Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

Let $S$ be the set of all $n$-tuples of real numbers, with the property that among the numbers $x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n$ the least is equal to $0$, and the greatest is equal to $1$. Determine $$\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).$$

The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.

Let $A_{12}$ denote the answer to problem $12$. There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}>B>C>D>0$ and \[\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C},\] where $\overline{A_{12}BCD}$ denotes the four digit base $10$ integer. Compute $B+C+D$.

How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$) of length $2013$ have letters in alphabetic order?

Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.

On a party with $99$ guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are $99$ chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair $c$. If some chair adjacent to $c$ is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to $c$ are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least $k$ chairs are occupied. Determine the largest $k$ for which Ann can reach the goal, regardless of Bob's play.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

$(\sqrt6 + \sqrt7)^{1000}$ in base ten has a tens digit of $a$ and a ones digit of $b$. Determine $10a + b$.

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

The 24 unshaded squares in the 5 × 5 grid below can be tiled with twelve 1 × 2 tiles. One such tiling is shown. Find the number of ways the grid can be tiled. [center][img]https://snag.gy/KMoPrF.jpg[/img][/center]

Given a set $S \subset R^+$, $S \ne \emptyset$ such that for all $a, b, c \in S$ (not necessarily distinct) then $a^3 + b^3 + c^3 - 3abc$ is rational number. Prove that for all $a, b \in S$ then $\frac{a - b}{a + b}$ is also rational.

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list] [*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or [*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down). [/list] Thus, for any $k$, the ant can choose to go to one of eight possible points. \\ Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having $n$ inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability $1/2$ independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let $p_n$ denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that $p_n\to 1$ if $n\to\infty$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on $AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$, respectively. Prove that $A,O_P,O_Q,R$ are concylic. [i]proposed by andychang[/i]

Let $P$ be a point on a circle $C_1$ and let $C_2$ be a circle with center $P$ that intersects $C_1$ at two points Q and R. The circle $C_3$, with center $Q$ and which passes through $R$, intersects $C_2$ at another point S, as in figure. Shows that $QS$ is tangent to $C_1$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/f48d414c68c33c4efaf4d6c8bebcf6f1fad4ba.png[/img]