Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$? $\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

In the country of Princetonia, there are an infinite number of cities, connected by roads. For every two distinct cities, there is a unique sequence of roads that leads from one city to the other. Moreover, there are exactly three roads from every city. On a sunny morning in early July, n tourists have arrived at the capital of Princetonia. They repeat the following process every day: in every city that contains three or more tourists, three tourists are picked and one moves to each of the three cities connected to the original one by roads. If there are $2$ or fewer tourists in the city, they do nothing. After some time, all tourists will settle and there will be no more changing cities. For how many values of n from $1$ to $2020$ will the tourists end in a configuration in which no two of them are in the same city?

Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as: $$a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},$$ where $A_{m}^{k}$ are defined by $$A_{m}^{k}=\binom{m}{k}\cdot k!.$$ Prove that $$a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.$$

Find all integers $n$ such that if \[ 1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n \] are the divisors of $n$, then the sequence \[ d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1} \] forms a permutation of an arithmetic progression. [i](Kritesh Dhakal, Nepal)[/i]

Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$. [i]Proposed by Tristan Shin[/i]

$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as \[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|. \] $ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

Let $\zeta_1, \zeta_2,\ldots$ be identically distributed, independent real-valued random variables with expected value $0$. Suppose that the $\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)$ logarithmic moment-generating function always exists for $\lambda\in\mathbb R$ ($\mathbb E$ is the expected value). Furthermore, let $G\colon\mathbb R \rightarrow \mathbb R$ be a function such that $G(x)\leq \min (|x|, x^2)$. Prove that for small $\gamma >0$ the following sequence is bounded: $$\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}$$ (translated by j___d)

Determine the largest number in the infinite sequence \[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]

Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and \[ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. \] Prove that \[ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. \]

Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$

The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$. [i]Proposed by Lewis Chen[/i]

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? [list] [*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; [*] $a_2 - a_1$ is not divisible by $m$. [/list] [i]Holden Mui[/i]