Found problems: 4275
Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.
Find all polynomials $f$ that satisfy the equation
\[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\]
for infinitely many real values of $x$.
Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.
Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$.
[i](Paolo Leonetti)[/i]
Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$.
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.
Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form
\[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$.
[i]Valentin Vornicu[/i]
Consider the equation $n^2+(n+1)^4=5(n+2)^3$
(a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation.
(b) Does the equation have a solution in positive integers?
Let be two (not necessarily distinct) roots of two rational polynoms (respectively) that are irreducible over the rationals. Prove that these polynoms have the same degree if the sum of those two roots is rational.
[i]Bogdan Enescu[/i]
$P(x)$ is a nonconstant polynomial with real coefficients and its all roots are real numbers. If there exist a $Q(x)$ polynomial with real coefficients that holds the equality for all $x$ real numbers
$(P(x))^{2}=P(Q(x))$,
then prove that all the roots of $P(x)$ are same.
Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?
Consider the infinite polynomial $G(x) = F_1x+F_2x^2 +F_3x^3 +...$ defined for $0 < x <\frac{\sqrt5 -1}{2}$ where Fk is the $k$th term of the Fibonacci sequence defined to be $F_k = F_{k-1} + F_{k-2}$ with $F_1 = 1$, $F_2 = 1$. Determine the value a such that $G(a) = 2$.
Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.
Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i]
For what is the smallest natural number $n$ there is a polynomial $P(x)$ with integer coefficients, having $m$ different integer roots, and at the same time the equation $P(x) = n$ has at least one integer solution if:
a) $m = 5$,
b) $ m = 6$?
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.
What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
[u]Round 1[/u]
[b]p1.[/b] Today, the date $4/9/16$ has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property?
[b]p2.[/b] What is the greatest integer less than $100$ whose digit sumis equal to its greatest prime factor?
[b]p3.[/b] In chess, a bishop can only move diagonally any number of squares. Find the number of possible squares a bishop starting in a corner of a $20\times 16$ chessboard can visit in finitely many moves, including the square it stars on.
[u]Round 2 [/u]
[b]p4.[/b] What is the fifth smallest positive integer with at least $5$ distinct prime divisors?
[b]p5.[/b] Let $\tau (n)$ be the number of divisors of a positive integer $n$, including $1$ and $n$. Howmany positive integers $n \le 1000$ are there such that $\tau (n) > 2$ and $\tau (\tau (n)) = 2$?
[b]p6.[/b] How many distinct quadratic polynomials $P(x)$ with leading coefficient $1$ exist whose roots are positive integers and whose coefficients sum to $2016$?
[u]Round 3[/u]
[b]p7.[/b] Find the largest prime factor of $112221$.
[b]p8.[/b] Find all ordered pairs of positive integers $(a,b)$ such that $\frac{a^2b^2+1}{ab-1}$ is an integer.
[b]p9.[/b] Suppose $f : Z \to Z$ is a function such that $f (2x)= f (1-x)+ f (1-x)$ for all integers $x$. Find the value of $f (2) f (0) +f (1) f (6)$.
[u]Round 4[/u]
[b]p10.[/b] For any six points in the plane, what is the maximum number of isosceles triangles that have three of the points as vertices?
[b]p11.[/b] Find the sum of all positive integers $n$ such that $\sqrt{n+ \sqrt{n -25}}$ is also a positive integer.
[b]p12.[/b] Distinct positive real numbers are written at the vertices of a regular $2016$-gon. On each diagonal and edge of the $2016$-gon, the sum of the numbers at its endpoints is written. Find the minimum number of distinct numbers that are now written, including the ones at the vertices.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.
Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.