Found problems: 4275
Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that:
\[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\]
$ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists.
[i]Proposed by Carl Schildkraut[/i]
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?
Let \[f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}\] be a polynomial with coefficients satisfying the conditions: \[0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.\] Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial
\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}
Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Consider the polynomial $$P\left(t\right)=t^3-29t^2+212t-399.$$ Find the product of all positive integers $n$ such that $P\left(n\right)$ is the sum of the digits of $n$.
Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$.
a) Compute the number of excellent polynomials with degree at most 3.
b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$.
c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.
For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.
Let $n$ be a natural number and the polynomial, $P(x)=x^n+n$.
$(a)$ Is it possible that for some odd number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$.
$(b)$ Is it possible that for some even number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$.
Reason your answers.
Find the largest $a$ for which there exists a polynomial
$$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$
with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$
Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)
Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.
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.)
The lengths of the altitudes of $\triangle{ABC}$ are the roots of the polynomial $x^3-34x^2+360x-1200.$ Find the area of $\triangle{ABC}.$
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?