Found problems: 5923
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
Compute the lowest positive integer $k$ such that none of the numbers in the sequence $$\{1, 1 +k, 1 + k + k^2
, 1 + k + k^2 + k^3, ... \}$$ are prime.
Let $(x_n)$, $n=1,2, \ldots $ be a sequence defined by $x_1=2008$ and
\begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*}
Let the sequence $a_n=x_n + \frac{1}{n} S_n$, $n=1,2,3, \ldots $ where $S_n$ $=$ $x_1+x_2 +\ldots +x_n$. Determine the values of $n$ for which the terms of the sequence $a_n$ are perfect squares of an integer.
Triangle $A_1 A_2 A_3$ has a right angle at $A_3$. A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ($n \geq 3$), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$.
(a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$, $n \geq 3$.
(b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$.
[b]p1.[/b] Find all prime factors of $8051$.
[b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$.
[b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G.
Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s?
[b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$?
[b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$).
[img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img]
[b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube?
PS. You had better use hide for answers.
Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
Prove that there are no positive integers $n$ and $k\le n$ such that the numbers
$$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.
Let $a_1, a_2, ... , a_{1000}$ be a sequence of integers such that $a_1=3, a_2=7$ and for all $n=2, 3, ... , 999$
$a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)$.
Find the number of indices $1\leq n\leq 1000$ for which $a_n+2018$ is a perfect square.
Prove that all subsets of a finite set can be arranged in a sequence in which every two successive subsets differ in exactly one element.
Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.
Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$
Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $
[i]Radu Gologan[/i]
Let $N$ be the number of sequences of words (not necessarily grammatically correct) that have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMATHBONANZA. Here is an example of a possible sequence:
[center]
N, NA, NZA, BNZA, BNAZA, BONAZA, BONANZA, CBONANZA, CABONANZA, CAMBONANZA, CAMABONANZA, CAMAHBONANZA, CCAMAHBONANZA, CCAMATHBONANZA.
[/center]
Estimate $\frac{N}{12!}$. An estimate of $E>0$ earns $\max(0,4-2\max(A/E,E/A))$ points, where $A$ is the actual answer. An estimate of $E=0$ earns $0$ points.
[i]2021 CCA Math Bonanza Lightning Round #5.3[/i]
Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 \equal{} b$ and $ x_{n \plus{} 1} \equal{} ax^{m}_{n} \plus{} b$, $ n \equal{} 1,2,...$. Prove that
(1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq \minus{} 2$;
(2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq\frac {(m \minus{} 1)^{m \minus{} 1}}{m^m}$.
Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and
$$kn+a_n=tm(m+1)+a_m$$
Let $f : N \to [0, \infty)$ be a function satisfying the following conditions:
a) $f(4)=2$
b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$.
Find $f(n)$ in closed form.
Let $n$ be a given positive integer.
(a) Do there exist $2n+1$ consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in the ascending order such that $a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}$?
(b) Do there exist consecutive positive integers $a_0,a+1,\ldots,a_{2n}$ in ascending order such that $a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2$?
(c) Do there exist consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in ascending order such that $a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3$?
[hide=Official Hint]You may study the function $f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3$ and prove that the equation $f(x)=0$ has a unique solution $x_n$ with $3n(n+1)<x_n<3n(n+1)+1$. You may use the identity $1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2$.[/hide]
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers $n$.
Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.
The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$.
a) Prove that the sequence $x_i$ has limit.
b) Can this limit be irrational if we have started with the rational number?
c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
Let $a_1, a_2, \dots$ be an infinite sequence of positive integers such that, for all positive integers $m$ and $n,$ we have that $a_{m+n}$ divides $a_ma_n-1.$ Prove that there exists an integer $C$ such that, for all positive integers $k>C,$ we have $a_k=1.$
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]
[i]Proposed by Pavle Martinović[/i]
Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.
For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits.
Hint: $\log 2$ is irrational number.
Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.