Found problems: 173
Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.
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$
Which of the following series are convergent?
$\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$
$\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$
$\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$
$\textbf{(D)}~\text{None of the above}$
by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.
What is the value of the expression
\[ \frac {1}{\log_2 100!} \plus{} \frac {1}{\log_3 100!} \plus{} \frac {1}{\log_4 100!} \plus{} \cdots \plus{} \frac {1}{\log_{100} 100!}?
\]$ \textbf{(A)}\ 0.01 \qquad \textbf{(B)}\ 0.1 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 10$
Find the value of $$\sum_{1\le a<b<c} \frac{1}{2^a3^b5^c}$$ (i.e. the sum of $\frac{1}{2^a3^b5^c}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$)
Give a geometric series $(a_n)$ with common ratio of $q$, let $b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}$, then sequence $(b_n)$
$\text{(A)}$ is an arithmetic sequence
$\text{(B)}$ is a geometric series with common ratio of $q$
$\text{(C)}$ is a geometric series with common ratio of $q^3$
$\text{(D)}$ is neither an arithmetic sequence nor a geometric series
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
For a real number $a$ and an integer $n(\geq 2)$, define
$$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$
Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real.
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]
Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$
Prove that the sequence $(a_n)$, where
$$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.
Let be $\gamma_1$ a circumference of radius $r$ and $P$ an exterior point that is at distance $a$ from the centre of $\gamma_1$. We build two tangent lines $r,s$ to $\gamma_1$ from $P$ and $\gamma_2$ is constructed as a smaller circumference, tangent to both lines and, also, tangent to $\gamma_1$. We construct inductively $\gamma_{n+1}$ as a tangent circumference to $\gamma_{n}$ and, also, tangent to $r$ and $s$. Determine:
a) The radius of $\gamma_2$.
b) The radius of $\gamma_n$.
c) The sum of the lengths of $\gamma_1, \gamma_2, \gamma_3, ...$.
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$
A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer.
[i]($3$ points)[/i]
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning.
Square ABCD has side length 7. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. SImilarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares.
[asy]
size(180);
pair[] A={(-1,-1),(-1,1),(1,1),(1,-1),(-1,-1)};
string[] X={"A","B","C","D"};
for(int k=0;k<10;++k)
{
for(int m=0;m<4;++m)
{
if(k==0) label("$"+X[m]+"$",A[m],A[m]);
if(k==1) label("$"+X[m]+"_1$",A[m],A[m]);
draw(A[m]--A[m+1]);
A[m]+=3/7*(A[m+1]-A[m]);
}
A[4]=A[0];
}[/asy]
Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that
$$\sum_{n=1}^{\infty} b_n$$
converges.
Let the series $$s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}$$ Find a real set on which this series is convergent, and then compute its sum. Find also $$\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)$$
Points $B$,$D$ , and $J$ are midpoints of the sides of right triangle $ACG$ . Points $K$, $E$, $I$ are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the first three are shown) and $ AC=CG=6 $, then the total area of the shaded triangles is nearest
[asy]
draw((0,0)--(6,0)--(6,6)--cycle);
draw((3,0)--(3,3)--(6,3));
draw((4.5,3)--(4.5,4.5)--(6,4.5));
draw((5.25,4.5)--(5.25,5.25)--(6,5.25));
fill((3,0)--(6,0)--(6,3)--cycle,black);
fill((4.5,3)--(6,3)--(6,4.5)--cycle,black);
fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black);
label("$A$",(0,0),SW);
label("$B$",(3,0),S);
label("$C$",(6,0),SE);
label("$D$",(6,3),E);
label("$E$",(6,4.5),E);
label("$F$",(6,5.25),E);
label("$G$",(6,6),NE);
label("$H$",(5.25,5.25),NW);
label("$I$",(4.5,4.5),NW);
label("$J$",(3,3),NW);
label("$K$",(4.5,3),S);
label("$L$",(5.25,4.5),S);[/asy]
$ \text{(A)}\ 6\qquad\text{(B)}\ 7\qquad\text{(C)}\ 8\qquad\text{(D)}\ 9\qquad\text{(E)}\ 10 $
Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$
Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$?
$
\textbf{(A)}\ 121
\qquad\textbf{(B)}\ 224
\qquad\textbf{(C)}\ 275
\qquad\textbf{(D)}\ 364
\qquad\textbf{(E)}\ 375
$
Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $
[b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $
[i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $
[i]Cătălin Țigăeru[/i]
Consider the harmonic series $\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots$. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms $\frac1k$.)