Found problems: 3597
2009 Princeton University Math Competition, 2
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
2012 IMO Shortlist, N5
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.
2023 BMT, 24
Define the sequence $s_0$, $s_1$, $s_2$,$ . . .$ by $s_0 = 0$ and $s_n = 3s_{n-1}+2$ for $n \ge 1$. The monic polynomial $f(x)$ defined as $$f(x) =\frac{1}{s_{2023}} \sum^{32}_{k=0} s_{2023+k}x^{32-k}$$ can be factored uniquely (up to permutation) as the product of $16$ monic quadratic polynomials $p_1$, $p_2$, $....$, $p_{16}$ with real coefficients, where $p_i(x) = x^2 + a_ix + b_i$ for $1\le i \le 16$. Compute the integer $N$ that minimizes
$$\left|N - \sum^{16}_{k=1} (a_k + b_k)\right|.$$
2005 Tournament of Towns, 4
For any function $f(x)$, define $f^1(x) = f(x)$ and $f^n (x) = f(f^{n-1}(x))$ for any integer $n \ge 2$. Does there exist a quadratic polynomial $f(x)$ such that the equation $f^n(x) = 0$ has exactly $2^n$ distinct real roots for every positive integer $n$?
[i](6 points)[/i]
1995 Turkey Team Selection Test, 1
Given real numbers $b \geq a>0$, find all solutions of the system
\begin{align*}
&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.
\end{align*}
2010 Princeton University Math Competition, 7
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
1995 IMO Shortlist, 3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
2010 All-Russian Olympiad, 4
Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true?
For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$.
(The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)
2011 Mediterranean Mathematics Olympiad, 1
A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
1991 China Team Selection Test, 1
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
2010 China Team Selection Test, 2
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
2017 Thailand TSTST, 5
Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.
2015 BMT Spring, 2
Let $g(x)=1+2x+3x^2+4x^3+\ldots$. Find the coefficient of $x^{2015}$ of $f(x)=\frac{g(x)}{1-x}$.
2023 Romanian Master of Mathematics, 5
Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that
$$\displaystyle P(x)=R(T(x))$$
2018 Singapore MO Open, 5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
2016 Korea Junior Math Olympiad, 5
$n \in \mathbb {N^+}$
Prove that the following equation can be expressed as a polynomial about $n$.
$$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$
1981 AMC 12/AHSME, 30
If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are
\[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2}
\]is
$ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$
$ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$
2005 Greece Team Selection Test, 1
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
2010 Bundeswettbewerb Mathematik, 4
In the following, let $N_0$ denotes the set of non-negative integers.
Find all polynomials $P(x)$ that fulfill the following two properties:
(1) All coefficients of $P(x)$ are from $N_0$.
(2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2020 GQMO, 1
Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities
\[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\]
hold for all real numbers $X$.
[i]Morteza Saghafian, Iran[/i]
2018 Hong Kong TST, 1
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?
1964 German National Olympiad, 6
Which of the following four statements are true and which are false?
a) If a polygon inscribed in a circle is equilateral, then it is also equiangular.
b) If a polygon inscribed in a circle is equiangular, then it is also equilateral.
c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular.
d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.
2008 Bulgarian Autumn Math Competition, Problem 9.3
Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.
2002 Kazakhstan National Olympiad, 6
Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.