Found problems: 5923
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that
\[2^{a_7}=2^{27} \cdot a_7.\]
What is the minimum possible value of $a_2$?
$\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$
Prove that for every real number $M$ there exists an infinite arithmetical progression of positive integers such that [list] [*] the common difference is not divisible by $10$, [*] the sum of digits of each term exceeds $M$. [/list]
How many finite sequances $x_1,x_2,\cdots,x_m$ are there such that $x_i=1$ or 2 and $\sum \limits_{i=1}^mx_i=10$ ?
[list=1]
[*] 89
[*] 73
[*] 107
[*] 119
[/list]
Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave."
[i]L. Fejes- Toth, E. Makai[/i]
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$.
Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.
Victor shuffles a standard 54-card deck then flips over cards one at a time onto a pile stopping after the first ace. However, if he ever reveals a joker he discards the entire pile, including the joker, and starts a new pile; for example, if the sequence of cards is 2-3-Joker-A, the pile ends with one card in it. Find the expected number of cards in the end pile.
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?
[i]By Omid Hatami[/i]
On each cell of a $200 \times 200$ grid, we place a car, which faces in one of the four cardinal directions.In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cellforward. If a move would cause a car to exit the grid, the car is removed instead. The cars are placedso that there exists a sequence of moves that eventually removes all the cars from the grid. Across all
such starting configurations, determine the maximum possible number of moves to do so.
Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$
The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$.
Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.
Find all $\{a_n\}_{n\ge 0}$ that satisfies the following conditions.
(1) $a_n\in \mathbb{Z}$
(2) $a_0=0, a_1=1$
(3) For infinitly many $m$, $a_m=m$
(4) For every $n\ge2$, $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$ $\mod n$
Let $f: [0, \infty) \to [0, \infty)$ be a continuous function with $f(0)>0$ and having the property $$x-y<f(y)-f(x) \le 0~\forall~0 \le x<y.$$ Prove that:
$a)$ There exists a unique $\alpha \in (0, \infty)$ such that $(f \circ f)(\alpha)=\alpha.$
$b)$ The sequence $(x_n)_{n \ge 1},$ defined by $x_1 \ge 0$ and $x_{n+1}=f(x_n)~\forall~n \in \mathbb{N}$ is convergent.
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.
[u]Round 5[/u]
[b]p13.[/b] Let $\{a\} _{n\ge 1}$ be an arithmetic sequence, with $a_ 1 = 0$, such that for some positive integers $k$ and $x$ we have $a_{k+1} = {k \choose x}$, $a_{2k+1} ={k \choose {x+1}}$ , and $a_{3k+1} ={k \choose {x+2}}$. Let $\{b\}_{n\ge 1}$ be an arithmetic sequence of integers with $b_1 = 0$. Given that there is some integer $m$ such that $b_m ={k \choose x}$, what is the number of possible values of $b_2$?
[b]p14.[/b] Let $A = arcsin \left( \frac14 \right)$ and $B = arcsin \left( \frac17 \right)$. Find $\sin(A + B) \sin(A - B)$.
[b]p15.[/b] Let $\{f_i\}^{9}_{i=1}$ be a sequence of continuous functions such that $f_i : R \to Z$ is continuous (i.e. each $f_i$ maps from the real numbers to the integers). Also, for all $i$, $f_i(i) = 3^i$. Compute $\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})$.
[u]Round 6[/u]
[b]p16.[/b] If $x$ and $y$ are integers for which $\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341$ and $x - y = 1$, then compute $x + y$.
[b]p17.[/b] Let $T_n$ be the number of ways that n letters from the set $\{a, b, c, d\}$ can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum $T_5 + T_6$.
[b]p18.[/b] McDonald plays a game with a standard deck of $52$ cards and a collection of chips numbered $1$ to $52$. He picks $1$ card from a fully shuffled deck and $1$ chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of $6$. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form $\frac{x^2 \cdot y}{z^3}$ such that $x, y$, and $z$ are relatively prime positive integers. What is $x + y + z$?
(NOTE: Use Ace as $1$, Jack as $11$, Queen as $12$, and King as $13$)
[u]Round 7[/u]
[b]p19.[/b] Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$.
[b]p20.[/b] $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$.
[b]p21.[/b] Define $\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1$ as the decimal representation of a four digit integer. You are given that $3^x5^y7^z2^t = \overline{xyz(t + 1)}$ where $x, y, z$, and t are non-negative integers such that $t$ is odd and $0 \le x, y, z,(t + 1) \le 9$. Compute$3^x5^y7^z$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
Let $(a_n)_{n\ge 1}$ be a sequence of positive numbers. If there is a constant $M > 0$ such that $a_2^2 + a_2^2 +\ldots + a_n^2 < Ma_{n+1}^2$ for all $n$, then prove that there is a constant $M ' > 0$ such that $a_1 + a_2 +\ldots + a_n < M ' a_{n+1}$ .
Let $n \ge 4$ be a natural number. Let $A_1A_2 \cdots A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 < \cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.
Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$.
Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$.
Does the conclusion of the problem remain true if the sets $X_n$ are infinite?
[i]Claudiu Raicu[/i]