Found problems: 492
2005 South East Mathematical Olympiad, 7
(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter.
(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition
\[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \]
Find the maximum value of $n$.
2012 Centers of Excellency of Suceava, 3
Prove that the sum of the squares of the medians of a triangle is at least $ 9/4 $ if the circumradius of the triangle, the area of the triangle and the inradius of the triangle (in this order) are in arithmetic progression.
[i]Dumitru Crăciun[/i]
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
1997 AMC 12/AHSME, 20
Which one of the following integers can be expressed as the sum of $ 100$ consecutive positive integers?
$ \textbf{(A)}\ 1,\!627,\!384,\!950\qquad \textbf{(B)}\ 2,\!345,\!678,\!910\qquad \textbf{(C)}\ 3,\!579,\!111,\!300\qquad \textbf{(D)}\ 4,\!692,\!581,\!470\qquad \textbf{(E)}\ 5,\!815,\!937,\!260$
2001 Poland - Second Round, 1
Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.
1978 Vietnam National Olympiad, 4
Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.
2019 Argentina National Olympiad, 5
There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression.
Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.
1967 IMO Longlists, 33
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
2003 Turkey Team Selection Test, 3
Is there an arithmetic sequence with
a. $2003$
b. infinitely many
terms such that each term is a power of a natural number with a degree greater than $1$?
2003 Finnish National High School Mathematics Competition, 5
Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance.
The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$
The one who obtains the progression wins.
Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$
1966 AMC 12/AHSME, 19
Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for:
$\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$
$\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$
2019 Saint Petersburg Mathematical Olympiad, 1
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
2012 NIMO Problems, 4
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
1982 AMC 12/AHSME, 8
By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 11\qquad
\textbf{(E)}\ 12$
2007 German National Olympiad, 5
Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.
1981 AMC 12/AHSME, 11
The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
$\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$
1969 AMC 12/AHSME, 9
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:
$\textbf{(A) }27\qquad
\textbf{(B) }27\tfrac14\qquad
\textbf{(C) }27\tfrac12\qquad
\textbf{(D) }28\qquad
\textbf{(E) }28\tfrac12$
2014 Postal Coaching, 5
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.
2005 Gheorghe Vranceanu, 3
Within an arithmetic progression of length $ 2005, $ find the number of arithmetic subprogressions of length $ 501 $ that don't contain the $ \text{1000-th} $ term of the progression.
1991 AMC 12/AHSME, 12
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon. The largest possible value of $m^{\circ}$ is
$ \textbf{(A)}\ 165^{\circ}\qquad\textbf{(B)}\ 167^{\circ}\qquad\textbf{(C)}\ 170^{\circ}\qquad\textbf{(D)}\ 175^{\circ}\qquad\textbf{(E)}\ 179^{\circ} $
2010 Purple Comet Problems, 15
In the number arrangement
\[\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}\]
what is the number that will appear directly below the number $2010$?
1960 Putnam, B4
Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.
2015 AMC 10, 7
How many terms are there in the arithmetic sequence $13, 16, 19, \dots, 70,73$?
$ \textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }24\qquad\textbf{(D) }60\qquad\textbf{(E) }61 $
2014 AMC 12/AHSME, 14
Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$?
$\textbf{(A) }-2\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }4\qquad
\textbf{(E) }6\qquad$
1983 IMO Longlists, 50
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?