Found problems: 3597
2003 Vietnam National Olympiad, 2
Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.
2021 Hong Kong TST, 4
Does there exist a nonzero polynomial $P(x)$ with integer coefficients satisfying both of the following conditions?
[list]
[*]$P(x)$ has no rational root;
[*]For every positive integer $n$, there exists an integer $m$ such that $n$ divides $P(m)$.
[/list]
2017 239 Open Mathematical Olympiad, 4
A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)
2017 South Africa National Olympiad, 6
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
2006 China Team Selection Test, 1
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties:
(1) $f(0)=0$ and $f(1)=1$; and.
(2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.
2014 International Zhautykov Olympiad, 1
Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $?
[i]Proposed by Alexander S. Golovanov, Russia[/i]
2012 Indonesia TST, 1
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 $0 \le i \le n$.)
PEN Q Problems, 7
Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
1968 Putnam, A5
Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$.
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.
2009 Putnam, B4
Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$
2015 Iran Team Selection Test, 1
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that
$$P(x)^3+Q(x)^3=x^{12}+1.$$
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.
1977 Poland - Second Round, 5
Let the polynomials $ w_n $ be given by the formulas: $$
w_1(x) = x^2 - 1, \quad w_{n+1}(x) = w_n(x)^2 - 1, \quad (n = 1, 2, \ldots)$$ and let $a$ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have?
2010 Tuymaada Olympiad, 3
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$.
1998 VJIMC, Problem 3
Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.
1986 National High School Mathematics League, 1
For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$.
Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.
2007 All-Russian Olympiad Regional Round, 11.2
Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.
2022 AIME Problems, 13
There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$
2008 Romania Team Selection Test, 5
Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]
2006 Bulgaria Team Selection Test, 3
[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$
[i] Ivan Landgev[/i]
1990 IMO Shortlist, 26
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
1963 Putnam, B1
For what integers $a$ does $x^2 -x+a$ divide $x^{13}+ x +90$ ?
2012 APMO, 1
Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.
1995 Turkey Team Selection Test, 1
Given real numbers $b \geq a>0$, find all solutions of the system
\begin{align*}
&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.
\end{align*}