Found problems: 3597
Kvant 2020, M2592
Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers?
[i]Proposed by F. Petrov[/i]
2016 Tournament Of Towns, 5
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial
on the blackboard has $37$ distinct positive roots. [i](8 points)[/i]
[i]Alexandr Kuznetsov[/i]
2012 Iran MO (3rd Round), 4
Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$
\[x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|\]
and the following polynomial has only one positive real root like $\beta$
\[x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.\]
And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.
2022 JHMT HS, 1
If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.
PEN S Problems, 6
Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.
2021 Moldova Team Selection Test, 1
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
[i]Brazitikos Silouanos, Greece[/i]
1999 National Olympiad First Round, 8
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
1990 APMO, 2
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
2008 Kazakhstan National Olympiad, 3
Let $ f(x,y,z)$ be the polynomial with integer coefficients. Suppose that for all reals $ x,y,z$ the following equation holds:
\[ f(x,y,z) \equal{} \minus{} f(x,z,y) \equal{} \minus{} f(y,x,z) \equal{} \minus{} f(z,y,x)
\]
Prove that if $ a,b,c\in\mathbb{Z}$ then $ f(a,b,c)$ takes an even value
2018 Harvard-MIT Mathematics Tournament, 10
Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.
1969 German National Olympiad, 6
Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.
2011 Putnam, B2
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$
2022 Greece Junior Math Olympiad, 1
(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients.
(b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$ find the maximum possible value of $a$.
2016 Peru IMO TST, 16
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
2023 IMC, 4
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2016 Saudi Arabia BMO TST, 1
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.
2022 Baltic Way, 3
We call a two-variable polynomial $P(x, y)$ [i]secretly one-variable,[/i] if there exist polynomials $Q(x)$ and $R(x, y)$ such that $\deg(Q) \ge 2$ and $P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are [i]secretly one-variable[/i], but $xy + 1$ is not).
Prove or disprove the following statement: If $P(x, y)$ is a polynomial such that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two non-constant polynomials, then $P$ is [i]secretly one-variable[/i].
[i]Note: All polynomials are assumed to have real coefficients. [/i]
1999 National Olympiad First Round, 12
\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\
{x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\
{x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array}
\]
The number of real triples $ \left(x,y,z\right)$ satisfying above system is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$
2018 Korea National Olympiad, 3
Denote $f(x) = x^4 + 2x^3 - 2x^2 - 4x+4$. Prove that there are infinitely many primes $p$ that satisfies the following.
For all positive integers $m$, $f(m)$ is not a multiple of $p$.
2009 China Team Selection Test, 2
Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.
2015 Iran Team Selection Test, 5
Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ :
$P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$
$n>1$
2011 Iran MO (3rd Round), 1
We define the recursive polynomial $T_n(x)$ as follows:
$T_0(x)=1$
$T_1(x)=x$
$T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$.
[b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$.
[b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$.
[i]Proposed by Morteza Saghafian[/i]
2001 Mongolian Mathematical Olympiad, Problem 1
Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying
$$p(x)p(4-x)=p(x(4-x))$$
1951 Moscow Mathematical Olympiad, 202
Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.