On a bookcase there are $ n \geq 3$ books side by side by different authors. A librarian considers the first and second book from left and exchanges them iff they are not alphabetically sorted. Then he is doing the same operation with the second and third book from left etc. Using this procedure he iterates through the bookcase three times from left to right. Considering all possible initial book configurations how many of them will then be alphabetically sorted?

What is the area of a triangle with side lengths $17$, $25$, and $26$? [i]2019 CCA Math Bonanza Lightning Round #3.2[/i]

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$

Let $ABC$ be a triangle such that the angular bisector of $\angle BAC$, the $B$-median and the perpendicular bisector of $AB$ intersect at a single point $X$. Let $H$ be the orthocenter of $ABC$. Show that $\angle BXH = 90^{\circ}$. [i]Proposed by usjl[/i]

Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once. At the beginning the soldier's back is faced at the edge of the grid.

A circle with radius $5$ is inscribed in a right triangle with hypotenuse $34$ as shown below. What is the area of the triangle? Note that the diagram is not to scale.

Let a neighborhood basis of a point $ x$ of the real line consist of all Lebesgue-measurable sets containing $ x$ whose density at $ x$ equals $ 1$. Show that this requirement defines a topology that is regular but not normal. [i]A. Csaszar[/i]

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property: if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations. (a) Show that, for each initial configuration, Harry stops after a finite number of operations. (b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$. [i]Proposed by David Altizio, USA[/i]

Given a $n\times n$ table with non-negative real entries such that the sums of entries in each column and row are equal, a player plays the following game: The step of the game consists of choosing $n$ cells, no two of which share a column or a row, and subtracting the same number from each of the entries of the $n$ cells, provided that the resulting table has all non-negative entries. Prove that the player can change all entries to zeros.

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

A ladder is a non-decreasing sequence $a_1, a_2, \dots, a_{2020}$ of non-negative integers. Diego and Pablo play by turns with the ladder $1, 2, \dots, 2020$, starting with Diego. In each turn, the player replaces an entry $a_i$ by $a_i'<a_i$, with the condition that the sequence remains a ladder. The player who gets $(0, 0, \dots, 0)$ wins. Who has a winning strategy? [i]Proposed by Violeta Hernández[/i]

Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.

For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements. ($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$. ($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$. ($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$. $ (a)$ Show that if all six of these statements are true then the hexagon is cyclic. $ (b)$ Prove that, in fact, five of the six statements suffice.

Let $X$ be a set with $n$ elements and $0 \le k \le n$. Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$). Let $b_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at most $k$ common components. (a) Show that $a_{n,k} \cdot b_{n,k-1} \le n!$. (b) Let $p$ be prime, and find the exact value of $a_{p,2}$.

The numbers $1,2,\dots,4022$ are placed to the cells of a $2 \times 2011$ chessboard in such a way that successive numbers should be inside cells with common sides. How many such arrangements are there? $\textbf{(A)}\ 16168444 \qquad\textbf{(B)}\ 12168440 \qquad\textbf{(C)}\ 10088242 \qquad\textbf{(D)}\ 8084224 \qquad\textbf{(E)}\ \text{None}$

Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave." [i]L. Fejes- Toth, E. Makai[/i]

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$, prove that $BE = CF$. (Oleksii Karliuchenko)

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $

Prove that $1^{2011}+2^{2011}+3^{2011}+...+2012^{2011} $ is divisible by $2025078$.