Found problems: 122
2012 Romania Team Selection Test, 2
Let $n$ be a positive integer. Find the value of the following sum \[\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},\] where $e_k\in\{0,1\}$ for $1\leq k \leq n$, and the sum $\sum_{(n)}$ is taken over all $2^n$ possible choices of $e_1,\ldots ,e_n$.
2007 Today's Calculation Of Integral, 191
(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$
(2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$
2013 Macedonian Team Selection Test, Problem 6
Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.
2015 Putnam, B4
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.
2000 National High School Mathematics League, 4
Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$?
$\text{(A)}$ It has no real roots.
$\text{(B)}$ It has two equal real roots.
$\text{(C)}$ It has two different real roots, and their product is positive.
$\text{(D)}$ It has two different real roots, and their product is negative.
1994 Poland - First Round, 12
The sequence $(x_n)$ is given by
$x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$
Prove that for all natural numbers $n \geq 1$ the following inequality holds
$x_1+x_2+...+x_n < 1$.
2009 AMC 10, 21
What is the remainder when $ 3^0\plus{}3^1\plus{}3^2\plus{}\ldots\plus{}3^{2009}$ is divided by $ 8$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 6$
2011 USAMTS Problems, 2
Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]
1989 IMO Longlists, 17
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and
\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]
Determine $ f \left( \frac{1}{7} \right).$
2009 Stanford Mathematics Tournament, 2
The pattern in the fi gure below continues inward in finitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.
[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
2016 AMC 10, 16
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$
$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt{5} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2006 Stanford Mathematics Tournament, 5
A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?
1990 National High School Mathematics League, 14
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 $a_{11}+a_{22}+\cdots+a_{nn}$.
1975 AMC 12/AHSME, 16
If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is 3, then the sum of the first two terms of the series is
$ \textbf{(A)}\ 1/3 \qquad
\textbf{(B)}\ 2/3 \qquad
\textbf{(C)}\ 8/3 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 9/2$
2010 Today's Calculation Of Integral, 659
Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$
2009 USAMTS Problems, 5
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.
1999 Harvard-MIT Mathematics Tournament, 4
Evaluate $\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}$, where $\cos\theta = \dfrac{1}{5}$.
2007 ITest, 5
Compute the sum of all twenty-one terms of the geometric series \[1+2+4+8+\cdots+1048576.\]
$\textbf{(A) }2097149\hspace{12em}\textbf{(B) }2097151\hspace{12em}\textbf{(C) }2097153$
$\textbf{(D) }2097157\hspace{12em}\textbf{(E) }2097161$
1951 AMC 12/AHSME, 11
The limit of the sum of an infinite number of terms in a geometric progression is $ \frac {a}{1 \minus{} r}$ where $ a$ denotes the first term and $ \minus{} 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is:
$ \textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$
1980 AMC 12/AHSME, 13
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\circ$ counterclockwise and travels $\frac 12$ a unit to $\left(1, \frac 12 \right)$. If it continues in this fashion, each time making a $90^\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
$\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$
1983 Iran MO (2nd round), 7
Find the sum $\sum_{i=1}^{\infty} \frac{n}{2^n}.$
2004 AMC 10, 18
A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 49\qquad
\textbf{(E)}\ 81$
2000 Turkey MO (2nd round), 1
Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$
2006 Miklós Schweitzer, 6
Let G (n) = max | A(n) |, where A(n) ranges over all subsets of {1,2,...,n} and contains no three-member geometric series, ie, there is no $x, y, z \in A$ such that x < y < z and xz = y^2. Prove that $\lim_{n \to \infty} \frac{G (n)}{n}$ exists.