Found problems: 3597
2002 AMC 10, 11
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
2010 Harvard-MIT Mathematics Tournament, 1
Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.
2024 Vietnam Team Selection Test, 6
Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. Determine all polynomials $Q(x) \in \mathbb{Z}[x]$, such that for every positive integer $n$, there exists a polynomial $R_n(x) \in \mathbb{Z}[x]$ satisfies
$$Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).$$
2009 IberoAmerican Olympiad For University Students, 6
Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials
$f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$
have integer coefficients.
Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$.
Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j) \right|=1$
2010 All-Russian Olympiad, 3
Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_1<x_3-x_2<....<x_n-x_{n-1}$. Prove that the maximum of the function $y=|P(x)|$ where $x$ is on the interval $[ x_1, x_n ]$, is in the interval $[x_n-1, x_n]$.
2022 Iran MO (3rd Round), 1
We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$.
For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.
1994 All-Russian Olympiad, 1
Let be given three quadratic polynomials:
$P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$.
Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.
2005 Tournament of Towns, 1
On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis.
[i](4 points)[/i]
1976 IMO Longlists, 43
Prove that if for a polynomial $P(x, y)$, we have
\[P(x - 1, y - 2x + 1) = P(x, y),\]
then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$
2010 Romanian Master of 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]
2013 Cuba MO, 5
Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.
2024 Thailand TST, 1
Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.
2012 India IMO Training Camp, 2
Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that
\[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]
1999 Czech and Slovak Match, 4
Find all positive integers $k$ for which the following assertion holds:
If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$, then \[F(0)= F(1) = \cdots =F(k+1).\]
1988 Bulgaria National Olympiad, Problem 6
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
2007 Grigore Moisil Intercounty, 2
Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.
2012 Ukraine Team Selection Test, 11
Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{ a_1, a_2, ..., a_k\}$ of positive integers we denote $S (A) = P (a_1) + P (a_2) + ... + P (a_k )$. The natural numbers $m, n$ are such that $m ^{d+ 1} | n$. Prove that the set $\{1, 2, ..., n\}$ can be subdivided into $m$ disjoint subsets $A_1, A_2, ..., A_m$ with the same number of elements such that $S (A_1) = S(A_2) = ... = S (A_m )$.
1975 IMO, 6
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2021 Harvard-MIT Mathematics Tournament., 3
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$
2013 Hitotsubashi University Entrance Examination, 5
Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$.
(1) Find the probability such that $s_n$ is divisible by 4.
(2) Find the probability such that $s_n$ is divisible by 6.
(3) Find the probability such that $s_n$ is divisible by 7.
Last Edited
Thanks, jmerry & JBL
1969 IMO Shortlist, 66
$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$
$(b)$ Formulate and prove the analogous result for polynomials of third degree.
2002 China Team Selection Test, 3
The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$.
Prove that $ \alpha\beta$ is not a perfect square.
2006 Iran MO (3rd Round), 6
a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that \[P(x)\mid Q(R(x)).\] b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that \[P(x)\mid Q(R(x)).\]
2010 AIME Problems, 6
Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.
2019 Belarusian National Olympiad, 10.5
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$.
[i](I. Voronovich)[/i]