Found problems: 5923
Consider a cube and let$ M, N$ be two of its vertices. Assign the number $1$ to these vertices and $0$ to the other six vertices. We are allowed to select a vertex and to increase with a unit the numbers assigned to the $3$ adjiacent vertices - call this a [i]movement[/i].
Prove that there is a sequence of [i]movements [/i] after which all the numbers assigned to the vertices of the cube became equal if and only if $MN$ is not a diagonal of a face of the cube.
Marius Ghergu, Dinu Serbanescu
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of integers. Define $a_n^{(0)} = a_n$ for all $n \in \mathbb{N}$. For all $M \geq 0$, we define $(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}$. We say that $(a_n)_{n \in \mathbb{N}}$ is $\textrm{(M + 1)-self-referencing}$ if there exists $k_1$ and $k_2$ fixed positive integers such that $a_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}$.
(a) Does there exist a sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$?
(a) Does there exist a stricly positive sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$?
A sequence $a_n$ is defined by $a_0 = 0$, and for all $n \ge 1$, $a_n = a_{n-1} + (-1)^n \cdot n^2$. Compute $a_{100}$
Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]
What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$?
(A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above
In a group of $2n$ students, each student has exactly $3$ friends within the group. The friendships are mutual and for each two students $A$ and $B$ which are not friends, there is a sequence $C_1, C_2, ..., C_r$ of students such that $A$ is a friend of $C_1$, $C_1$ is a friend of $C_2$, et cetera, and $C_r$ is a friend of $B$.
Every student was asked to assess each of his three friendships with: "acquaintance", "friend" and "BFF". It turned out that each student either gave the same assessment to all of his friends or gave every assessment exactly once.
We say that a pair of students is in conflict if they gave each other different assessments. Let $D$ be the set of all possible values of the total number of conflicts.
Prove that $|D| \geq 3n$ with equality if and only if the group can be partitioned into two subsets such that each student is separated from all of his friends.
In arithmetic sequence $(a_n)$, $3a_8=5a_{13},a_1>0$. Define $S_n=\sum_{i=1}^n a_i$, then the largest number in $(S_n)$ is
$\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}$
Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $
Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$.
Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.
Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote
\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]
a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$.
Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.
The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \).
a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \).
b) Does there exist a non-constant bounded sequence with the above properties?
(A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)
A sequence $x_1,x_2,x_3,...$ has the following properties:
(a) $1 = x_1 < x_2 < x_3 < ...$
(b) $x_{n+1} \le 2n$ for all $n \in N$.
Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.
Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).
Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]
is a multiple of $p$.
[/list]
$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information.
[I]Proposed by Dominic Yeo, United Kingdom.[/i]
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?
In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]