Found problems: 3597
TNO 2008 Senior, 5
Consider the polynomial with real coefficients:
\[ p(x) = a_{2008}x^{2008} + a_{2007}x^{2007} + \dots + a_1x + a_0 \]
and it is given that its coefficients satisfy:
\[ a_i + a_{i+1} = a_{i+2}, \quad i \in \{0,1,2,\dots,2006\} \]
If $p(1) = 2008$ and $p(-1) = 0$, compute $a_{2008} - a_0$.
2021 China Team Selection Test, 4
Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that
$$f(a) \equiv g(a+m_p) \pmod p$$
holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that
$$f(x)=g(x+r).$$
2013 Baltic Way, 2
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies
\[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]
Determine the maximal value of $kn$.
2013 AMC 12/AHSME, 25
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $
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.
1997 Vietnam Team Selection Test, 3
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
2019 Canadian Mathematical Olympiad Qualification, 3
Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$.
(a) Find the value of $a^3 + b^3 + c^3$
(b) Find all possible values of $a^2b + b^2c + c^2a$
2015 AMC 10, 12
Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$?
$ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $
2001 National Olympiad First Round, 16
The polynomial $P(x)=x^3+ax+1$ has exactly one solution on the interval $[-2,0)$ and has exactly one solution on the interval $(0,1]$ where $a$ is a real number. Which of the followings cannot be equal to $P(2)$?
$
\textbf{(A)}\ \sqrt{17}
\qquad\textbf{(B)}\ \sqrt[3]{30}
\qquad\textbf{(C)}\ \sqrt{26}-1
\qquad\textbf{(D)}\ \sqrt {30}
\qquad\textbf{(E)}\ \sqrt [3]{10}
$
2012 Putnam, 6
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
2018 Singapore Senior Math Olympiad, 3
Determine the largest positive integer $n$ such that the following statement is true:
There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.
2016 IMO Shortlist, N3
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
1991 Arnold's Trivium, 45
Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.
1976 IMO Shortlist, 12
The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.
2013 BMT Spring, 5
Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2001 Romania National Olympiad, 1
Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.
2006 IMC, 6
The scores of this problem were:
one time 17/20 (by the runner-up)
one time 4/20 (by Andrei Negut)
one time 1/20 (by the winner)
the rest had zero... just to give an idea of the difficulty.
Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.
2020/2021 Tournament of Towns, P1
Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide?
[i]Boris Frenkin[/i]
2014 Flanders Math Olympiad, 4
Let $P(x)$ be a polynomial of degree $5$ and suppose that a and b are real numbers different from zero. Suppose the remainder when $P(x)$ is divided by $x^3 + ax + b$ equals the remainder when $P(x)$ is divided by $x^3 + ax^2 + b$. Then determine $a + b$.
2007 IMC, 1
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.
1989 Greece National Olympiad, 3
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.
2012 USAMTS Problems, 5
An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$.
Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic.
[i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i]
[asy]
size(100);
defaultpen(linewidth(0.8));
for(int i=0;i<=4;i=i+1)
draw((i,0)--(i,4));
for(int i=0;i<=4;i=i+1)
draw((0,i)--(4,i));
[/asy]
2016 IFYM, Sozopol, 6
Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation:
$P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.
2015 Tournament of Towns, 3
Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$.
Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$.
[i]($6$ points)[/i]