Found problems: 3597
Let $ f(0) \equal{} f(1) \equal{} 0$ and
\[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\]
Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$
Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n$, we have that $P(n)$ is not zero and $\frac{P(\overline{nn})}{P(n)}$ is an integer, where $\overline{nn}$ is the integer obtained upon concatenating $n$ with itself.
We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is [i]good[/i] if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\]
[i]Proposed by Mohammad Ahmadi[/i]
Polynomial $ f\left(x\right)$ satisfies $ \left(x \minus{} 1\right)f\left(x \plus{} 1\right) \minus{} \left(x \plus{} 2\right)f\left(x\right) \equal{} 0$ for every $ x\in \Re$. If $ f\left(2\right) \equal{} 6$, $ f\left({\tfrac{3}{2}} \right) \equal{} ?$
$\textbf{(A)}\ -6 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \frac {15}{8} \qquad\textbf{(E)}\ \text{None}$
A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.
a) Consider the function $$f(x,y) = x^2 + xy + y^2$$ Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $$f((x-m),(y-n)) \le 1/2$$
b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le 1/2$.
Prove that in fact, $$g(x,y) \le 1/3$$
Find all the points $(x,y)$, where $g(x,y)=1/3$.
c) Consider function $$f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)$$
Find any $c$ such that $g_a(x,y) \le c$.
Try to obtain the closest estimation.
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
For positive integers $i$ and $N$, let $k_{N,i}$ be the $i$th smallest positive integer such that the polynomial $\frac{x^2}{2023} + \frac{N_x}{7} - k_{N,i}$ has integer roots. Compute the minimum positive integer $N$ satisfying the condition $\frac{k_{N,2023}}{k_{N,1000}}< 3$.
Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be
$\max \left( 0, 25 \min \left( \frac{A}{E} , \frac{E}{A}\right)^{\frac32}\right)$, rounded to the nearest integer.
Let $ f(0) \equal{} f(1) \equal{} 0$ and
\[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\]
Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.
a) if P,Q are similar, then $P(2007)-Q(2007)$ is even
b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.
Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$.
(A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies
\[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]
Determine the maximal value of $kn$.
We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$.
[i]Proposed by Victor Wang[/i]
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder.
[i]Proposed by Yahya Motevassel[/i]
Find the sum of the coefficients of the polynomial $(63x-61)^4$.
Does there exist an integer $n \ge 3$ and an arithmetic sequence $a_0, a_1, ... , a_n$ such that the polynomial $a_nx^n +... + a_1x + a_0$ has $n$ roots which also form an arithmetic sequence?
The polynomial $ (x\plus{}y)^9$ is expanded in decreasing powers of $ x$. The second and third terms have equal values when evaluated at $ x\equal{}p$ and $ y\equal{}q$, where $ p$ and $ q$ are positive numbers whose sum is one. What is the value of $ p$?
$ \textbf{(A)}\ 1/5 \qquad
\textbf{(B)}\ 4/5 \qquad
\textbf{(C)}\ 1/4 \qquad
\textbf{(D)}\ 3/4 \qquad
\textbf{(E)}\ 8/9$
For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.
If $ a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial
\[ \left(1\plus{}x\plus{}x^2\right)^{25}\]
show that $ a_0\plus{}a_2\plus{}a_4\plus{}\cdots\plus{}a_{50}$ is even.
Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.
Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.