Found problems: 5923
a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence.
b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$
c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by$$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$.Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.
Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that:
1. Every positive integer appears in the sequence at least once, and;
2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$.
[i](Proposed by Ho Janson)[/i]
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
.
Consider the infinite sequence $\{a_i\}$ that extends the pattern
\[1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \dots\]
Formally, $a_i = i-T(i)$ for all $i \geq 1$, where $T(i)$ represents the largest triangular number less than $i$ (triangle numbers are integers of the form $\frac{k(k+1)}2$ for some nonnegative integer $k$). Find the number of indices $i$ such that $a_i = a_{i + 2020}$.
[i]Proposed by Gabriel Wu[/i]
For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that $x_n = ax_{n-1} + b$ for each $n \ge 2$. Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$.
(Jaromír Šimša)
Determine if there are positive integers $a, b$ such that all terms of the sequence defined by
\[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.
Given the sequence $(a_n)$, in which $a_n =\frac14 n^4 - 10n^2(n - 1)$, with $n = 0, 1, 2,...$ Determine the smallest term of the sequence.
Let $(s_n)_{n\geq 1}$ be a sequence given by $s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}$ with $\lim \limits_{n \to \infty}s_n=s=$Ioachimescu constant and $(a_n)_{n\geq 1}$ , $(b_n)_{n\geq 1}$ be a positive real sequences such that $$\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+$$Compute$$\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}$$
A sequence $\{a_n\}_{n\geq 1}$ of positive reals is defined by the rule $a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2$ for integers $n>2$ together with the initial values $a_1=8$ and $a_2=64$ and $a_3=1024$. Compute \[\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}\]
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$
The sequences of natural numbers $p_n$ and $q_n$ are given such that
$$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$
Prove that $p_n$ and $q_m$ are coprime for any m and n.
Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.
Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation:
\[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \]
The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$
Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$?
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.
[i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.
Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$.
[i]Proposed by Ivan Chan Kai Chin[/i]
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that
$\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$
Let
$y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural.
Prove that $(y_n)$ is an increasing and divergent sequence.
The volume of a certain rectangular solid is $ 8 \text{ cm}^3$, its total surface area is $ 32 \text{ cm}^3$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
$ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 44$
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that
\[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \]
and
\[ a_1 + \dots + a_n = b_1 + \dots + b_n. \]
Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.
A sequence of integers $x_1, x_2, \dots, x_k$ is called fibtastic if the difference between any two consecutive elements in the sequence is a Fibonacci number.
The integers from $1$ to $2024$ are split into two groups, each written in increasing order.
Group A is $a_1, a_2, \dots, a_m$ and Group B is $b_1, b_2, \dots, b_n.$
Find the largest integer $M$ such that we can guarantee that we can pick $M$ consecutive elements from either Group A or Group B which form a fibtastic sequence.
As an illustrative example, if a group of numbers is $2, 4, 11, 12, 13, 16, 18, 27, 29, 30,$ the
longest fibtastic sequence is $11, 12, 13, 16, 18,$ which has length $5.$
Welcome to the [b]USAYNO[/b], a twelve-part question where each part has a yes/no answer. If you provide $C$ correct answers, your score on this problem will be $\frac{C}{6}$.
Your answer should be a twelve-character string containing `Y' (for yes) and `N' (for no). For instance if you think a, c, and f are `yes' and the rest are `no', you should answer YNYNNYNNNNNN.
(a) Is there a positive integer $n$ such that the sum of the digits of $2018n+1337$ in base $10$ is $2018$ more than the sum of the digits of $2018n+1337$ in base $4$?
(b) Is there a fixed constant $\theta$ such that for all triangles $ABC$ with $$2018AB^2=2018CA^2+2017CA\cdot CB+2018CB^2,$$ one of the angles of $ABC$ is $\theta$?
(c) Adam lists out every possible way to arrange the letters of ``CCACCACCA'' (including the given arrangement) at $1$ arrangement every $5$ seconds. Madam lists out every possible way to arrange the letters of ``CCACCAA'' (including the given arrangement) at $1$ arrangement every $12$ seconds. Does Adam finish first?
(d) Do there exist real numbers $a,b,c$, none of which is the average of the other two, such that \[\frac{1}{b+c-2a}+\frac{1}{c+a-2b}+\frac{1}{a+b-2c}=0?\]
(e) Let $f\left(x\right)=\frac{2^x-2}{x}-1$. Is there an integer $n$ such that $$f\left(n\right),f\left(f\left(n\right)\right),f\left(f\left(f\left(n\right)\right)\right),\ldots$$ are all integers?
(f) In an elementary school with $2585$ students and $159$ classes (every student is in exactly one class), each student reports the size of their class. The principal of the school takes the average of all of these numbers and calls it $X$. The principal then computes the average size of each class and calls it $Y$. Is it necessarily true that $X>Y$?
(g) Six sticks of lengths $3$, $5$, $7$, $11$, $13$, and $17$ are put together to form a hexagon. From a point inside the hexagon, a circular water balloon begins to expand and will stop expanding once it hits any stick. Is it possible that once the balloon stops expanding, it is touching each of the six sticks?
(h) A coin is biased so that it flips heads and tails (and only heads or tails) each with a positive rational probability (not necessarily $\frac{1}{2}$). Is it possible that on average, it takes exactly twice as long to flip two heads in a row as it is to flip two tails in a row?
(i) Does there exist a base $b$ such that $2018_b$ is prime?
(j) Does there exist a sequence of $2018$ distinct real numbers such that no $45$ terms (not necessarily consecutive) can be examined, in order, and be in strictly increasing or strictly decreasing order?
(k) Does there exist a scalene triangle $ABC$ such that there exist two distinct rectangles $PQRS$ inscribed in $\triangle{ABC}$ with $P\in AB$, $Q,R\in BC$, $S\in AC$ such that the angle bisectors of $\angle{PAS}$, $\angle{PQR}$, and $\angle{SRQ}$ concur?
(l) For three vectors $\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3$ with $\mathbf{u}_i=\left(x_{i,1},x_{i,2},x_{i,3},x_{i,4}\right)$, define \[f\left(\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\right)=1-\displaystyle\prod_{j=1}^4\left(1+\left(x_{2,j}-x_{3,j}\right)^2+\left(x_{3,j}-x_{1,j}\right)^2+\left(x_{1,j}-x_{2,j}\right)^2\right).\] Are there any sequences $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_{18}$ of distinct vectors with four components, with all components in $\left\{1,2,3\right\}$, such that \[\displaystyle\prod_{1\leq i<j<k\leq18}f\left(\mathbf{v}_i,\mathbf{v}_j,\mathbf{v}_k\right)\equiv1\pmod3?\]
[i]2018 CCA Math Bonanza Lightning Round #5.4[/i]
We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.