Found problems: 3597
2014 Postal Coaching, 3
Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.
2010 AMC 12/AHSME, 24
The set of real numbers $ x$ for which
\[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\]
is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals?
$ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$
2016 CCA Math Bonanza, I9
Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$).
[i]2016 CCA Math Bonanza Individual #9[/i]
2013 Bosnia Herzegovina Team Selection Test, 2
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.
Prove that all terms of this sequence are perfect squares.
2014 PUMaC Individual Finals A, 2
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
2010 Vietnam Team Selection Test, 3
Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$
2011 Israel National Olympiad, 2
Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.
2025 China Team Selection Test, 1
Show that the polynomial over variables $x,y,z$
\[
x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y)
\]
can't be written as a finite sum of squares of real polynomials over $x,y,z$.
2016 IMO Shortlist, N1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
2019 IMAR Test, 2
Let $ f_1,f_2,f_3,f_4 $ be four polynomials with real coefficients, having the property that
$$ f_1 (1) =f_2 (0), \quad f_2 (1) =f_3 (0),\quad f_3 (1) =f_4 (0),\quad f_4 (1) =f_1 (0) . $$
Prove that there exists a polynomial $ f\in\mathbb{R}[X,Y] $ such that
$$ f(X,0)=f_1(X),\quad f(1,Y) =f_2(Y) ,\quad f(1-X,1) =f_3(X),\quad f(0,1-Y)=f_4(Y) . $$
2010 Mediterranean Mathematics Olympiad, 4
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$
2011 Romanian Masters In Mathematics, 2
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties:
(1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$;
(2) the degree of $f$ is less than $n$.
[i](Hungary) Géza Kós[/i]
1972 Vietnam National Olympiad, 1
Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$).
i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$.
Thus we can write $y$ as a function of $x, y = T_n(x)$.
Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$.
From this it follows that $T_n(x)$ is a polynomial of degree $n$.
ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.
2009 IMS, 4
In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node.
Find all $ \lambda$ for which this is possible.
2016 ISI Entrance Examination, 2
Consider the polynomial $ax^3+bx^2+cx+d$ where $a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even.Prove that not all of its roots are rational..
2005 VJIMC, Problem 3
Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that
$$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.
2007 Romania Team Selection Test, 3
The problem is about real polynomial functions, denoted by $f$, of degree $\deg f$.
a) Prove that a polynomial function $f$ can`t be wrriten as sum of at most $\deg f$ periodic functions.
b) Show that if a polynomial function of degree $1$ is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are "wild").
c) Show that every polynomial function of degree $1$ can be written as sum of two periodic functions.
d) Show that every polynomial function $f$ can be written as sum of $\deg f+1$ periodic functions.
e) Give an example of a function that can`t be written as a finite sum of periodic functions.
[i]Dan Schwarz[/i]
2017 China Team Selection Test, 4
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.
2008 Indonesia TST, 1
A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$
2014 AIME Problems, 9
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
2005 USAMTS Problems, 4
Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)
1993 Poland - First Round, 5
Prove that if the polynomial $x^3 + ax^2 + bx + c$ has three distinct real roots, so does the polynomial
$x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)$.
2015 All-Russian Olympiad, 4
You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?
2011 Canadian Open Math Challenge, 12
Let $f(x)=x^2-ax+b$, where $a$ and $b$ are positive integers.
(a) Suppose that $a=2$ and $b=2$. Determine the set of real roots of $f(x)-x$, and the set of real roots of $f(f(x))-x$.
(b) Determine the number of positive integers $(a,b)$ with $1\le a,b\le 2011$ for which every root of $f(f(x))-x$ is an integer.
1988 China Team Selection Test, 4
There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.