Found problems: 4275
Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$
[i]G. Tusnady[/i]
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 $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$.
a)Prove that $(BA)^3=(BA)^2$.
b)Find $BA$.
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.
$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$
Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.
We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that:
$i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$
$ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$
$iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$
Find all [i]interesting [/i]$n$.
[i]Proposed by Mojtaba Zare Bidaki[/i]
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i]
Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$.
[i]Proposed by Tobi Moektijono, Indonesia[/i]
Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
Let $n, \in \mathbb {N^*} , n\ge 3$
a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $
b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.
$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.
Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers.
[i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.
[b]p1.[/b] A large square is cut into four smaller, congruent squares. If each of the smaller squares has perimeter $4$, what was the perimeter of the original square?
[b]p2.[/b] Pie loves to bake apples so much that he spends $24$ hours a day baking them. If Pie bakes a dozen apples in one day, how many minutes does it take Pie to bake one apple, on average?
[b]p3.[/b] Bames Jond is sent to spy on James Pond. One day, Bames sees James type in his $4$-digit phone password. Bames remembers that James used the digits $0$, $5$, and $9$, and no other digits, but he does not remember the order. How many possible phone passwords satisfy this condition?
[b]p4.[/b] What do you get if you square the answer to this question, add $256$ to it, and then divide by $32$?
[b]p5.[/b] Chloe the Horse and Flower the Chicken are best friends. When Chloe gets sad for any reason, she calls Flower, so Chloe must remember Flower's $3$ digit phone number, which can consist of any digits $0-5$. Given that the phone number's digits are unique and add to $5$, the number does not start with $0$, and the $3$ digit number is prime, what is the sum of all possible phone numbers?
[b]p6.[/b] Anuj has a circular pizza with diameter $A$ inches, which is cut into $B$ congruent slices, where $A$,$B$ are positive integers. If one of Anuj's pizza slices has a perimeter of $3\pi + 30$ inches, find $A + B$.
[b]p7.[/b] Bob really likes to study math. Unfortunately, he gets easily distracted by messages sent by friends. At the beginning of every minute, there is an $\frac{6}{10}$ chance that he will get a message from a friend. If Bob does get a message from a friend, there is a $\frac{9}{10}$ chance that he will look at the message, causing him to waste $30$ seconds before resuming his studying. If Bob doesn't get a message from a friend, there is a $\frac{3}{10}$ chance Bob will still check his messages hoping for a message from his friends, wasting $10$ seconds before he resumes his studying. What is the expected number of minutes in $100$ minutes for which Bob will be studying math?
[b]p8.[/b] Suppose there is a positive integer $n$ with $225$ distinct positive integer divisors. What is the minimum possible number of divisors of n that are perfect squares?
[b]p9.[/b] Let $a, b, c$ be positive integers. $a$ has $12$ divisors, $b$ has $8$ divisors, $c$ has $6$ divisors, and $lcm(a, b, c) = abc$. Let $d$ be the number of divisors of $a^2bc$. Find the sum of all possible values of $d$.
[b]p10.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 17$, $BC = 28$, $AC = 25$. Let the altitude from $A$ to $BC$ and the angle bisector of angle $B$ meet at $P$. Given the length of $BP$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a + b + c$.
[b]p11.[/b] Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-5x+3 = 0$. Let $S = a^4b+ab^4+a^4c+ac^4+b^4c+bc^4$. Find $|S|$.
[b]p12.[/b] Call a number palindromeish if changing a single digit of the number into a different digit results in a new six-digit palindrome. For example, the number $110012$ is a palindromeish number since you can change the last digit into a $1$, which results in the palindrome $110011$. Find the number of $6$ digit palindromeish numbers.
[b]p13.[/b] Let $P(x)$ be a polynomial of degree $3$ with real coecients and leading coecient $1$. Let the roots of $P(x)$ be $a$, $b$, $c$. Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ such that $gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
[b]p14.[/b] Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, $AC = 9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c) = gcd(b, d) = 1$ and $b$,$ d$ are not divisible by the square of any prime. Find $a + b + c + d$.
[b]p15.[/b] Many students in Mr. Noeth's BC Calculus class missed their first test, and to avoid taking a makeup, have decided to never leave their houses again. As a result, Mr. Noeth decides that he will have to visit their houses to deliver the makeup tests. Conveniently, the $17$ absent students in his class live in consecutive houses on the same street. Mr. Noeth chooses at least three of every four people in consecutive houses to take a makeup. How many ways can Mr. Noeth select students to take makeups?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$.
Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.
For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with
\[\omega(n)\ge\omega(P(n)).\]
Greece (Minos Margaritis - Iasonas Prodromidis)
Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$.
Prove that $\det{ (A^2+A+xI_2) } = x$.
Find all functions $f: \mathbb N \to \mathbb N$ for which
\[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\]
holds for all positive integers $n$.
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$:
\[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]