How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$.

In the sequence of numbers 1, 3, 2, ... each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \minus{}1$

[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles. [img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img] [b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by: $D(n) = n$ if $1 \le n \le 9$ $D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$ where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ . [b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ . [b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians. [img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img] [b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

Consider an $m$-by-$n$ grid of unit squares, indexed by $(i, j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i, j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i, j)$ with $i \leq m-1$ and $j \leq n-1$ and the squares $(i+1, j),(i, j+1)$, and $(i+1, j+1)$ are unoccupied, then a legal move is to slide the coin from $(i, j)$ to $(i+1, j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?

The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$

Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.

Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?

Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.

Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\cdots+98+99+100\] in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?

Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

Let $a_1,a_2,\ldots$ be a sequence defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\geq 1$. Find \[\sum_{n=1}^\infty \dfrac{a_n}{4^{n+1}}.\]

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: [center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center] [center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center] Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy. [i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a $2020$-term strictly increasing geometric sequence with an integer common ratio . Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_0$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_0$, what is the closest integer to $\log_{2} d$?