Found problems: 492
A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.
Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
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.
Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.
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.
Determine the values that \(n\) can take so that the equation in \( x \) $$ x^4-(3n+2)x^2+n^2=0$$ has four different real roots \( x_1\), \(x_2\), \(x_3\) and \(x_4\) in arithmetic progression. That is, they satisfy that $$x_4-x_3=x_3-x_2=x_2-x_1$$
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.
Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.
Let $S$ be the set of the reciprocals of the first $2016$ positive integers and $T$ the set of all subsets of $S$ that form arithmetic progressions. What is the largest possible number of terms in a member of $T$?
[i]2016 CCA Math Bonanza Lightning #3.4[/i]
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?
$ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 38\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 44$
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
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]
A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.
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 $a_{11}+a_{22}+\cdots+a_{nn}$.
Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.
Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.
Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that:
1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences;
2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences,
Find the smallest possible value of $k$ (as a function of $n$.)
Note: The common differences might not be positive.
Proposed by Chu-Lan Kao
Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.
Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $
Study the convergence of the sequences
$$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$
and calculate their limits.
[i]Dumitru Acu[/i]
Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.
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$
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?
The cubic equation $x^3+2x-1=0$ has exactly one real root $r$. Note that $0.4<r<0.5$.
(a) Find, with proof, an increasing sequence of positive integers $a_1 < a_2 < a_3 < \cdots$ such that
\[\frac{1}{2}=r^{a_1}+r^{a_2}+r^{a_3}+\cdots.\]
(b) Prove that the sequence that you found in part (a) is the unique increasing sequence with the above property.