Found problems: 3597
1983 AMC 12/AHSME, 18
Let $f$ be a polynomial function such that, for all real $x$,
\[f(x^2 + 1) = x^4 + 5x^2 + 3.\]
For all real $x$, $f(x^2-1)$ is
$ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $
1994 AIME Problems, 3
The function $f$ has the property that, for each real number $x,$ \[ f(x)+f(x-1) = x^2. \] If $f(19)=94,$ what is the remainder when $f(94)$ is divided by 1000?
1999 Vietnam Team Selection Test, 2
Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$.
[b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar.
[b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.
1997 China Team Selection Test, 1
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:
[b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}
x^2 + a_{2n}, a_0 > 0$;
[b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n\\
n\end{array} \right) a_0 a_{2n}$;
[b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.
2025 Korea Winter Program Practice Test, P8
Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$
$$x^{2p} - 2px^m - p^2x^n - 1$$
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
1970 IMO Longlists, 48
Let a polynomial $p(x)$ with integer coefficients take the value $5$ for five different integer values of $x.$ Prove that $p(x)$ does not take the value $8$ for any integer $x.$
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
2005 Czech-Polish-Slovak Match, 3
Find all integers $n \ge 3$ for which the polynomial
\[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\]
can be written as a product of two non-constant polynomials with integer coefficients.
2021 JHMT HS, 8
For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2005 District Olympiad, 1
a) Prove that if $x,y>0$ then
\[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \]
b) Prove that if $a,b,c$ are positive real numbers, then
\[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]
1965 Swedish Mathematical Competition, 5
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
2020 Spain Mathematical Olympiad, 1
A polynomial $p(x)$ with real coefficients is said to be [i]almeriense[/i] if it is of the form:
$$
p(x) = x^3+ax^2+bx+a
$$
And its three roots are positive real numbers in arithmetic progression. Find all [i]almeriense[/i] polynomials such that $p\left(\frac{7}{4}\right) = 0$
2014 China Girls Math Olympiad, 4
For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold:
(1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$
(2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$
(3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$
Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]
2008 Harvard-MIT Mathematics Tournament, 10
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
1995 AIME Problems, 5
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$
2005 SNSB Admission, 3
Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $
Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $
1992 Baltic Way, 10
Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
(i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$,
(ii) $ p(x)\ge0$ for all $ x$,
(iii) $ p(0)\equal{}1$
(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$.
2022 Kazakhstan National Olympiad, 4
Let $P(x)$ be a polynomial with positive integer coefficients such that $deg(P)=699$. Prove that if $P(1) \le 2022$, then there exist some consecutive coefficients such that their sum is $22$, $55$, or $77$.
2024/2025 TOURNAMENT OF TOWNS, P5
Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.
2014 Iran MO (3rd Round), 6
$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\]
(a) Prove that there are a finite number of natural numbers in range of $f$.
(b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions.
(c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers.
Time allowed for this problem was 105 minutes.
2023 Iberoamerican, 6
Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].
1968 Putnam, A6
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
2016 Taiwan TST Round 3, 5
Let $f(x)$ be the polynomial with integer coefficients ($f(x)$ is not constant) such that
\[(x^3+4x^2+4x+3)f(x)=(x^3-2x^2+2x-1)f(x+1)\]
Prove that for each positive integer $n\geq8$, $f(n)$ has at least five distinct prime divisors.
1997 Korea National Olympiad, 6
Find all polynomial $P(x,y)$ for any reals $x,y$ such that
(i) $x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})$
(ii) $(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).$