Found problems: 766
Let $D_n$ be the determinant of order $n$ of which the element in the $i$-th row and the $j$-th
column is $|i-j|.$ Show that $D_n$ is equal to
$$(-1)^{n-1}(n-1)2^{n-2}.$$
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :
$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and
$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$
Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and
\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]
for all integers \(k \geq 1\). Determine all positive integers \(n\) such that
\[
\frac{a_n}{n}
\]
is an integer.
Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
Let $n\geq 4$ and $y_1,\dots, y_n$ real with
$$\sum_{k=1}^n y_k=\sum_{k=1}^n k y_k=\sum_{k=1}^n k^2y_k=0$$
and
$$y_{k+3}-3y_{k+2}+3y_{k+1}-y_k=0$$
for $1\leq k\leq n-3$. Prove that
$$\sum_{k=1}^n k^3y_k=0$$
A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$
$
\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$
Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]
We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$.
(a) Show that there are infinitely many good numbers.
(b) Show that if $n$ is a good number, then $7 \nmid n$.
Consider a circular necklace with $\displaystyle{2013}$ beads. Each bead can be paintes either green or white. A painting of the necklace is called [i]good[/i] if among any $\displaystyle{21}$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.
[b]Note.[/b] Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.
[i]Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.[/i]
The sequence $(x_n)$ is defined by formulas
$$
x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$
Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{
\begin{array}{ll}
x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\
& \\
x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\
\end{array}
\right. $$
where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.
There are $11$ points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?
Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.
Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that
\[
\frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k
\]
for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$
Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.
Vincent has a fair die with sides labeled $1$ to $6$. He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If Vincent rolls a different value than his previous roll, he records the value and continues rolling. If Vincent rolls the same value, he stops, does \emph{not} record his final roll, and computes the average of his previously recorded rolls. Given that Vincent first rolled a $1$, let $E$ be the expected value of his result. There exist rational numbers $r,s,t > 0$ such that $E = r-s\ln t$ and $t$ is not a perfect power. If $r+s+t = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Sean Li[/i]
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.