Found problems: 4275
Let $a_0,a_1,\dots,a_n \in \mathbb N$. Prove that there exist positive integers $b_0,b_1,\dots,b_n$ such that for $0 \leq i \leq n : a_i \leq b_i \leq 2a_i$ and polynomial \[P(x) = b_0 + b_1 x + \dots + b_n x^n\] is irreducible over $\mathbb Q[x]$.
(10 points)
Let $P(x) = x^{2022} + x^{1011} + 1$. Which of the following polynomials divides $P(x)$?
$\textbf{(A)}~x^2 - x + 1\qquad\textbf{(B)}~x^2 + x + 1\qquad\textbf{(C)}~x^4 + 1\qquad\textbf{(D)}~x^6 - x^3 + 1\qquad\textbf{(E)}~x^6 + x^3 + 1$
Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$
Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that:
$a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$;
$b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.
Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$?
Post no:[size=300]$100$[/size]
For each non-negative $a$, consider the equation $$x^3 + ax - a^3 - 29 = 0.$$ Let $x_o$ be the positive root of this equation. Prove that for all $a > 0$ such a root exists. What is the smallest value of $x_o$?
A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions.
$$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$
Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$.
Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$
$|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.
Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$.
[i]Proposed by Mohammad Ahmadi[/i]
Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$
A polynomial $P(x)$ of degree $3$ has three distinct real roots.
Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.
Let $p$ be an[color=#FF0000] odd [/color]prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that
\[f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).\]
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
[i]2016 CCA Math Bonanza Individual #13[/i]
Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.
Let $S$ denote the sum of the 2011th powers of the roots of the polynomial $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$. How many ones are in the binary expansion of $S$?
[i]Author: Alex Zhu[/i]
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?
Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.
Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .
Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.
Consider the polynomials
\[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\]
\[g(p) = p^3 + 2p + m.\]
Find all integral values of $m$ for which $f$ is divisible by $g$.
Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[
\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i]
[i]25 problems for 30 minutes[/i]
[b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game?
[b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$
What is the value of $x$ in Sammy's equation?
[b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars?
[b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line?
[b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$?
[b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$
[b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw?
[b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT?
[b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many out ts can he wear?
[b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number?
[b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip?
[b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers?
[b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at?
[b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches?
[b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color?
[b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$?
[b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$?
[b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$?
[b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral?
[b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more a ordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$?
[b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$.
[b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$.
[b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$?
[b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees?
[b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.