Found problems: 85335
Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]
Given a circumscribed trapezium $ ABCD$ with circumcircle $ \omega$ and 2 parallel sides $ AD,BC$ ($ BC<AD$). Tangent line of circle $ \omega$ at the point $ C$ meets with the line $ AD$ at point $ P$. $ PE$ is another tangent line of circle $ \omega$ and $ E\in\omega$. The line $ BP$ meets circle $ \omega$ at point $ K$. The line passing through the point $ C$ paralel to $ AB$ intersects with $ AE$ and $ AK$ at points $ N$ and $ M$ respectively. Prove that $ M$ is midpoint of segment $ CN$.
Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.
A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2).
Is it possible, starting with $9$ white coins, to have $9$ black coins?.
prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths.
[i]proposed by Sina Rezayi[/i]
Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements.
[b]a)[/b] $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real.
[b]b)[/b] $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $
[i]Laurențiu Panaitopol[/i]
Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left(
\begin{array}{cc}
1 & 2 \\
3 & 6 \end{array}
\right)$
Patrick tosses four four-sided dice, each numbered $1$ through $4$. What's the probability their product is a multiple of four?
Alice and Bob are playing the following game:
They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden).
Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression.
Which player has a winning strategy?
Determine the natural numbers a, b, c s.t. :
$ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $
The challenge here is to come up with as basic solution as possible.
There is house with $2^n$ rooms and every room has one light bulb and light switch. But wiring was connected wrong, so light switch can turn on light in some another room. Master want to find what switch connected to every light bulb. He use next practice: he send some workers in the some rooms, then they turn on switches in same time, then they go to master and tell him, in what rooms light bulb was turned on.
a) Prove that $2n$ moves is enough to find, how switches are connected to bulbs.
b) Is $2n-1$ moves always enough ?
Consider the closed plane curves $C_i$ and $C_o,$ their respective lengths $|C_i|$ and $|C_o|,$ the closed surfaces $S_i$ and $S_o,$ and their respective areas $|S_i|$ and $|S_o|.$ Assume that $C_i$ lies inside $C_o$ and $S_i$ inside $S_o.$ (Subscript $i$ stands for "inner," $o$ for "outer.") Prove the correct assertions among the following four, and disprove the others.
(i) If $C_i$ is convex, $|C_i| \le |C_o|.$
(ii) If $S_i$ is convex, $|S_i| \le |S_o|.$
(iii) If $C_o$ is the smallest convex curve containing $C_i,$ then $|C_o| \le |C_i|.$
(iv) If $S_o$ is the smallest convex surface containing $S_i,$ then $|S_o| \le |S_i|.$
You may assume that $C_i$ and $C_o$ are polygons and $S_i$ and $S_o$ polyhedra.
Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$
Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$
[Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]
Determine whether there exist ten sets $A_1$, $A_2$, $\dots$, $A_{10}$ such that
(i) each set is of the form $\{a,b,c\}$, where $a \in \{1,2,3\}$, $b \in \{4,5,6\}$, $c \in \{7,8,9\}$,
(ii) no two sets are the same,
(iii) if the ten sets are arranged in a circle $(A_1, A_2, \dots, A_{10})$, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: $A_{10}$ is adjacent to $A_1$.)
Find all functions $ f: \mathbb{Z} \minus{} \{ 0 \} \rightarrow \mathbb{Q}$ satisfying:
$ f \left( \frac{x\plus{}y}{3} \right)\equal{}\frac {f(x)\plus{}f(y)}{2},$ whenever $ x,y,\frac{x\plus{}y}{3} \in \mathbb{Z} \minus{} \{ 0 \}.$
Let $w, x, y, z$ be integers from $0$ to $3$ inclusive. Find the number of ordered quadruples of $(w, x, y, z)$ such that $5x^2 + 5y^2 + 5z^2 - 6wx-6wy -6wz$ is divisible by $4$.
[i]Proposed by Timothy Qian[/i]
How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$
A circle is divided into $2n$ equal by $2n$ points. Ali draws $n+1$ arcs, of length $1,2,\ldots,n+1$. Prove that we can find two arcs, such that one of them is inside in the other one.
Let $P$ be a point in rectangle $ABCD$ such that the area of $PAB$ is $20$ and the area of $PCD$ is $24$. Find the area of $ABCD$.
Let $x $ and $y $ be positive integers such that
$[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$.Prove that one of the two numbers $x $ and $y $ divide the other.
(Here $[a,b] $ denote the least common multiple of $a $ and $b $).
Proposed by Dusan Djukic.
Prove that $$x^4 + 2x^2 + 2x + 2$$ is not the product of two polynomials $x^2 + ax + b$ and $x^2 + cx + d$ in which $a$, $b$, $c$, $d$ are integers.
A teacher gives each student in the class the following task in their exercise book .
"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."
Prove that each student would obtain the same result .
[img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img]
( A . K . Tolpygo, Kiev)
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$