Found problems: 492
Let $a,b,c>0.$ If $\frac 1a,\frac 1b,\frac 1c$ are in arithmetic progression, and if $a^2+b^2,b^2+c^2,c^2+a^2$ are in geometric progression, show that $a=b=c.$
Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.
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 $M < A +\sqrt{B}$.
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power.
Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.
$ 90\plus{}91\plus{}92\plus{}93\plus{}94\plus{}95\plus{}96\plus{}97\plus{}98\plus{}99\equal{}$
\[ \textbf{(A)}\ 845 \qquad
\textbf{(B)}\ 945 \qquad
\textbf{(C)}\ 1005 \qquad
\textbf{(D)}\ 1025 \qquad
\textbf{(E)}\ 1045
\]
From a fixed set formed by the first consecutive natural numbers, find the number of subsets having exactly three elements, and these in arithmetic progression.
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 $N$.
Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.
The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$,
\[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \]
Find $a_{2004}$.
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.
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
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]
Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.
Let $d(n)$ denote the number of divisors of a positive integer $n$. If $k$ is a given odd number, prove that there exist an increasing arithmetic progression in positive integers $(a_1,a_2,\ldots a_{2019}) $ such that $gcd(k,d(a_1)d(a_2)\ldots d(a_{2019})) =1$
Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$.
Prove that the elements of $S$ may be arranged in an arithmetic progression.
This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.
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 $a_1 = a_2 = . . . = a_n$.
Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$
Let $d$ be a positive integer.
The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ .
Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression.
Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.
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 $x$?
$\text{(A)}\ 3\qquad\text{(B)}\ 6 \qquad\text{(C)}\ 9 \qquad\text{(D)}\ 17\qquad\text{(E)}\ 20$
The first item and common difference of an arithmetic sequence are nonnegative intengers. The number of items is not less than $3$, and the sum of all items is $97^2$. Then the number of such sequences is
$\text{(A)}2\qquad\text{(B)}3\qquad\text{(C)}4\qquad\text{(D)}5$
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.
Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that
$W(k,2)=\Omega (2^{\frac{k}{2}})$.
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.
Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.