Found problems: 5923
Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)?
Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.
Divide a regular hexagon into $24$ identical small equilateral triangles as shown. To each of the $19$ points that are vertices of at least one of the equilateral triangles, a different number is assigned. Find the maximum possible number of small equilateral triangles with their vertices having numbers in ascending order going clockwise.
[asy]
size(5.5cm);
int n=6;
pair[] V= sequence(new pair(int i){return dir(360*i/n);}, n);
V.cyclic=true;
for(int i=0;i<n;++i){
draw(V[i]--V[i+1],black+0.7bp);
draw(V[i]--V[i+3],black+0.7bp);
draw(midpoint(V[i]--V[i+1])--midpoint(V[i+2]--V[i+3]),black+0.7bp);
}
[/asy]
Let $b_1,b_2,... $ be a sequence of positive real numbers such that for each $ n\ge 1$, $$b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}$$
Show that there is a positive integer $M$ such that $$\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}$$
We define the sequence $a_n=(2n)^2+1$ for each natural number $n$. We will call one number [i]bad[/i], if there don’t exist natural numbers $a>1$ and $b>1$ such that $a_n=a^2+b^2$. Prove that the natural number $n$ is [i]bad[/i], if and only if $a_n$ is prime.
For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$.
Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$.
For example, $a_3=a_2+1=2,a_4=a_3+1=3$.
[b](a)[/b] Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$.
[b](b)[/b] Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.
Let $C$ denote the set of points $(x, y) \in R^2$ such that $x^2 + y^2 \le1$. A sequence $A_i = (x_i, y_i), |i \ge¸ 0$ of points in $R^2$ is ‘centric’ if it satisfies the following properties:
$\bullet$ $A_0 = (x_0, y_0) = (0, 0)$, $A_1 = (x_1, y_1) = (1, 0)$.
$\bullet$ For all $n\ge 0$, the circumcenter of triangle $A_nA_{n+1}A_{n+2}$ lies in $C$.
Let $K$ be the maximum value of $x^2_{2012} + y^2_{2012}$ over all centric sequences. Find all points $(x, y)$ such that $x^2 + y^2 = K$ and there exists a centric sequence such that $A_{2012} = (x, y)$.
Let $k\ge t\ge 2$ positive integers. For integers $n\ge k$ let $p_n$ be the probability that if we choose $k$ from the first $n$ positive integers randomly, any $t$ of the $k$ chosen integers have greatest common divisor $1$. Let qn be the probability that if we choose $k-t+1$ from the first $n$ positive integers the product is not divisible by a perfect $t^{th}$ power that is greater then $1$.
Prove that sequences $p_n$ and $q_n$ converge to the same value.
Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$
\[x_n=2x_{n-1}-4x_{n-2}+3.\]
Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
Let be $x_1=\frac{1}{\sqrt[n+1]{n!}}$ and $x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}$ for all $n\in \mathbb{N}^*$ and $f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}$ where $$f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}$$Prove that the sequence $(a_n)_{n\geq1}$ when $a_n=\int \limits_{x_1}^{x_2}f(x)dx$ is convergent and compute $$\lim \limits_{n \to \infty}a_n$$
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 $n\geq2$.
The sequence $(p_n)_{n\in\mathbb N}$ is defined by $p_1=2$ and, for $n\ge2$, $p_n$ is the largest prime factor of $p_1p_2\cdots p_{n-1}+1$. Show that $p_n\ne5$ for all $n$.
Does there exist a real sequence $\{a_n\}_{n=1}^\infty$ such that
$$a_na_{n+1}\ge a_{n+2}^2 +1$$
for all $n\ge 1$?
For positive integers $a>b>1$, define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers.
[i]V. Senderov[/i]
The sequences of positive integers $1,a_2,a_3,\ldots$ and $1,b_2,b_3,\ldots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.
Let $n$ be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?
A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that
\[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \]
Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.
Compute the number of sequences of real numbers $a_1, a_2, a_3, \dots, a_{16}$ satisfying the condition that for every positive integer $n$,
\[
a_1^n + a_2^{2n} + \dots + a_{16}^{16n} = \left \{ \begin{array}{ll} 10^{n+1} + 10^n + 1 & \text{for even } n \\ 10^n - 1 & \text{for odd } n \end{array} \right. .
\][i]Proposed by Evan Chen[/i]
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that
\[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\]
[i]Proposed by usjl[/i]
(a) Let $k,n\ge 1$.Find the number of sequences $\phi=S_0,S_1,\ldots,S_k$ of subsets of $[n]=\{1,2,3,\ldots,n\}$ if for all $1\le i\le k$ we have either (i)$S_{i-1}\subset S_i$ and $|S_i-S_{i-1}|$,or (ii)$S_i\subset S_{i-1}$ and $|S_{i-1}-S_i|=1$.
(b) Suppose that we add the additional condition that $S_k=\phi$.Show that now the number $f_k(n)$ of sequences is given by$f_k(n)=\frac{1}{2^n}\sum_{i=0}^n\binom ni (n-2i)^k$.
Note that $f_k(n)=0$ if $k$ is odd.
Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} \equal{} x_{1995}$, such that
\[ x_{i \minus{} 1} \plus{} \frac {2}{x_{i \minus{} 1}} \equal{} 2x_{i} \plus{} \frac {1}{x_{i}},
\]
for all $ i \equal{} 1,\cdots ,1995$.
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
$ \textbf{(A)}\ \frac{1509}{8} \qquad
\textbf{(B)}\ \frac{1509}{32} \qquad
\textbf{(C)}\ \frac{1509}{64} \qquad
\textbf{(D)}\ \frac{1509}{128} \qquad
\textbf{(E)}\ \frac{1509}{256} $
Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$?
(A)$6$
(B)$7$
(C)$8$
(D)$9$