Found problems: 492
Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]
In arithmetic sequence $(a_n)$, $3a_8=5a_{13},a_1>0$. Define $S_n=\sum_{i=1}^n a_i$, then the largest number in $(S_n)$ is
$\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}$
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?
In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.
How many ways are there to put the cards in the three boxes so that the trick works?
Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.
Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$
The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled?
[i]Proposed by A. Golovanov[/i]
Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?
$\textbf{(A) }9\qquad
\textbf{(B) }18\qquad
\textbf{(C) }27\qquad
\textbf{(D) }36\qquad
\textbf{(E) }45\qquad$
The first four terms in an arithmetic sequence are $ x \plus{} y$, $ x \minus{} y$, $ xy$, and $ x/y$, in that order. What is the fifth term?
$ \textbf{(A)}\ \minus{}\frac{15}{8} \qquad
\textbf{(B)}\ \minus{}\frac{6}{5} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \frac{27}{20} \qquad
\textbf{(E)}\ \frac{123}{40}$
Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
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 $d$?
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 4006$
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?
$\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$
We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$.
Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.
[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below:
[asy]
size(250);defaultpen(linewidth(0.4));
dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin);
int i;
for(i=0; i<360; i=i+60) {
dot(1*dir(i)^^2*dir(i)^^3*dir(i));
dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i));
dot(shift(1, sqrt(3))*1*dir(i));
dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i));
dot(shift(-12,0)*origin+sqrt(3)*dir(i+30));
dot(shift(-24,0)*origin+1*dir(i));
}
label("$1$", (-36, -5), S);
label("$7$", (-24, -5), S);
label("$19$", (-12, -5), S);
label("$37$", (0, -5), S);
label("Centered Hexagonal Numbers", (-18,-10), S);[/asy]
Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.
For real numbers $x,y,z$, $3x,4y,5z$ are geometric series, $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are arithmetic sequence. Then $\frac{x}{z}+\frac{z}{x}=$________.