The set $\{1,2,\dots\>,n\}$ is called $P$. The function $f: P \to \{1,2,\dots\>,m\}$ satisfies \[f(A\cap B)=\min (f(A), f(B)).\] What is the relationship between the number of possible functions $f$ with the sum $\displaystyle \sum_{j=1}^m j^n$? There is a nice and easy solution to this. Too bad I did not think of it...

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

Let $n$ be a natural number. The set $A{}$ of natural numbers has the following property: for any natural number $m\leqslant n$ in the set $A{}$ there is a number divisible by $m{}$. What is the smallest value that the sum of all the elements of the set $A{}$ can take? [i]Proposed by A. Kuznetsov[/i]

Lines $\mathit{L}_1,\mathit{L}_2,\dots,\mathit{L}_{100}$ are distinct. All lines $\mathit{L}_{4n}$, $n$ a positive integer, are parallel to each other. All lines $\mathit{L}_{4n-3}$, $n$ a positive integer, pass through a given point $\mathit{A}$. The maximum number of points of intersection of pairs of lines from the complete set $\{\mathit{L}_1,\mathit{L}_2,\dots,\mathit{L}_{100}\}$ is $\textbf{(A) }4350\qquad\textbf{(B) }4351\qquad\textbf{(C) }4900\qquad\textbf{(D) }4901\qquad \textbf{(E) }9851$

Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$. a) Prove that $f$ attains its minimum and its maximum. b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or local minimum or maximum.

An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table.

[b]p33.[/b] Lines $\ell_1$ and $\ell_2$ have slopes $m_1$ and $m_2$ such that $0 < m_2 < m_1$. $\ell'_1$ and $\ell'_2$ are the reflections of $\ell_1$ and $\ell_2$ about the line $\ell_3$ defined by $y = x$. Let $A = \ell_1 \cap \ell_2 = (5, 4)$, $B = \ell_1 \cap \ell_3$, $C = \ell'_1 \cap \ell'_2$ and $D = \ell_2 \cap \ell_3$. If $\frac{4-5m_1}{-5-4m_1} = m_2$ and $\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41$, compute the area of quadrilateral $ABCD$. [b]p34.[/b] Suppose $S(m, n) = \sum^m_{i=1}(-1)^ii^n$. Compute the remainder when $S(2020, 4)$ is divided by $S(1010, 2)$. [b]p35.[/b] Let $N$ be the number of ways to place the numbers $1, 2, ..., 12$ on a circle such that every pair of adjacent numbers has greatest common divisor $1$. What is $N/144$? (Arrangements that can be rotated to yield each other are the same). [b]p36.[/b] Compute the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose circles $\mathit{W}_1$ and $\mathit{W}2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.* [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(10.1cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; /* image dimensions */ pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773); /* draw figures */ draw(circle(O_2, 4.000000000000000)); draw(circle(O_1, 3.000000000000000)); draw((4.048892687647541,4.413249028538064)--B_2); draw(B_2--A_2); draw(A_2--(4.048892687647541,4.413249028538064)); draw((4.048892687647541,4.413249028538064)--B_1); draw(B_1--A_1); draw(A_1--(4.048892687647541,4.413249028538064)); /* dots and labels */ dot(O_2,dotstyle); label("$O_2$", (7.788512439159622,6.243082420501817), NE * labelscalefactor); dot(O_1,dotstyle); label("$O_1$", (2.298205165350667,6.929370829727937), NE * labelscalefactor); dot(M,dotstyle); label("$M$", (4.383466101076183,8.935444641311980), NE * labelscalefactor); dot((4.048892687647541,4.413249028538064),dotstyle); label("$N$", (3.855551940133015,3.761885864068922), NE * labelscalefactor); dot(B_2,dotstyle); label("$B_2$", (10.19052187145104,9.463358802255147), NE * labelscalefactor); dot(A_2,dotstyle); label("$A_2$", (11.80066006232771,4.659339937672310), NE * labelscalefactor); dot(B_1,dotstyle); label("$B_1$", (1.981456668784765,10.09685579538695), NE * labelscalefactor); dot(A_1,dotstyle); label("$A_1$", (0.08096568938935705,3.973051528446190), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] Show that \[\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.\] *[size=80]Given a triangle ABC, the incentre of the triangle is defined to be the intersection of the angle bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram. Note also that the diagram serves only as an aid and is not necessarily drawn to scale.[/size]

Find the sum of all integers $n$ such that \[n^4+n^3+n^2+n+1\] is a perfect square.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

Consider a square of sidelength $ n$ and $ (n\plus{}1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.

We call a set $ A$ a good set if it has the following properties: 1. $ A$ consists circles in plane. 2. No two element of $ A$ intersect. Let $ A,B$ be two good sets. We say $ A,B$ are equivalent if we can reach from $ A$ to $ B$ by moving circles in $ A$, making them bigger or smaller in such a way that during these operations each circle does not intersect with other circles. Let $ a_{n}$ be the number of inequivalent good subsets with $ n$ elements. For example $ a_{1}\equal{} 1,a_{2}\equal{} 2,a_{3}\equal{} 4,a_{4}\equal{} 9$. [img]http://i5.tinypic.com/4r0x81v.png[/img] If there exist $ a,b$ such that $ Aa^{n}\leq a_{n}\leq Bb^{n}$, we say growth ratio of $ a_{n}$ is larger than $ a$ and is smaller than $ b$. a) Prove that growth ratio of $ a_{n}$ is larger than 2 and is smaller than 4. b) Find better bounds for upper and lower growth ratio of $ a_{n}$.

How many five digit numbers are divisible by $3$ and contain the digit $6$?

Let $\{a_n \}_{n=1}^{\infty}$ be a sequence such that $a_1=\frac 12$ and \[a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.\] Prove that for every positive integer $n,$ we have $\sum_{k=1}^n a_k <1.$

Find the smallest positive integer that is a multiple of $18$ and whose digits can only be $4$ or $7$.

Is it possible to find $2008$ infinite arithmetical progressions such that there exist finitely many positive integers not in any of these progressions, no two progressions intersect and each progression contains a prime number bigger than $2008$?

Let $x$, $y$, and $z$ be nonnegative integers that are less than or equal to 100. Suppose that $x + y + z$, $xy + z$, $x + yz$, and $xyz$ are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples $(x, y, z)$.

Find all real numbers $\alpha$ for which the equality \[\lfloor \sqrt{n}+\sqrt{n+\alpha}\rfloor =\lfloor \sqrt{4n+1}\rfloor\] holds for all positive integers $n$.

Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$. [list] a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$. b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$. [/list]

Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.

Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?

Rumburak kidnapped $31$ members of party $A$ , $28$ members of party $B$, $23$ members of party $C$, $19$ members of Party $D$ and each of them in a separate cell. After work out occasionally they could walk in the yard and talk. Once three people started to talk to each other members of three different parties, Rumburak re-registered them to the fourth party as a punishment.(They never talked to each other more than three kidnapped.) a) Could it be that after some time all were abducted by members of one party? Which? b) Determine all four positive integers of which the sum is $101$ and which as the numbers of kidnapped members of the four parties allow the Rumburaks all of them became members of one party over time.

How many permutations $(s_1, s_2,...,s_n) $of $(1,2 ,...,n)$ are there satisfying the condition $s_i > s_j$ for all $i \ge j + 3$ when $n = 5$ and when $n = 7$?

Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.