Found problems: 122
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 $a_{11}+a_{22}+\cdots+a_{nn}$.
Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$
Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$
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.
Evaluate $\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}$, where $\cos\theta = \dfrac{1}{5}$.
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 $S?$
$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt{5} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
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]
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 $T(x)$
A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of 5 units due east. Each night the bug rotates $60 ^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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.
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
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).$
Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.
Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$?
$ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$
Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.
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?
$\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$
Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
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 $S?$
$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt{5} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
Compute the sum of all twenty-one terms of the geometric series \[1+2+4+8+\cdots+1048576.\]
$\textbf{(A) }2097149\hspace{12em}\textbf{(B) }2097151\hspace{12em}\textbf{(C) }2097153$
$\textbf{(D) }2097157\hspace{12em}\textbf{(E) }2097161$
In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student.
Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.
Find, as a function of $\, n, \,$ the sum of the digits of
\[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \]
where each factor has twice as many digits as the previous one.
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]