Found problems: 3597
2011 Bogdan Stan, 1
Consider the multiplicative group $ \left\{ \left.A_k:=\left(\begin{matrix} 2^k& 2^k\\2^k& 2^k\end{matrix}\right)\right| k\in\mathbb{Z} \right\} . $
[b]a)[/b] Prove that $A_xA_y=A_{x+y+1} , $ for all integers $ x,y. $
[b]b)[/b] Show that, for all integers $ t, $ the multiplicative group $ \left\{ A_{jt-1}|j\in\mathbb{Z} \right\} $ is a subgroup of $ G. $
[b]c)[/b] Determine the linear integer polynomials $ P $ for which it exists an isomorphism $ \left(
G,\cdot \right)\stackrel{\eta}{\cong}\left( \mathbb{Z} ,+ \right) $ such that $ \eta\left( A_k \right) =P(k). $
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2017 VJIMC, 1
Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by
\[F(x)=\sum_{n=1}^{\infty} a_nx^n\]
and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.
2011 USA Team Selection Test, 3
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum
\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]
Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.
2011 All-Russian Olympiad, 1
Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$.
2018 Nordic, 4
Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?
1985 Traian Lălescu, 1.2
Prove that all real roots of the polynomial
$$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$
are positive.
1998 Vietnam Team Selection Test, 1
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.
2024 China Team Selection Test, 23
$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$.
[i]Proposed by Yijun Yao[/i]
2010 Contests, 1
suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)
Russian TST 2016, P1
For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?
2011 AIME Problems, 8
Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.
2007 ISI B.Math Entrance Exam, 5
Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$.
$(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$.
$(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.
1981 Romania Team Selection Tests, 1.
Let $P(X)=aX^3-\frac16 X$ where $a\in\mathbb{R}$.
[b]1)[/b] Determine $a$ such that for every $\alpha\in\mathbb{Z}$ we have $P(\alpha)\in\mathbb{Z}$.
[b]2)[/b] Show that if $a$ is irrational then for every $0<u<v<1$ there exists $n\in\mathbb{Z}$ such that
\[u<P(n)-\lfloor P(n)\rfloor <v.\]
Generalize the problem!
1985 IMO, 3
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}). \]
2002 Vietnam Team Selection Test, 2
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
2019 Hanoi Open Mathematics Competitions, 3
Let $a$ and $b$ be real numbers, and the polynomial $P(x) =ax + b$ such that $P(2)- P(1)= 3$:
Compute the value of $P(5)- P(0)$.
[b]A.[/b] $11$ [b]B.[/b] $13$ [b]C.[/b] $15$ [b]D.[/b] $17$ [b]E.[/b] $19$
2023 Iran Team Selection Test, 5
Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.
[i]Proposed by Hesam Rajabzadeh [/i]
2002 Spain Mathematical Olympiad, Problem 1
Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$:
$P(x^2-y^2) = P(x+y)P(x-y)$.
2008 Silk Road, 4
Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist
$ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$
2011 IMO Shortlist, 6
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
[i]Proposed by Oleksiy Klurman, Ukraine[/i]
2012 ELMO Problems, 3
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
1986 All Soviet Union Mathematical Olympiad, 439
Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.
2015 Tournament of Towns, 5
Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding $2015$ but all coefficients of their product have absolute values not exceeding $1$?
[i]($10$ points)[/i]
2011 Math Prize For Girls Problems, 11
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?