Found problems: 3597
Let $x_1\leq x_2\leq x_3\leq x_4$ be real numbers. Prove that there exist polynomials of degree two $P(x)$ and $Q(x)$ with real coefficients such that $x_1$, $x_2$, $x_3$ and $x_4$ are the roots of $P(Q(x))$ if and only if $x_1+x_4=x_2+x_3$.
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.
[i]Proposed by Masud Shafaie[/i]
Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions
$Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.
Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)
Define the sequence $s_0$, $s_1$, $s_2$,$ . . .$ by $s_0 = 0$ and $s_n = 3s_{n-1}+2$ for $n \ge 1$. The monic polynomial $f(x)$ defined as $$f(x) =\frac{1}{s_{2023}} \sum^{32}_{k=0} s_{2023+k}x^{32-k}$$ can be factored uniquely (up to permutation) as the product of $16$ monic quadratic polynomials $p_1$, $p_2$, $....$, $p_{16}$ with real coefficients, where $p_i(x) = x^2 + a_ix + b_i$ for $1\le i \le 16$. Compute the integer $N$ that minimizes
$$\left|N - \sum^{16}_{k=1} (a_k + b_k)\right|.$$
The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation
\[
x^4 + *x^3 + *x^2 + *x + * = 0
\]
with a positive real number. Who has a winning strategy if the goals of the players are:
a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots?
b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?
Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$?
[i]S. Tokarev[/i]
Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.
Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?
Let $ p(x)\equal{}a_0 \plus{}a_1 x\plus{}...\plus{}a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)\equal{}2$ and $ p(16)\equal{}8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)\equal{}4$.
Find all integers $n$ for which the polynomial $p(x) = x^5 -nx -n -2$ can be represented as a product of two non-constant polynomials with integer coefficients.
How many 7-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$?
For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?
Let $p(x)$ and $q(x)$ be two polynomials, both of which have their sum of coefficients equal to $s.$ Let $p,q$ satisfy $p(x)^3-q(x)^3=p(x^3)-q(x^3).$ Show that
(i) There exists an integer $a\geq1$ and a polynomial $r(x)$ with $r(1)\neq0$ such that
\[p(x)-q(x)=(x-1)^ar(x).\]
(ii) Show that $s^2=3^{a-1},$ where $a$ is described as above.
Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent:
$ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers.
$ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.
Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$,
\[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$.
[i]Calvin Deng.[/i]
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.
Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $
Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$. Prove that $0<a<1<b<3<c<4$
[hide="Solution"]
Let $abc=k$, then $a,b,c\ (a<b<c)$ are the roots of cubic equation $x^3-6x^2+9x-k=0\Longleftrightarrow x(x-3)^2=k$
that is to say, $a,b,c\ (a<b<c)$ are the $x$-coordinates of the interception of points between $y=x(x-3)^2$ and
$y=k$.
$y=x(x-3)^2$ have local maximuml value of $4$ at $x=1$ and local minimum value of $0$ at $x=3$.
Since the $x$-coordinate of the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at
local maximum point $(1,4)$ is a point $(4,4)$,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and
$y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are
$0<a<1,1<b<3,3<c<4
$,we are done.[/hide]
If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is:
$ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$
(a) Prove that $x^4+3x^3+6x^2+9x+12$ cannot be expressed as product of two polynomials of degree 2 with integers coefficients.
(b) $2n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.
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.