Found problems: 3597
2014 Harvard-MIT Mathematics Tournament, 12
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
1987 Austrian-Polish Competition, 2
Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.
2013 Stanford Mathematics Tournament, 7
Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.
2009 Croatia Team Selection Test, 1
Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.
MathLinks Contest 1st, 2
Let $f$ be a polynomial with real coefficients such that for each positive integer n the equation $f(x) = n$ has at least one rational solution. Find $f$.
2017 Korea Winter Program Practice Test, 3
Do there exist polynomials $f(x)$, $g(x)$ with real coefficients and a positive integer $k$ satisfying the following condition? (Here, the equation $x^2 = 0$ is considered to have $1$ distinct real roots. The equation $0 = 0$ has infinitely many distinct real roots.)
For any real numbers $a, b$ with $(a,b) \neq (0,0)$, the number of distinct real roots of $a f(x) + b g(x) = 0$ is $k$.
2017 Australian MO, 1
Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:
(a) $P(2017)=2016$ and
(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.
1958 AMC 12/AHSME, 40
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals:
$ \textbf{(A)}\ \frac{13}{27}\qquad
\textbf{(B)}\ 33\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ \minus{}17$
1980 IMO, 2
Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies
\[x \in \mathbb R \iff p(x) \in \mathbb R.\]
Show that $n=1$
2005 All-Russian Olympiad Regional Round, 11.5
Prove that for any polynomial $P$ with integer coefficients and any natural number $k$ there exists a natural number $n$ such that $P(1) + P(2) + ...+ P(n)$ is divisible by $k$.
2025 All-Russian Olympiad, 9.5
Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2018 Bosnia and Herzegovina Team Selection Test, 2
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
1991 Flanders Math Olympiad, 2
(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$.
(b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.
1991 IberoAmerican, 5
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
2006 Harvard-MIT Mathematics Tournament, 9
Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]
2024 IFYM, Sozopol, 1
Does there exist a polynomial \( P(x,y) \) in two variables with real coefficients, such that the following two conditions hold:
1) \( P(x,y) = P(x, x-y) = P(y-x, y) \) for any real numbers \( x \) and \( y \);
2) There does not exist a polynomial \( Q(z) \) in one variable with real coefficients such that \( P(x,y) = Q(x^2 - xy + y^2) \) for any real numbers \( x \) and \( y \)?
1996 IMO Shortlist, 4
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$.
(b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.
1981 IMO, 3
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
1963 Dutch Mathematical Olympiad, 4
One considers for $n > 2$ the polynomial: $$(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n$$
Show that the degree of this polynomial is $2n - 2$.
The polynomial is written in the form $$a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}$$
Prove that $a_2+a_3+...+a_{2n-2}=0$
1985 Traian Lălescu, 1.1
$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations
$$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$
Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.
1986 AIME Problems, 5
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
2004 Estonia Team Selection Test, 1
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.
2010 Contests, 2
Positive rational number $a$ and $b$ satisfy the equality
\[a^3 + 4a^2b = 4a^2 + b^4.\]
Prove that the number $\sqrt{a}-1$ is a square of a rational number.
2001 China Team Selection Test, 1
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer