Found problems: 3597
2016 Mathematical Talent Reward Programme, SAQ: P 1
Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .
2005 Baltic Way, 4
Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
2019 IOM, 6
Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that
\[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\]
[i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]
2008 Indonesia TST, 1
A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$
1981 AMC 12/AHSME, 30
If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are
\[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2}
\]is
$ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$
$ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$
1982 IMO Longlists, 16
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
2010 Germany Team Selection Test, 3
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
2000 German National Olympiad, 2
For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.
1986 Miklós Schweitzer, 8
Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers.
(i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which
$$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$
and
$$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$
where the constant $c$ depends only on the numbers $a_i, b_i$.
(ii) Prove that, in general, (*) cannot be replaced by the relation
$$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$
[J. Szabados]
2019 AMC 12/AHSME, 17
Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$
2010 Contests, 2
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
2001 Spain Mathematical Olympiad, Problem 1
Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that:
$P(x) = b + (x-a)Q((x-a)^2)).$
1975 All Soviet Union Mathematical Olympiad, 217
Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients;
Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
2011 Morocco National Olympiad, 2
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :
$\left\{\begin{matrix}
x+y+z+t=4\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}
\end{matrix}\right.$
2021 JHMT HS, 10
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of
\[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \]
can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$
1982 Poland - Second Round, 1
Prove that if $ c, d $ are integers with $ c \neq d $, $ d > 0 $ then the equation $$ x^3 - 3cx^2 - dx + c = 0$$
has no more than one rational root.
PEN E Problems, 5
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
2010 Germany Team Selection Test, 3
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.
[i]Proposed by Jozsef Pelikan, Hungary[/i]
1953 Poland - Second Round, 1
Prove that the equation
$$ (x - a) (x - c) + 2 (x - b) (x - d) = 0,$$
in which $ a < b < c < d $, has two real roots.
2012 Ukraine Team Selection Test, 11
Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{ a_1, a_2, ..., a_k\}$ of positive integers we denote $S (A) = P (a_1) + P (a_2) + ... + P (a_k )$. The natural numbers $m, n$ are such that $m ^{d+ 1} | n$. Prove that the set $\{1, 2, ..., n\}$ can be subdivided into $m$ disjoint subsets $A_1, A_2, ..., A_m$ with the same number of elements such that $S (A_1) = S(A_2) = ... = S (A_m )$.
2023 VN Math Olympiad For High School Students, Problem 6
Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$
a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number.
b) $x^{2^n}+1,$ with $n$ is a positive integer.
1972 Miklós Schweitzer, 6
Let $ P(z)$ be a polynomial of degree $ n$ with complex coefficients, \[ P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .\] Prove that every root of $ P(z)$ in the closed unit disc has multiplicity at most $ c\sqrt{n}$, where $ c\equal{}c(M) >0$ is a constant depending only on $ M$.
[i]G. Halasz[/i]
2008 Bulgaria Team Selection Test, 3
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
2003 IMC, 3
Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)