Found problems: 373
Given a positive integer number $n$, determine the minimum of
\[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\]
as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$.
[i]Kvant Magazine[/i]
Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove:
(i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$
(ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Suppose we wish to pick a random integer between $1$ and $N$ inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between $0$ and $1$, then multiplying the result by $N$ and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping the flipping process after we have enough flips to determine the ceiling of the number. For instance, if $N=3$, we could conclude that the number is $2$ after flipping $.011_2$, but $.010_2$ is inconclusive.
Suppose $N=2014$. The expected number of flips for such a process is $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $100m+n$.
[i]Proposed by Lewis Chen[/i]
Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$. Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that
[list]
[*] $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$, and
[*] $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.
[/list]
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
If $C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)$, prove the identity
\[C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}\]
and then evaluate the sum
\[S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.\]
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.
[i]Proposed by Connor Gordon[/i]
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \][i]Proposed by Evan Chen[/i]
Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]
What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$
$\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$
Calculate the sum
$$ 2^n+2^{n-1}\cos\alpha +2^{n-2} \cos2\alpha +\cdots +2\cos (n-1)\alpha +\cos n\alpha , $$
where $ \alpha $ is a real number and $ n $ a natural one.
[i]Dan Negulescu[/i]
If $T_{n} = 1 + 2 + 3 + \ldots + n$ and \[P_{n} = \frac{T_{2}}{T_{2} - 1} \cdot \frac{T_{3}}{T_{3} - 1} \cdot \frac{T_{4}}{T_{4} - 1} \cdot\,\, \cdots \,\,\cdot \frac{T_{n}}{T_{n} - 1}\quad\text{for }n = 2,3,4,\ldots,\] then $P_{1991}$ is closest to which of the following numbers?
$ \textbf{(A)}\ 2.0\qquad\textbf{(B)}\ 2.3\qquad\textbf{(C)}\ 2.6\qquad\textbf{(D)}\ 2.9\qquad\textbf{(E)}\ 3.2 $
Compute the sum of the digits of \[\prod_{n=0}^{2018}\left(10^{2\cdot 3^n} - 10^{3^n} + 1\right)\left(10^{2\cdot 3^n} + 10^{3^n} + 1\right).\]
Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]
Find the value of $2(\sin2^{\circ}\tan1^{\circ}+\sin4^{\circ}\tan1^{\circ}+\cdots+\sin178^{\circ}\tan 1^{\circ})$
Compute \[ \displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}. \]
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$.
a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$
b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$
For a given integer $n\ge 2$, let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n$. Prove that if every prime that divides $m$ also divides $n$, then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k$.
[i]Proposed by Ivan Borsenco[/i]
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$, the value of $$\sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}}$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}}
.\]
The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$