Found problems: 5923
Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$.
[color=#008000]Moderator says: and the original source for this one is Richard Stanley, [i]Enumerative Combinatorics[/i] vol. 1 (1st edition), exercise 1.15.[/color]
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.
Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a [i]table[/i] denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called [i]balanced[/i] if one diagonal pair of points are coloured white and the other diagonal pair black.
Determine all possible values of \(k \geq 2\) for which there exists a colouring of a \(k\times 2019\) grid with no balanced tables.
[i]proposed by the ICMC Problem Committee[/i]
Find all pairs of natural numbers $(n, k)$ with the following property:
Given a $k\times k$ array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of $n$.
Note: A path from the left to the right edge is a sequence of cells of the array $a_1, a_2, ... , a_m$ so that $a_1$ is a cell of the leftmost column, $a_m$ is the cell of the rightmost column, and $a_{i}$, $a_{i+1}$ share an edge for all $i = 1, 2, ... , m -1$.
For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$
Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits?
[i]Proposed by A. Shapovalov[/i]
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$
\[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \]
Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$
Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$
The sum
$$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$
$ $ \\
can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$.
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_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 $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$.
Show that the sequence $\{s_k\}$ is convergent.
Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties:
(i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$;
(ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then
$ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$;
(iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$.
A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties:
If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$;
If $ s_i\geq 0$, then $ f(S)\geq 0$;
$ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$;
$ f(ST)\equal{}f(S)f(T)$,
$ f(S')\equal{}f(S)$
Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.
$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by
\[a_0 = a, \quad a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.\]
Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that
\[0 < n(A - a_n) < A^3.\]
Is it possible to complete the following square knowning that each row and column make an aritmetic progression?
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
Let $(a_n)^\infty_{n=1}$ be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms $a_n,a_{n+1},a_{n+2},a_{n+3}$ belongs to the same sequence. Prove that the sequence $\frac{a_{n+1}}{a_n}$ converges and find all possible values of its limit.
For all positive integers $ x$, let
\[ f(x) \equal{} \begin{cases}1 & \text{if }x \equal{} 1 \\
\frac x{10} & \text{if }x\text{ is divisible by 10} \\
x \plus{} 1 & \text{otherwise}\end{cases}\]and define a sequence as follows: $ x_1 \equal{} x$ and $ x_{n \plus{} 1} \equal{} f(x_n)$ for all positive integers $ n$. Let $ d(x)$ be the smallest $ n$ such that $ x_n \equal{} 1$. (For example, $ d(100) \equal{} 3$ and $ d(87) \equal{} 7$.) Let $ m$ be the number of positive integers $ x$ such that $ d(x) \equal{} 20$. Find the sum of the distinct prime factors of $ m$.
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$
Let $a_1, a_2, \ldots$ be a strictly increasing sequence of positive integers, such that for any positive integer $n$, $a_n$ is not representable in the for $\sum_{i=1}^{n-1}c_ia_i$ for $c_i \in \{0, 1\}$. For every positive integer $m$, let $f(m)$ denote the number of $a_i$ that are at most $m$. Show that for any positive integers $m, k$, we have that $$f(m) \leq a_k+\frac{m} {k+1}.$$
Tim has a multiset of positive integers. Let $c_i$ be the number of occurrences of numbers that are [i]at least[/i] $i$ in the multiset. Let $m$ be the maximum element of the multiset. Tim calls a multiset [i]spicy[/i] if $c_1, \dots, c_m$ is a sequence of strictly decreasing powers of $3$. Tim calls the [i]hotness[/i] of a spicy multiset the sum of its elements. Find the sum of the hotness of all spicy multisets that satisfy $c_1 = 3^{2020}$. Give your answer $\pmod{1000}$. (Note: a multiset is an unordered set of numbers that can have repeats)
[i]Proposed by Timothy Qian[/i]
Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.