Found problems: 492
2013 Online Math Open Problems, 4
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i]
2003 Rioplatense Mathematical Olympiad, Level 3, 2
Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?
2014 India Regional Mathematical Olympiad, 2
The roots of the equation
\[ x^3-3ax^2+bx+18c=0 \]
form a non-constant arithmetic progression and the roots of the equation
\[ x^3+bx^2+x-c^3=0 \]
form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.
VMEO IV 2015, 12.3
Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$
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]
2010 Purple Comet Problems, 6
Evaluate the sum $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 \cdots + 208 + 209 - 210.$
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.
2000 239 Open Mathematical Olympiad, 5
Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.
2012 India IMO Training Camp, 3
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
1987 AMC 12/AHSME, 3
How many primes less than $100$ have $7$ as the ones digit? (Assume the usual base ten representation)
$\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ 8$
1999 National High School Mathematics League, 1
Give a geometric series $(a_n)$ with common ratio of $q$, let $b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}$, then sequence $(b_n)$
$\text{(A)}$ is an arithmetic sequence
$\text{(B)}$ is a geometric series with common ratio of $q$
$\text{(C)}$ is a geometric series with common ratio of $q^3$
$\text{(D)}$ is neither an arithmetic sequence nor a geometric series
1973 Bulgaria National Olympiad, Problem 2
Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that:
$$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$
2010 Baltic Way, 12
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.
$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.
$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.
2019 Teodor Topan, 3
Let be a natural number $ m\ge 2. $
[b]a)[/b] Let be $ m $ pairwise distinct rational numbers. Prove that there is an ordering of these numbers such that these are terms of an arithmetic progression.
[b]b)[/b] Given that for any $ m $ pairwise distinct real numbers there is an ordering of them such that they are terms of an arithmetic sequence, determine the number $ m. $
[i]Bogdan Blaga[/i]
2018 China Northern MO, 8
2 players A and B play the following game with A going first: On each player's turn, he puts a number from 1 to 99 among 99 equally spaced points on a circle (numbers cannot be repeated), and the player who completes 3 consecutive numbers that forms an arithmetic sequence around the circle wins the game. Who has the winning strategy? Explain your reasoning.
2023 AMC 10, 23
An arithmetic sequence has $n \geq 3$ terms, initial term $a$ and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term which was off by $1$. The sum of the terms was $222$. What was $a + d + n$
$\textbf{(A) } 24 \qquad \textbf{(B) } 20 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 28 \qquad \textbf{(E) } 26$
2014 Online Math Open Problems, 2
Suppose $(a_n)$, $(b_n)$, $(c_n)$ are arithmetic progressions. Given that $a_1+b_1+c_1 = 0$ and $a_2+b_2+c_2 = 1$, compute $a_{2014}+b_{2014}+c_{2014}$.
[i]Proposed by Evan Chen[/i]
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.
2004 AIME Problems, 9
A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n-1}$, $a_{2n}$, $a_{2n+1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000. Find $n+a_n$.
2023 AMC 12/AHSME, 17
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
$\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$
2018 China Team Selection Test, 4
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
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.
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.)
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$
1999 Finnish National High School Mathematics Competition, 2
Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$
Prove that \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.\]
2018 Saudi Arabia IMO TST, 2
a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$.
b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.