Found problems: 4275
Suppose $P$ is a quintic polynomial with real coefficients with $P(0)=2$ and $P(1)=3$ such that $|z|=1$ whenever $z$ is a complex number satisfying $P(z) = 0$. What is the smallest possible value of $P(2)$ over all such polynomials $P$?
Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct?
[i]Proposed by Ivan Chan Kai Chin[/i]
Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$:
$P(x^2-y^2) = P(x+y)P(x-y)$.
Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\]
[i]J. Szabados[/i]
Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$.
(1) Find the probability such that $s_n$ is divisible by 4.
(2) Find the probability such that $s_n$ is divisible by 6.
(3) Find the probability such that $s_n$ is divisible by 7.
Last Edited
Thanks, jmerry & JBL
Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
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 $x$.
Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.
If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.
The third-degree polynomial $P(x)=x^3+2x^2-3x-5$ has the three roots $a$, $b$ and $c$. State a third degree polynomial with roots $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$.
Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$
Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point.
[i]Vasile Pop[/i]
[u]Set 8[/u]
[b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends.
[b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid.
[b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings
such that any integer $ n \ge 2$ has that
(a) If $n$ is prime, then at least one digit of $n$ is not blue.
(b) If $n$ is composite, then at least one digit of $n$ is not green.
PS. You should use hide for answers.
Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.
For each non-constant integer polynomial $P(x)$, let’s define $$M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.$$
1. Find the minimum value of $M_{P(x)}$ when deg $P(x) = 1$.
2. Suppose that $P(x) \in Z[x]$ when deg $P(x) = n$ and $2 \le n \le 2022$. Prove that $M_{P(x)} \ge 1011$.
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
Find all polynomials \(P\) with real coefficients satisfying
$$P(\frac{1}{1+x})=\frac{1}{1+P(x)}$$
for all real numbers \(x\neq -1\)
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .
A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove :
a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.
b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds:
\[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\]
[i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)
Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$
Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of
\begin{tabular}{|ccc|}
$b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\
$n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\
$n(b - a)$ & $n^2(a - b)$ & $a + n^3b$
\end{tabular}
A polynomial $P$ in $n$ variables and real coefficients is called [i]magical[/i] if $P(\mathbb{N}^n)\subset \mathbb{N}$, and moreover the map $P: \mathbb{N}^n \to \mathbb{N}$ is a bijection. Prove that for all positive integers $n$, there are at least
\[n!\cdot (C(n)-C(n-1))\]
magical polynomials, where $C(n)$ is the $n$-th Catalan number.
Here $\mathbb{N}=\{0,1,2,\dots\}$.