Found problems: 85335
A table has $m$ rows and $n$ columns. The following permutations of its $mn$ elements are permitted: an arbitrary permutation leaving each element in the same row (a“horizontal move”) and an arbitrary permutation leaving each element in the same column (a “vertical move”). Find the number $k$ such that any permutation of $mn$ elements can be obtained by $k$ permitted moves but there exists a permutation that cannot be achieved in less than $k$ moves.
(A Andjans, Riga0
The exchange rates of the Dollar and the German mark during the week changed as follows:
$\begin{tabular}{|l|l|l|}
\hline
& Dollar & Mark \\ \hline
Monday & 4000 rub. & 2500 rub. \\ \hline
Tuesday & 4500 rub. & 2800 rub.\\ \hline
Wednesday & 5000 rub. & 2500 rub.\\ \hline
Thursday & 4500 rub. & 3000 rub.\\ \hline
Friday & 4000 rub. & 2500 rub.\\ \hline
Saturday & 4500 rub. & 3000 rub.\\ \hline
\end{tabular}$
What percentage was the maximum possible increase in capital this week by playing on changes in the exchange rates of these currencies? (The initial capital was in rubles. The final capital should also be in rubles. During the week, the available money can be distributed as desired into rubles, dollars and marks. The selling and purchasing rates are considered the same.)
Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?
A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). The points of tangency of the circumcircles of triangles \(ABE\) and \(CDE\) with their common external tangents lie on a circle \(\omega\). The points of tangency of the circumcircles of triangles \(ADE\) and \(BCE\) with their common external tangents lie on a circle \(\gamma\). Prove that the centers of circles \(\omega\) and \(\gamma\) coincide.
Find the sum of the four smallest prime divisors of $2016^{239} - 1$.
Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$
In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.
Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.
There are two kinds of creatures living in the flatland of Pentagonia: the Spires ($S$) and the Bones ($B$). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$.
Every year on [i]Great Day of Division[/i] (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$'s and a $B$; each $B$ in an $S$ and a $B$. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$-being. Deaths do not occur.
Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.
Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$?
[i]Proposed by Adam Bertelli[/i]
Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.
Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.
Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy:
\[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\]
Find the maximum and minimum value of the sum $S=x_1+x_2.$
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
Let $n$ be a positive integer.
Find the number of sequences $a_1,a_2,...,a_k$ of different numbers from $\{ 1,2,3,...,n\}$ with the following property:
for every number $a$ of the sequence (except the first one) there exists a previous number $b$ such that their difference is $1$ (so $a-b= \pm 1$)
Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers.
Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that:
[b](1)[/b] $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$.
[b](2)[/b] The number of all pairs of integers $(i,j)$ satisfying $1\leq i<j\leq m$ and $a_i>a_j$ is even.
For example, when $n=4$, the number of all such lists $(a_1,a_2,\cdots,a_m)$ is $6$, and these lists are $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$.
Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
Solve the following system (over the real numbers):
\[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\
3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]
A square $ n \times n $, ($ n> 2 $) contains nonzero real numbers. It is known that every number is exactly $ k $ times smaller than the sum of all the numbers in its row or sum of all number in its column. For which real numbers $ k $ is this possible?
Consider a sequence of words each consisting of two letters, $A$ and $B$ . The first word is "$A$" , while the second word is "$B$" . The $k$-th word is obtained from the ($k -2$)-nd by writing after it the ($k -1$)th one. (So the first few elements of the sequence are "$A$" , "$B$" ,"$AB$" , "$BAB$" , "$ABBAB$" . ) Does there exist in this sequence a "periodical" word, i.e. a word of the form $P P P ... P$ , where $P$ is a word , repeated at least once?
(Remark: For instance, the word $BABBBABB$ is of the form $PP$ , in which $P$ is repeated exactly once . )
(A. Andjans, Riga)
Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.
Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that
$$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.