Found problems: 3597
2012 District Olympiad, 2
Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that:
[b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $
[b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $
2005 Putnam, B1
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$
(Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)
1977 Canada National Olympiad, 4
Let
\[p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0\]
and
\[q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0\]
be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.
2020/2021 Tournament of Towns, P2
Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?
2007 IberoAmerican Olympiad For University Students, 5
Determine all pairs of polynomials $f,g\in\mathbb{C}[x]$ with complex coefficients such that the following equalities hold for all $x\in\mathbb{C}$:
$f(f(x))-g(g(x))=1+i$
$f(g(x))-g(f(x))=1-i$
1980 IMO Longlists, 5
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.
2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.
2016 Postal Coaching, 1
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.
2020-2021 OMMC, 12
Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$
1989 IMO Longlists, 39
Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices that the frequency with which each sum occurs is the same as if it were the sum of two natural numbers drawn at random from the range 1 to 4. What can he deduce about the numbers on the balls?
1985 IMO Longlists, 33
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
1998 Vietnam Team Selection Test, 1
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.
1989 USAMO, 3
Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.
2014 IFYM, Sozopol, 4
Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and
$(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.
2017 NMTC Junior, 5
(a) Prove that $x^4+3x^3+6x^2+9x+12$ cannot be expressed as product of two polynomials of degree 2 with integers coefficients.
(b) $2n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.
Russian TST 2015, P1
Let $P(x, y)$ and $Q(x, y)$ be polynomials in two variables with integer coefficients. The sequences of integers $a_0, a_1,\ldots$ and $b_0, b_1,\ldots$ satisfy \[a_{n+1}=P(a_n,b_n),\quad b_{n+1}=Q(a_n,b_n)\]for all $n\geqslant 0$. Let $m_n$ be the number of integer points of the coordinate plane, lying strictly inside the segment with endpoints $(a_n,b_n)$ and $(a_{n+1},b_{n+1})$. Prove that the sequence $m_0,m_1,\ldots$ is non-decreasing.
2019 Israel Olympic Revenge, P1
A polynomial $P$ in $n$ variables and real coefficients is called [i]magical[/i] if $P(\mathbb{N}^n)\subset \mathbb{N}$, and moreover the map $P: \mathbb{N}^n \to \mathbb{N}$ is a bijection. Prove that for all positive integers $n$, there are at least
\[n!\cdot (C(n)-C(n-1))\]
magical polynomials, where $C(n)$ is the $n$-th Catalan number.
Here $\mathbb{N}=\{0,1,2,\dots\}$.
2010 Romanian Masters In Mathematics, 6
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]
2012 Iran Team Selection Test, 1
Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity.
[i]Proposed by Mr.Etesami[/i]
PEN Q Problems, 12
Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
1968 IMO Shortlist, 16
A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$
1995 VJIMC, Problem 2
Let $f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}$ and $f_k=f_{2n-k}$ for each $k$. Prove that $f(z)=z^ng(z+z^{-1})$, where $g$ is a polynomial of degree $n$.
2005 IberoAmerican Olympiad For University Students, 7
Prove that for any integers $n,p$, $0<n\leq p$, all the roots of the polynomial below are real:
\[P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j\]
2006 Flanders Math Olympiad, 1
(a) Solve for $\theta\in\mathbb{R}$: $\cos(4\theta) = \cos(3\theta)$
(b) $\cos\left(\frac{2\pi}{7}\right)$, $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$.
2014 Contests, 3.
For each positive integer $n$, determine the smallest possible value of the polynomial
$$
W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx.
$$