Found problems: 4275
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$
for any complex number $x.$
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]
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities:
$$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$
Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time
$-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.
Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.
[b]Randy:[/b] "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?''
[b]Rachel:[/b] "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.''
[b]Randy:[/b] "That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn't be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.''
[b]Rachel:[/b] "Interesting. Now figure out how old I am.''
[b]Randy:[/b] "Instead, I will guess your age and substitute it for $x$ in your quadratic equation $\dots$ darn, that gives me $-55$, and not $0$.''
[b]Rachel:[/b] "Oh, leave me alone!''
(1) Prove that Jimmy is two years old.
(2) Determine Rachel's age.
The polynomial $ P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $ y$-intercept of the graph of $ y \equal{} P(x)$ is 2, what is $ b$?
$ \textbf{(A)} \ \minus{} 11 \qquad \textbf{(B)} \ \minus{} 10 \qquad \textbf{(C)} \ \minus{} 9 \qquad \textbf{(D)} \ 1 \qquad \textbf{(E)} \ 5$
We say that a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar.
(a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \).
(b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.
[b]p1.[/b] How many two-digit primes have a units digit of $3$?
[b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once.
[b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna?
[b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$?
[b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$.
[b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$?
[b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer.
[b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$?
[b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
[b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ .
[b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more?
[b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$.
[b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
[b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function.
Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$.
[b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\]
where $n$ and $k$ are positive integers.
Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant?
(A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.)
[i]Nikolai Beluhov[/i]
Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.
Consider a polynomial $P(x) = x^4+x^3-3x^2+x+2$. Prove that at least one of the coefficients of $(P(x))^k$, ($k$ is any positive integer) is negative. (5)
A pair $(f,g)$ of degree $2$ real polynomials is called [i]foolish[/i] if $f(g(x)) = f(x) \cdot g(x)$ for all real $x.$ How many positive integers less than $2023$ can be a root of $g(x)$ for some foolish pair $(f,g)$?
Let $q_{0},q_{1},...$ be a sequence of integers such that
a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and
b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$.
Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.
Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b \mid f(a)+f(b)$.
Can every polynomial with integer coefficients be expressed as a sum of cubes
of polynomials with integer coefficients?
[hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]
Let $p$, $q$, and $r$ be the three roots of the polynomial $x^3 -2x^2 + 3x - 2023$. Suppose that the polynomial $x^3 + Bx^2 +Mx + T$ has roots $p + q$, $p + r$, and $q + r$ for real numbers $B$, $M$, and $T$. Compute $B -M + T$.
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
[b]p1.[/b] Suppose $a_1 \cdot 2 = a_2 \cdot 3 = a_3$ and $a_1 + a_2 + a_3 = 66$. What is $a_3$?
[b]p2.[/b] Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the $7$ cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is $20$ cm$^2$ . He fills the bottle with water up to the $5$ cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the $7$ cm mark. Find the total volume (in cm$^3$) of this bottle.
[img]https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png[/img]
[b]p3.[/b] If $P$ is a quadratic polynomial with leading coefficient $ 1$ such that $P(1) = 1$, $P(2) = 2$, what is $P(10)$?
[b]p4.[/b] Let ABC be a triangle with $AB = 1$, $AC = 3$, and $BC = 3$. Let $D$ be a point on $BC$ such that $BD =\frac13$ . What is the ratio of the area of $BAD$ to the area of $CAD$?
[b]p5.[/b] A coin is flipped $ 12$ times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms.
[b]p6.[/b] Moor pours $3$ ounces of ginger ale and $ 1$ ounce of lime juice in cup $A$, $3$ ounces of lime juice and $ 1$ ounce of ginger ale in cup $B$, and mixes each cup well. Then he pours $ 1$ ounce of cup $A$ into cup $B$, mixes it well, and pours $ 1$ ounce of cup $B$ into cup $A$. What proportion of cup $A$ is now ginger ale? Express your answer as a common fraction in lowest terms.
[b]p7.[/b] Determine the maximum possible area of a right triangle with hypotenuse $7$. Express your answer as a common fraction in lowest terms.
[b]p8.[/b] Debbie has six Pusheens: $2$ pink ones, $2$ gray ones, and $2$ blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so?
[b]p9.[/b] How many nonnegative integer pairs $(a, b)$ are there that satisfy $ab = 90 - a - b$?
[b]p10.[/b] What is the smallest positive integer $a_1...a_n$ (where $a_1, ... , a_n$ are its digits) such that $9 \cdot a_1 ... a_n = a_n ... a_1$, where $a_1$, $a_n \ne 0$?
[b]p11.[/b] Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs $2$ square meters of land and $4$ gallons of spring water to grow, matsutake mushrooms need $3$ square meters of land and $3$ gallons of spring water, and daikon need $ 1$ square meter of land and $ 1$ gallon of spring water to grow. Wasabi sell for $60$ per root, matsutake mushrooms sell for $60$ per mushroom, and daikon sell for $2$ per root. If Justin has $500$ gallons of spring water and $400$ square meters of land, what is the maximum amount of money, in dollars, he can make?
[b]p12.[/b] A [i]prim [/i] number is a number that is prime if its last digit is removed. A [i]rime [/i] number is a number that is prime if its first digit is removed. Determine how many numbers between $100$ and $999$ inclusive are both prim and rime numbers.
[b]p13.[/b] Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a $2\times 2\times 2$ cube, what is the volume of the resulting solid?
[img]https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png[/img]
[b]p14.[/b] If a parallelogram with perimeter $14$ and area $ 12$ is inscribed in a circle, what is the radius of the circle?
[b]p15.[/b] Take a square $ABCD$ of side length $1$, and draw $\overline{AC}$. Point $E$ lies on $\overline{BC}$ such that $\overline{AE}$ bisects $\angle BAC$. What is the length of $BE$?
[b]p16.[/b] How many integer solutions does $f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017$ have?
[b]p17.[/b] Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing?
[b]p18.[/b] Let $x$ be a positive integer with a remainder of $2$ when divided by $3$, $3$ when divided by $4$, $4$ when divided by $5$, and $5$ when divided by $6$. What is the smallest possible such $x$?
[b]p19[/b]. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is $ 1$, and the upper base of the trapezoid is $ 1$, what is the area of the trapezoid?
[b]p20.[/b] Ray is blindfolded and standing $ 1$ step away from an ice cream stand. Every second, he has a $1/4$ probability of walking $ 1$ step towards the ice cream stand, and a $3/4$ probability of walking $ 1$ step away from the ice cream stand. When he is $0$ steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].