Found problems: 492
2013 All-Russian Olympiad, 2
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
PEN A Problems, 2
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.
1993 Romania Team Selection Test, 1
Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$.
Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.
2011 Moldova Team Selection Test, 1
Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.
2015 Benelux, 4
Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain?
(An [i]arithmetic progression[/i] is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the [i]common difference[/i] of the arithmetic progression.)
2010 Contests, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
2006 Grigore Moisil Urziceni, 2
Let be an infinite sequence $ \left( c_n \right)_{n\ge 1} $ of positive real numbers, with $ c_1=1, $ and satisfying
$$ c_{n+1}-\frac{1}{c_{n+1}} =c_n+\frac{1}{c_n} , $$
for all natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a natural number $ k $ such that the sequence $ \left( c_n^k+\frac{1}{c_n^k} \right)_{n\ge 1} $ is an arithmetic one.
[b]b)[/b] there exist two sequences $ \left( u_n \right)_{n\ge 1} ,\left( v_n \right)_{n\ge 1} $ of nonegative integers such that $ c_n=\sqrt{u_n} +\sqrt{v_n} , $ for any natural number $ n. $
1970 AMC 12/AHSME, 15
Lines in the xy-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation
$\textbf{(A) }3x-2y-1=0\qquad\textbf{(B) }4x-5y+8=0\qquad\textbf{(C) }5x+2y-23=0\qquad$
$\textbf{(D) }x+7y-31=0\qquad \textbf{(E) }x-4y+13=0$
1976 Euclid, 2
Source: 1976 Euclid Part A Problem 2
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The sum of the series $2+5+8+11+14+...+50$ equals
$\textbf{(A) } 90 \qquad \textbf{(B) } 425 \qquad \textbf{(C) } 416 \qquad \textbf{(D) } 442 \qquad \textbf{(E) } 495$
2008 Tournament Of Towns, 4
Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?
1959 AMC 12/AHSME, 18
The arithmetic mean (average) of the first $n$ positive integers is:
$ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $
2009 IMO, 3
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
2021 Regional Olympiad of Mexico Southeast, 2
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.
2003 Gheorghe Vranceanu, 2
Prove that with $ n\ge 1 $ distinct numbers we can form an arithmetic progression if and only if there are exactly $ n-1 $ distinct elements in the set of positive differences between any two of these numbers.
1990 National High School Mathematics League, 14
Here are $n^2$ numbers:
$a_{11},a_{12},a_{13},\cdots,a_{1n}\\
a_{21},a_{22},a_{23},\cdots,a_{2n}\\
\cdots\\
a_{n1},a_{n2},a_{n3},\cdots,a_{nn}$
Numbers in each line are arithmetic sequence, numbers in each column are geometric series.
If $a_{24}=1,a_{42}=\frac{1}{8},a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+\cdots+a_{nn}$.
PEN O Problems, 35
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
2004 Italy TST, 2
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
1973 AMC 12/AHSME, 26
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$
2000 IMO Shortlist, 1
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.
How many ways are there to put the cards in the three boxes so that the trick works?
1991 IMO Shortlist, 16
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
2000 AMC 12/AHSME, 14
When the mean, median, and mode of the list
\[ 10, 2, 5, 2, 4, 2, x\]are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $ x$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 20$
2010 Dutch IMO TST, 2
Let $A$ and $B$ be positive integers. De fine the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2013 Baltic Way, 15
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
2008 China Team Selection Test, 3
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.
2015 Estonia Team Selection Test, 1
Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.