Found problems: 5802
Given a real number $t\ge3$, suppose a polynomial $f\in\mathbb R[x]$ satisfies
$$\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.$$Prove that $\deg f\ge n$.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
$$
f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3
$$
for all integers $a, b, c$.
Find all positive integers $n$ such that
$$n(2^n-1)$$
is a perfect square
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$.
[i]Proposed by Ankan Bhattacharya, USA[/i]
Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
Given a positive integer $n$, determine all functions $f$ from the first $n$ positive integers to the positive integers, satisfying the following two conditions: [b](1)[/b] $\sum_{k=1}^{n}{f(k)}=2n$; and [b](2)[/b] $\sum_{k\in K}{f(k)}=n$ for no subset $K$ of the first $n$ positive integers.
Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.
If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.
Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
Let $ a_1<a_2<\cdots<a_n $ be positive integers such that for every distinct $1\leq{i,j}\leq{n}$ we have $ a_j-a_i $ divides $ a_i $.
Prove that
\[ ia_j\leq{ja_i} \qquad \text{ for } 1\leq{i}<j\leq{n} \]
Let $n$ be a positive integer.
Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.
A sequence $P=\left \{ a_{n} \right \}$ is called a $ \text{Permutation}$ of natural numbers (positive integers) if for any natural number $m,$ there exists a unique natural number $n$ such that $a_n=m.$
We also define $S_k(P)$ as:
$S_k(P)=a_{1}+a_{2}+\cdots +a_{k}$ (the sum of the first $k$ elements of the sequence).
Prove that there exists infinitely many distinct $ \text{Permutations}$ of natural numbers like $P_1,P_2, \cdots$ such that$:$
$$\forall k, \forall i<j: S_k(P_i)|S_k(P_j)$$
Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that
\[n+f(m)\mid f(n)+nf(m)\]
for all $m,n\in \mathbb{N}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.
Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.)
[i]Proposed by Steven Liu[/i]
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
Let $S$ be a set of rational numbers with the following properties:
(a) $\frac12$ is an element in $S$,
(b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$.
Prove that $S$ contains all rational numbers in the interval $(0, 1)$.
Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.