Let $S_n$ be the sum of the first $n$ term of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$? $\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.

Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?

Right triangle $ABC$ has a right angle at $A.$ Points $D$ and $E$ respectively lie on $\overline{AC}$ and $\overline{BC}$ so that $\angle BDA \cong \angle CDE.$ If the lengths $DE,$ $DA,$ $DC,$ and $DB,$ in this order, form an arithmetic sequence of distinct positive integers, then the set of all possible areas of $\triangle ABC$ is a subset of the positive integers. Compute the smallest element in this set that is greater than $1000.$

[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below: [asy] size(250);defaultpen(linewidth(0.4)); dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin); int i; for(i=0; i<360; i=i+60) { dot(1*dir(i)^^2*dir(i)^^3*dir(i)); dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i)); dot(shift(1, sqrt(3))*1*dir(i)); dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i)); dot(shift(-12,0)*origin+sqrt(3)*dir(i+30)); dot(shift(-24,0)*origin+1*dir(i)); } label("$1$", (-36, -5), S); label("$7$", (-24, -5), S); label("$19$", (-12, -5), S); label("$37$", (0, -5), S); label("Centered Hexagonal Numbers", (-18,-10), S);[/asy] Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.

Let $M=\{\frac{1}{n}|n\in\mathbb{N}\}$. Numbers $a_1,a_2,\ldots,a_l$ from an [i]arithmetic progression of maximum length[/i] $l$ $(l\geq 3)$ if they verify the properties: a) numbers $a_1,a_2,\ldots,a_l$ from a finite arithmetic progression; b) there is no number $b\in M$ such that numbers $b,a_1,a_2,\ldots,a_l$ or $a_1,a_2,\ldots,a_l, b$ form a finite arithmetic progression. For example numbers $\frac{1}{6},\frac{1}{3},\frac{1}{2}\in M$ form an arithmetic progression of maximum length $3$. a) FInd an arithmetic progression of maximum length $1998$. b) Prove that there exist maximum arithmetic progressions of any length $l \geq 3$.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 8$

Let $s_1, s_2, s_3$ be the respective sums of $n$, $2n$, $3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on: $ \textbf{(A)}\ a \text{ } \text{and} \text{ } d\qquad\textbf{(B)}\ d \text{ } \text{and} \text{ } n\qquad\textbf{(C)}\ a \text{ } \text{and} \text{ } n\qquad\textbf{(D)}\ a, d, \text{ } \text{and} \text{ } n\qquad$ $\textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n $

In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.

Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$. [i]Mircea Lascu, Dorel Mihet[/i]

Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.

A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order. The numbers \[4,6,13,27,50,84\] are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,

Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.

An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$? $ \textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$? [i]2017 CCA Math Bonanza Team Round #1[/i]

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

Let $a_1,a_2 ,\ldots, a_n$ be an arithmetic progression of integers such that $i|a_i$ for $i=1, 2,\ldots ,n-1$ and $n\nmid a_n$. Prove that $n$ is a prime power.

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$