Found problems: 3597
2020 Tournament Of Towns, 4
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:
a) in exactly one way;
b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov
1969 IMO Longlists, 54
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
2019 Thailand TST, 3
Determine all polynomials $P (x, y), Q(x, y)$ and $R(x, y)$ with real coefficients satisfying $$P (ux + vy, uy + vx) = Q(x, y)R(u, v)$$
for all real numbers $u, v, x$ and $y$.
2016 Tournament Of Towns, 6
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.
2013 All-Russian Olympiad, 2
Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out?
[i]A. Golovanov[/i]
2020 Iran MO (3rd Round), 2
Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)
1991 Arnold's Trivium, 91
Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.
1969 Canada National Olympiad, 7
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
2021 Korea Winter Program Practice Test, 7
Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below.
[b]Condition[/b]: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$
2006 Pre-Preparation Course Examination, 4
Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.
1996 AIME Problems, 5
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a, b,$ and $c,$ and that the roots of $x^3+rx^2+sx+t=0$ are $a+b, b+c,$ and $c+a.$ Find $t.$
2003 Putnam, 1
Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?
2003 IMO, 6
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
2014 IMAC Arhimede, 4
Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$.
If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.
2011 USA Team Selection Test, 6
A polynomial $P(x)$ is called [i]nice[/i] if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that
\[Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}\]
is nice as well.
1971 IMO Longlists, 55
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying
\[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]
2002 SNSB Admission, 2
Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left(
a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left(
\mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.
2019 Belarusian National Olympiad, 10.3
The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all possible values of $P_1(1,1,\ldots,1)$.
[i](A. Yuran)[/i]
2002 Iran MO (2nd round), 5
Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define
\[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\]
Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$
2005 Postal Coaching, 14
Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$
2012 Math Prize For Girls Problems, 17
How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation
\[
a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2 \, ?
\]
2019 Iran Team Selection Test, 1
Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$:
$$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$
[i]Proposed by Sina Saleh[/i]
2016 India Regional Mathematical Olympiad, 3
Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$
2007 Iran MO (3rd Round), 6
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$.
[img]http://i14.tinypic.com/4tk6mnr.png[/img]
a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.
b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.
2024 ISI Entrance UGB, P2
Suppose $n\ge 2$. Consider the polynomial \[Q_n(x) = 1-x^n - (1-x)^n .\]
Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.