Found problems: 492
2009 IMO Shortlist, 6
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]
2007 Kurschak Competition, 2
Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$, one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.
1997 Korea National Olympiad, 8
For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.
2007 Harvard-MIT Mathematics Tournament, 12
Let $A_{11}$ denote the answer to problem $11$. Determine the smallest prime $p$ such that the arithmetic sequence $p,p+A_{11},p+2A_{11},\cdots$ begins with the largest number of primes.
There is just one triple of possible $(A_{10},A_{11},A_{12})$ of answers to these three problems. Your team will receive credit only for answers matching these. (So, for example, submitting a wrong answer for problem $11$ will not alter the correctness of your answer to problem $12$.)
2022 AMC 10, 20
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence?
$\textbf{(A) }190\qquad\textbf{(B) }194\qquad\textbf{(C) }198\qquad\textbf{(D) }202\qquad\textbf{(E) }206$
2002 Estonia National Olympiad, 1
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.
2013 AIME Problems, 15
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions
(a) $0\leq A<B<C\leq99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences.
Find $N$.
1993 Romania Team Selection Test, 3
Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.
2015 NIMO Problems, 2
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]
2012 Regional Competition For Advanced Students, 3
In an arithmetic sequence, the di fference of consecutive terms in constant. We consider sequences of integers in which the di fference of consecutive terms equals the sum of the differences of all preceding consecutive terms.
Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?
1994 China Team Selection Test, 2
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
1973 Kurschak Competition, 1
For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?
1999 AIME Problems, 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
2009 Math Prize For Girls Problems, 11
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$?
2012 AIME Problems, 2
The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.
2004 Regional Olympiad - Republic of Srpska, 3
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
2012-2013 SDML (High School), 14
A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$.
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
2012-2013 SDML (Middle School), 6
How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.)
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$
2004 USAMTS Problems, 4
The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$
2013 Harvard-MIT Mathematics Tournament, 2
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
1996 All-Russian Olympiad, 5
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
1978 IMO Longlists, 26
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
1993 Turkey Team Selection Test, 1
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.
2002 All-Russian Olympiad Regional Round, 10.1
What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}\plus{}1$ is prime for $ k \equal{} 1, 2, . . . , n$?