Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers. [b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $ [b]b)[/b] $ S_5 $ is not a perfect square.

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.

The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb N$ of cardinality $l$, an infinite set $T \subset \mathbb N$, and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$, the sum $t+ \sum_{y \in Y} y$ belongs to $B_k.$

Let $ f(0) \equal{} f(1) \equal{} 0$ and \[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\] Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$

Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.

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? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$? $\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$ [asy] size(3.85cm); label("$X$",(2.5,2.1),N); for (int i=0; i<=5; ++i) draw((i,0)--(i,5), linewidth(.5)); for (int j=0; j<=5; ++j) draw((0,j)--(5,j), linewidth(.5)); void draw_num(pair ll_corner, int num) { label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt)); } draw_num((0,0), 17); draw_num((4, 0), 81); draw_num((0, 4), 1); draw_num((4,4), 25); void foo(int x, int y, string n) { label(n, (x+0.5,y+0.5), p = fontsize(19pt)); } foo(2, 4, " "); foo(3, 4, " "); foo(0, 3, " "); foo(2, 3, " "); foo(1, 2, " "); foo(3, 2, " "); foo(1, 1, " "); foo(2, 1, " "); foo(3, 1, " "); foo(4, 1, " "); foo(2, 0, " "); foo(3, 0, " "); foo(0, 1, " "); foo(0, 2, " "); foo(1, 0, " "); foo(1, 3, " "); foo(1, 4, " "); foo(3, 3, " "); foo(4, 2, " "); foo(4, 3, " "); [/asy]

Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.

Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(9cm); for (int x=0; x<=11; ++x) draw((x, 0) -- (x, 5), linewidth(.5pt)); for (int y=0; y<=5; ++y) draw((0, y) -- (11, y), linewidth(.5pt)); filldraw((0,4)--(0,3)--(2,3)--(2,4)--cycle, gray, gray); filldraw((1,1)--(1,2)--(3,2)--(3,1)--cycle, gray, gray); filldraw((4,1)--(4,4)--(5,4)--(5,1)--cycle, gray, gray); filldraw((7,0)--(7,3)--(6,3)--(6,0)--cycle, gray, gray); filldraw((7,4)--(7,5)--(6,5)--(6,4)--cycle, gray, gray); filldraw((8,1)--(8,2)--(10,2)--(10,1)--cycle, gray, gray); filldraw((9,4)--(9,3)--(11,3)--(11,4)--cycle, gray, gray); draw((0,0)--(11,0)--(11,5)--(0,5)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } foo(1, 2, "10"); foo(4, 0, "31"); foo(5, 0, "26"); foo(10, 0, "59"); foo(0, 4, "3"); foo(7, 4, "59"); [/asy]

Let $a_1=0$, $a_2=1$, and $a_{n+2}=a_{n+1}+a_n$ for all positive integers $n$. Show that there exists an increasing infinite arithmetic progression of integers, which has no number in common in the sequence $\{a_n\}_{n \ge 0}$.

Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$. What is the smallest ghost that is not a multiple of $p$? [i]Proposed by Guerrero[/i]

Let $n\geq 3$ an integer. Determine whether there exist permutations $(a_1,a_2, \ldots, a_n)$ of the numbers $(1,2,\ldots, n)$ and $(b_1, b_2, \ldots, b_n)$ of the numbers $(n+1,n+2,\ldots, 2n)$ so that $(a_1b_1, a_2b_2, \ldots a_nb_n)$ is a strictly increasing arithmetic progression.

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Suppose an infinite non-constant arithmetic progression of integers contains $1$ in it. Prove that there are an infinite number of perfect cubes in this progression. (A [i]perfect cube[/i] is an integer of the form $k^3$, where $k$ is an integer. For example, $-8$, $0$ and $1$ are perfect cubes.)

Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?

Let $n \ge 4$ be a natural number. Let $A_1A_2
\cdots
A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 <
\cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots
A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.