Found problems: 4275
The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that
\[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \]
for every real number $x$ and every positive integer $n$.
Let $f (x) = x^2 + px + q$, where $p, q$ are integers.
Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
Jenn draws a scalene triangle, and measures the heights from each of the vertices to its opposite side. She discovers that the three height lengths are all roots of the polynomial $x^3 - 3.9 x^2 + 4.4 x - 1.2.$ What is the length of the inradius of the triangle?
$\textbf{(A) }\frac{3}{13}\qquad\textbf{(B) }\frac{3}{11}\qquad\textbf{(C) }\frac{2}{7}\qquad\textbf{(D) }\frac{8}{15}\qquad\textbf{(E) }\frac{9}{14}$
Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.
Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality
\[a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.\]
Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.
Find the number of positive integers $n$ less than $2017$ such that
\[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \]
is an integer.
The [i]minimal polynomial[/i] of a complex number $r$ is the unique polynomial with rational coefficients of minimal degree with leading coefficient $1$ that has $r$ as a root. If $f$ is the minimal polynomial of $\cos\frac\pi7$, what is $f(-1)$?
What is the smallest value of \( k \) such that for any polynomial \( f(x) \) of degree $100$ with real coefficients, there exists a polynomial \( g(x) \) of degree at most \( k \) with real coefficients such that the graphs of \( y = f(x) \) and \( y = g(x) \) intersect at exactly $100$ points? \\
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
Suppose the roots of
$$x^4 - 3x^2 + 6x - 12 = 1$$
are $\alpha$, $\beta$, $\gamma$ , and $\delta$. What is the value of
$$\frac{\alpha+ \beta+ \gamma }{\delta^2}+\frac{\alpha+ \delta+ \gamma}{\beta^2}+\frac{\alpha+ \beta+ \delta}{\gamma^2}+\frac{\delta+ \beta+ \gamma }{\alpha^2}?$$
How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?
Let $a,b,c,m$ be integers, where $m>1$. Prove that if
$$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?
Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$
Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that
(i) $P(0) = 1$.
(ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$.
Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.
Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].
[i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.
Nikhil constructs a list of all polynomial pairs $(a(x), b(x))$ with real coefficients such that $a(x)$ has higher degree than $b(x)$ and $a(x)^2 + b(x)^2 = x^{10} + 1$. Danielle takes Nikhil’s list and adds all polynomial pairs that satisfy the same conditions but have complex coefficients. If Nikhil’s original list had $N$ pairs and Danielle added $D$ pairs, compute $D - N$.
Let $p(x)$ be a polynomial of degree $2022$ such that:
$$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$
Find $p(2023)$.
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Let $Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]$ be a polynomial with integer coefficients. For an odd prime number $p$ we define the polynomial $Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.$
Assume that there exist infinitely primes $p$ such that
$$\frac{Q_{p}(x)-Q(x)}{p}$$
is an integer for all $x \in \mathbb{Z}$. Determine the largest possible value of $Q(2023)$ over all such polynomials $Q$.
[i]Authored by Nikola Velov[/i]
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$.
[hide] Stronger one.
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]
Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.