[u]Chessboards [/u] Joe B. is playing with some chess pieces on a $6\times 6$ chessboard. Help him find out some things. [b]p1.[/b] Joe B. first places the black king in one corner of the board. In how many of the $35$ remaining squares can he place a white bishop so that it does not check the black king? [b]p2.[/b] Joe B. then places a white king in the opposite corner of the board. How many total ways can he place one black bishop and one white bishop so that neither checks the king of the opposite color? [b]p3.[/b] Joe B. now clears the board. How many ways can he place $3$ white rooks and $3$ black rooks on the board so that no two rooks of opposite color can attack each other? [b]p4.[/b] Joe B. is frustrated with chess. He breaks the board, leaving a $4\times 4$ board, and throws $3$ black knights and $3$ white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.) [b]p5.[/b] Suppose that at some point Joe B. has placed $2$ black knights on the original board, but gets bored of chess. He now decides to cover the $34$ remaining squares with $17$ dominos so that no two overlap and the dominos cover the entire rest of the board. For how many initial arrangements of the two pieces is this possible? Note: Chess is a game played with pieces of two colors, black and white, that players can move between squares on a rectangular grid. Some of the pieces move in the following ways: $\bullet$ Bishop: This piece can move any number of squares diagonally if there are no other pieces along its path. $\bullet$ Rook: This piece can move any number of squares either vertically or horizontally if there are no other pieces along its path. $\bullet$ Knight: This piece can move either two squares along a row and one square along a column or two squares along a column and one square along a row. $\bullet$ King: This piece can move to any open adjacent square (including diagonally). If a piece can move to a square occupied by a king of the opposite color, we say that it is checking the king. If a piece moves to a square occupied by another piece, this is called attacking.

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?

Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$. Proposed by Karl Czakler

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

Show that: a) There are infinitely many positive integers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integers (for instance, $2$ is one such number as $5^2=3^2+4^2$). b) If $n$ is a positive integer which is not a perfect square, and if $x$ is an integer number such that $x^2+(x+1)^2+...+(x+n-1)^2$ is a perfect square, then there are infinitely many positive integers $y$ such that $y^2+(y+1)^2+...+(y+n-1)^2$ is a perfect square.

Given rectangle $ R_1$ with one side $ 2$ inches and area $ 12$ square inches. Rectangle $ R_2$ with diagonal $ 15$ inches is similar to $ R_1.$ Expressed in square inches the area of $ R_2$ is: $ \textbf{(A)}\ \frac92 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ \frac{135}{2} \qquad \textbf{(D)}\ 9 \sqrt{10} \qquad \textbf{(E)}\ \frac{27 \sqrt{10}}{4}$

$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.

Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n) ${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $

Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ [i]naughty[/i] if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is [i]not[/i] in $T$. Prove that 1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements; 2) If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.

The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$. [i]Proposed by David Brodsky[/i]

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

What is $\frac{2^3+2^3}{2^{-3}+2^{-3}}?$ ${ \textbf{(A)}\ \ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 64 $

In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]

In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.

One considers for $n > 2$ the polynomial: $$(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n$$ Show that the degree of this polynomial is $2n - 2$. The polynomial is written in the form $$a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}$$ Prove that $a_2+a_3+...+a_{2n-2}=0$

$a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers.

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

A criminal is at point $X$, and three policemen at points $A, B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A, B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)? by V.Protasov

Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$? $ \textbf{(A)}\ 3\sqrt 2 \qquad\textbf{(B)}\ 3\sqrt 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ \dfrac {16}3 \qquad\textbf{(E)}\ 6 $

$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations $$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$ Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.