A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then: $ \textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac12 S_4 $

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

The first four terms in an arithmetic sequence are $ x \plus{} y$, $ x \minus{} y$, $ xy$, and $ x/y$, in that order. What is the fifth term? $ \textbf{(A)}\ \minus{}\frac{15}{8} \qquad \textbf{(B)}\ \minus{}\frac{6}{5} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac{27}{20} \qquad \textbf{(E)}\ \frac{123}{40}$

How many numbers are there that appear both in the arithmetic sequence $10, 16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$

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.

Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length $1$, then its perimeter is $a + \sqrt{b}4$, where $ a$ and $b$ are positive integers. What is the value of $a + b$?

There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$. [i]Proposed by David Altizio and Tony Kim[/i]

A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 81$

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

A donkey suffers an attack of hiccups and the first hiccup happens at $\text{4:00}$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $\text{700th}$ hiccup occur? $\textbf{(A) }$ $15$ seconds after $\text{4:58}$ $\textbf{(B) }$ $20$ seconds after $\text{4:58}$ $\textbf{(C)}$ $25$ seconds after $\text{4:58}$ $\textbf{(D) }$ $30$ seconds after $\text{4:58}$ $\textbf{(E) }$ $35$ seconds after $\text{4:58}$

How many sets of two or more consecutive positive integers have a sum of 15? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?

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.

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]

Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.

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 $0$).

Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?

If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals $\textbf{(A) }\frac{2}{3}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad \textbf{(E) }\frac{3}{2}$

Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

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]

Prove that the arithmetic sequence $5, 11, 17, 23, 29, \ldots$ contains infinitely many primes.