Found problems: 766
Let $f(x)$ be a function satisfying
\[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\]
Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.
Find all positive integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots +
\frac{a_{n-1}}{a_n},
\]
where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.
In how many ways can a $2\times n$ grid be covered by
(a) 2 monominoes and $n-1$ dominoes
(b) 4 monominoes and $n-2$ dominoes.
Let $x_1=1$, $x_2$, $x_3$, $\ldots$ be a sequence of real numbers such that for all $n\geq 1$ we have \[ x_{n+1} = x_n + \frac 1{2x_n} . \] Prove that \[ \lfloor 25 x_{625} \rfloor = 625 . \]
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer?
$\textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 19 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 21$
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and
\[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\]
Show that for all $k$,
\[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\]
where $[x]$ denotes the greatest integer not exceeding $x.$
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
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(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
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(-2,2)--(-3,3);
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draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
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label("$F_1$",(0,-4));
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label("$F_3$",(12,-4));
label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
Let $a_0 = 1998$ and $a_{n+1} =\frac{a_n^2}{a_n +1}$ for each nonnegative integer $n$.
Prove that $[a_n] = 1994- n$ for $0 \le n \le 1000$
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
The numbers $a_1,a_2,...,a_n$ ( $n\ge 3$) satisfy the relations $$a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)$$ Prove that the numbers $a_1,a_2,...,a_n$ are non-negative.
A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.
(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$.
(b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.
Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
Let $a_1>0$ and for $n \ge 1$ define
\[a_{n+1}=a_n+\frac{1}{a_1+a_2+\dots+a_n}.\]
Prove that
\[\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.\]
Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$
Determine all integers $m$ such that every term of the sequence is a square.
Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$ and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations
$b_1 = a_1$ ,
$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,
$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.
Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$