Found problems: 4275
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$.
Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm.
(translated by Miklós Maróti)
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]
In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.
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.
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of
degree $n$ with integer coefficients. Prove that
$$
\max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}.
$$
Proposed by Géza Kós, Eötvös University, Budapest
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.
Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots.
b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.
Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.
Suppose that for polynomials \( P, Q, R \) with positive integer coefficients, the following two conditions hold:
\(\bullet\) The constant terms of \( P, Q, R \) are equal.
\(\bullet\) For all real numbers \( x \), the following relations hold:
\[
P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))).
\]
Prove that for every real number \( x \), \( P(x) = Q(x) = R(x) \).
Proposed by Soroush Behroozifar & Ali Nazarboland