Found problems: 27
Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.
The sum of $ 18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is
$ \textbf{(A)}\ 169 \qquad
\textbf{(B)}\ 225 \qquad
\textbf{(C)}\ 289 \qquad
\textbf{(D)}\ 361 \qquad
\textbf{(E)}\ 441$
Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown.
\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular}
Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 50 \qquad
\textbf{(D)}\ 62 \qquad
\textbf{(E)}\ 74$
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
Prove that there are 100 natural number $a_1 < a_2 < ... < a_{99} < a_{100}$ ( $ a_i < 10^6$) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ?
$A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}$
$2A = \{2a_i \vert 1\leq i\leq 100\}$
$A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}$
$A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}$
$2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}$
(20 ponits )
A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression.
Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then:
$ \textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac12 S_4 $
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 $S$?
$\textbf{(A) }9\qquad
\textbf{(B) }18\qquad
\textbf{(C) }27\qquad
\textbf{(D) }36\qquad
\textbf{(E) }45\qquad$
If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence
$ \textbf{(A)}\ \text{which is a G.P} \qquad$
$ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$
$ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$
$ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$
$ \textbf{(E)}\ \text{none of these}$
If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$?
$\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
There are infinitely many ordered pairs $(m,n)$ of positive integers for which the sum
\[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\]
is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.
Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.
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]
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
If the following instructions are carried out by a computer, which of $X$ will be printed because of instruction $5$?
$1.$ Start $X$ at $3$ and $S$ at $0$
$2.$ Increase the value of $X$ by $2$.
$3.$ Increase the value of $S$ by the value of $X$.
$4.$ If $S$ is at least $10000$, then go to instsruction $5$; otherwise, go to instruction $2$ and proceed from there.
$5.$ Print the value of $X$.
$6.$ Stop.
$\text{(A)} \ 19 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ 23 \qquad \text{(D)} \ 199 \qquad \text{(E)} \ 201$
The arithmetic mean (average) of the first $n$ positive integers is:
$ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $
The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.
Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be five consecutive terms in an arithmetic sequence, and suppose that $ a \plus{} b \plus{} c \plus{} d \plus{} e \equal{} 30.$ Which of the following can be found?
$ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$
In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
If $ x_{k\plus{}1} \equal{} x_k \plus{} \frac12$ for $ k\equal{}1, 2, \dots, n\minus{}1$ and $ x_1\equal{}1,$ find $ x_1 \plus{} x_2 \plus{} \dots \plus{} x_n.$
$ \textbf{(A)}\ \frac{n\plus{}1}{2} \qquad
\textbf{(B)}\ \frac{n\plus{}3}{2} \qquad
\textbf{(C)}\ \frac{n^2\minus{}1}{2} \qquad
\textbf{(D)}\ \frac{n^2\plus{}n}{4} \qquad
\textbf{(E)}\ \frac{n^2\plus{}3n}{4}$
The first term of an arithmetic series of consecutive integers is $ k^2 \plus{} 1$. The sum of $ 2k \plus{} 1$ terms of this series may be expressed as:
$ \textbf{(A)}\ k^3 \plus{} (k \plus{} 1)^3\qquad
\textbf{(B)}\ (k \minus{} 1)^3 \plus{} k^3\qquad
\textbf{(C)}\ (k \plus{} 1)^3\qquad \\
\textbf{(D)}\ (k \plus{} 1)^2\qquad
\textbf{(E)}\ (2k \plus{} 1)(k \plus{} 1)^2$
In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is:
$\text{(A)} \ 3 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \frac{27}{19} \qquad \text{(D)} \ \frac{13}9 \qquad \text{(E)} \ \frac{23}{38}$
Find the sum of the arithmetic series
\[ 20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40 \]
$ \textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000 $