Found problems: 4275
Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$.
[i]Alternative formulation[/i]:
Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial
\[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}.
\]
Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$
[i]Proposed by Federico Ardila, Colombia[/i]
If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals:
$ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $
Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$
$$x^{2p} - 2px^m - p^2x^n - 1$$
A sequence of polynomials is defined by the recursion $P_1(x) = x+1$ and$$P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2}$$for all $n \geq 2$. Find the remainder when $P_{2022}(1)$ is divided by $1000$.
[i]Proposed by [b]treemath[/b][/i]
For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots?
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?
For any positive integer $n$, we define
$$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$
Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)
[b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.
[b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)
Find the sum of the non-repeated roots of the polynomial $P(x) = x^6-5x^5-4x^4-5x^3+8x^2+7x+7$.
Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system
\[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial
$f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients
and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?
Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying
(i) $A$ and $B$ are disjoint;
(ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$.
Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)
Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty.
Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]
Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative.
Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $
[i]Cătălin Țigăeru[/i]
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]