Found problems: 4275
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$.
Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.
Find all positive integers $k$ for which the following assertion holds:
If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$, then \[F(0)= F(1) = \cdots =F(k+1).\]
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.)
[i]David Yang.[/i]
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$. Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$, let $p(x)$ the product of the smallest and largest root of $f(x)-s$. Determine the smallest possible value that $p(s)$ for $s\in S$.
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition:
$$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$
for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$
Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.
Consider the set $F$ of all polynomials whose coefficients are in the set of $\{0,1\}$. Let $q(x) = x^3 + x +1$. The number of polynomials $p(x)$ in $F$ of degree $14$ such that the product $p(x)q(x)$ is also in $F$ is:
A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero?
$ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$
Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials.
[i]Marius Cavachi[/i]
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$
Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation:
\[Q(a+b) = \frac{P(a) - P(b)}{a - b}\]
$\forall a, b \in \mathbb{Z}^+$ and $a>b$
Let $t$ be a positive real number such that $t^4 + t^{-4} = 2023$. Determine the value of $t^3 + t^{-3}$ in the form of $a\sqrt b$, where $a$ and $b$ are positive integers.
The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$. Compute the greatest possible sum of the other two roots.
[i]2015 CCA Math Bonanza Lightning Round #2.4[/i]
Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$.
[i]Proposed by Sardor Gafforov, Uzbekistan[/i]
Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).
a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
[i]Victor Wang.[/i]
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
Find all values of $a$ such that the two polynomials \[x^2+ax-1\qquad\text{and}\qquad x^2-x+a\] share at least 1 root.
[i]2018 CCA Math Bonanza Individual Round #7[/i]
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) $(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).$