Olympiad problems

1973 AMC 12/AHSME, 28

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}$

1998 USAMTS Problems, 2

Tags: quadratic , arithmetic series , quadratic formula , algebra

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.

2008 ITest, 44

Tags: arithmetic series , relatively prime , floor function , probability , number theory

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 $a+b$.

2018 China Team Selection Test, 2

Tags: arithmetic series , number of divisors , number theory

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.

2014 Iran MO (3rd Round), 6

Tags: arithmetic series , modular arithmetic , number theory

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 )

1969 AMC 12/AHSME, 33

Tags: ratio , arithmetic series

Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is: $\textbf{(A) }4:3\qquad \textbf{(B) }3:2\qquad \textbf{(C) }7:4\qquad \textbf{(D) }78:71\qquad \textbf{(E) }\text{undetermined}$

1962 AMC 12/AHSME, 32

Tags: arithmetic series , induction

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}$

2006 Purple Comet Problems, 18

Tags: arithmetic series

In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?

2018 China Team Selection Test, 2

Tags: arithmetic series , number of divisors , number theory

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.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: algebra , arithmetic sequence , quadratic , arithmetic series , quadratic formula , polynomial , function

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

2017-2018 SDML (Middle School), 5

Tags: arithmetic series

If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$? $\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$

1966 AMC 12/AHSME, 18

Tags: arithmetic sequence , arithmetic series

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: $\text{(A)} \ 3 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \frac{27}{19} \qquad \text{(D)} \ \frac{13}9 \qquad \text{(E)} \ \frac{23}{38}$

1959 AMC 12/AHSME, 33

Tags: harmonic progression , arithmetic series , arithmetic sequence , algebra

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 $

1959 AMC 12/AHSME, 18

Tags: algebra , arithmetic sequence , arithmetic mean , arithmetic series

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} $

2002 AMC 12/AHSME, 13

Tags: arithmetic series

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$

1958 AMC 12/AHSME, 37

Tags: arithmetic series

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$

2014 AMC 12/AHSME, 15

Tags: arithmetic sequence , arithmetic series

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 $S$? $\textbf{(A) }9\qquad \textbf{(B) }18\qquad \textbf{(C) }27\qquad \textbf{(D) }36\qquad \textbf{(E) }45\qquad$

2007 AMC 12/AHSME, 7

Tags: arithmetic sequence , arithmetic series

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$

2010 Stanford Mathematics Tournament, 12

Tags: arithmetic series

Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...$ Find $n$ such that the first $n$ terms sum up to $2010$.

1994 AMC 12/AHSME, 14

Tags: arithmetic series

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 $

1995 AMC 12/AHSME, 27

Tags: greatest common divisor , number theory , arithmetic series , modular arithmetic

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 $100$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 74$

1961 AMC 12/AHSME, 26

Tags: system of equations , arithmetic series , algebra

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 } $

2006 Mathematics for Its Sake, 3

Tags: real analysis , arithmetic sequence , arithmetic series , study convergence , sequence , limit of sequence

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]

2008 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations , arithmetic series

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$.

1950 AMC 12/AHSME, 17

Tags: algebra , derivative , calculus , quadratic , system of equations , arithmetic series

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$