately move the centaur. The one who cannot make a move loses the game. Who has a winning strategy?
[img]https://cdn.artofproblemsolving.com/attachments/c/6/f61c186413b642b5b59f3947bc7a108c772d27.png[/img]
Three vertices of a parallelogram are $(2,-4),(-2,8),$ and $(12,7.)$ Determine the sum of the three possible x-coordinates of the fourth vertex.
Suppose that real number $x$ satisfies $$\sqrt{49-x^2}-\sqrt{25-x^2}=3.$$ What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$?
$
\textbf{(A) }8 \qquad
\textbf{(B) }\sqrt{33}+8\qquad
\textbf{(C) }9 \qquad
\textbf{(D) }2\sqrt{10}+4 \qquad
\textbf{(E) }12 \qquad
$
Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and
$$n^q-q$$
is an integer.
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?
Let $n$ be a positive integer such that $n^5+n^3+2n^2+2n+2$ is a perfect cube. Prove that $2n^2+n+2$ is not a perfect cube.
[i]Proposed by Anastasija Trajanova[/i]
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.
Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$
Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$?
${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$
Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.
Prove that the square of any integer cannot end with two fives.
Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.
Points $K,L,M$ on the sides $AB,BC,CA$ respectively of a triangle $ABC$ satisfy $\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}$. Show that the triangles $ABC$ and $KLM$ have a common orthocenter if and only if $\triangle ABC$ is equilateral.
A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes.
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$
$ABC$ is an acute triangle and $AD$, $BE$ and $CF$ are the altitudes, with $H$ being the point of intersection of these altitudes. Points $A_1$, $B_1$, $C_1$ are chosen on rays $AD$, $BE$ and $CF$ respectively such that $AA_1 = HD$, $BB_1 = HE$ and $CC_1 =HF$. Let $A_2$, $B_2$ and $C_2$ be midpoints of segments $A_1D$, $B_1E$ and $C_1F$ respectively. Prove that $H$, $A_2$, $B_2$ and $C_2$ are concyclic.
(Mykhailo Barkulov)