Found problems: 3597
2010 N.N. Mihăileanu Individual, 1
Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.
2014 Belarus Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
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$.
1997 Federal Competition For Advanced Students, Part 2, 3
For every natural number $n$, find all polynomials $x^2+ax+b$, where $a^2 \geq 4b$, that divide $x^{2n} + ax^n + b$.
2012 Kyoto University Entry Examination, 4
(1) Prove that $\sqrt[3]{2}$ is irrational.
(2) Let $P(x)$ be a polynomoal with rational coefficients such that $P(\sqrt[3]{2})=0$. Prove that $P(x)$ is divisible by $x^3-2$.
35 points
2024 Brazil Cono Sur TST, 4
In the cartesian plane, consider the subset of all the points with both integer coordinates. Prove that it is possible to choose a finite non-empty subset $S$ of these points in such a way that any line $l$ that forms an angle of $90^{\circ},0^{\circ},135^{\circ}$ or $45^{\circ}$ with the positive horizontal semi-axis intersects $S$ at exactly $2024$ points or at no points.
1982 AMC 12/AHSME, 12
Let $f(x) = ax^7+bx^3+cx-5$, where $a,b$ and $c$ are constants. If $f(-7) = 7$, the $f(7)$ equals
$\textbf {(A) } -17 \qquad \textbf {(B) } -7 \qquad \textbf {(C) } 14 \qquad \textbf {(D) } 21\qquad \textbf {(E) } \text{not uniquely determined}$
2010 Canada National Olympiad, 5
Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.
2014 Costa Rica - Final Round, 6
The sequences $a_n$, $b_n$ and $c_n$ are defined recursively in the following way:
$a_0 = 1/6$, $b_0 = 1/2$, $c_0 = 1/3,$
$$a_{n+1}= \frac{(a_n + b_n)(a_n + c_n)}{(a_n - b_n)(a_n - c_n)},\,\,
b_{n+1}= \frac{(b_n + a_n)(b_n + c_n)}{(b_n - a_n)(b_n - c_n)},\,\,
c_{n+1}= \frac{(c_n + a_n)(c_n + b_n)}{(c_n - a_n)(c_n - b_n)}$$
For each natural number $N$, the following polynomials are defined:
$A_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$
$B_n(x) =b_o+a_1 x+ ...+ a_{2N}x^{2N}$
$C_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$
Assume the sequences are well defined.
Show that there is no real $c$ such that $A_N(c) = B_N(c) = C_N(c) = 0$.
1967 Spain Mathematical Olympiad, 8
To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]
2015 IMC, 10
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of
degree $n$ with integer coefficients. Prove that
$$
\max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}.
$$
Proposed by Géza Kós, Eötvös University, Budapest
1985 ITAMO, 13
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
2013 Brazil National Olympiad, 3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
2007 Balkan MO Shortlist, A7
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number.
1995 AIME Problems, 2
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]
2013 IFYM, Sozopol, 5
Find all polynomilals $P$ with real coefficients, such that
$(x+1)P(x-1)+(x-1)P(x+1)=2xP(x)$
2016 Azerbaijan Team Selection Test, 2
A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values.
Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?
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]
2009 Kazakhstan National Olympiad, 6
Let $P(x)$ be polynomial with integer coefficients.
Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
2001 India IMO Training Camp, 3
Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that
\[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]
2011 ELMO Shortlist, 8
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
2010 Turkey MO (2nd round), 2
For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i]
Find the number of [i]good polynomials.[/i]
2009 Ukraine National Mathematical Olympiad, 4
Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds
\[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\]
[b]Note.[/b] $P^2(k)=\left( P(k) \right)^2.$
1992 IMO Longlists, 82
Let $f(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m$ and $g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n$ be two polynomials with real coefficients such that for each real number $x, f(x)$ is the square of an integer if and only if so is $g(x)$. Prove that if $n +m > 0$, then there exists a polynomial $h(x)$ with real coefficients such that $f(x) \cdot g(x) = (h(x))^2.$
[hide="Remark."]Remark. The original problem stated $g(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n$, but I think the right form of the problem is what I wrote.[/hide]
1978 IMO Longlists, 14
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.