#### Practice (6)

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Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.$$ Find $|f(0)|$.

Steve says to Jon, 'I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $$P(x) = 2x^3-2ax^2+(a^2-81)x-c$$

for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?' After some calculations, Jon says, 'There is more than one such polynomial.' Steve says, 'You're right. Here is the value of $a$.' He writes down a positive integer and asks, 'Can you tell me the value of $c$?' Jon says, 'There are still two possible values of $c$.' Find the sum of the two possible values of $c$.

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{2014} x^3 - 4029x^2 + 2 = 0$. Find $x_2(x_1+x_3)$.

Let $m$ be the largest real solution to the equation$$\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$$There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.

Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$. If $f(-1) = -1$, $f(2)=-4$, $f(-3) = -9$, and $f(4) = -16$. Find $f(1)$.

Solve in positive integers $x^2 - 4xy + 5y^2 = 169$.

Solve in integers the question $x+y=x^2 -xy + y^2$.

Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$

Prove the product of $4$ consecutive positive integers is a perfect square minus $1$.

For any arithmetic sequence whose terms are all positive integers, show that if one term is a perfect square, this sequence must have infinite number of terms which are perfect squares.

Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.

Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that $m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.$

Suppose $x$ is in the interval $[0, \frac{\pi}{2}]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$.

Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers $x$, and suppose $P(11) = 181$. Find $P(16)$.

Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

The zeros of the function $f(x) = x^2-ax+2a$ are integers. What is the sum of the possible values of $a$?

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$ ?

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $24.50$. How much would $8$ adult tickets and $6$ child tickets cost?

The quadratic equation $x^2+ px + 2p = 0$ has solutions $x = a$ and $x = b$. If the quadratic equation $x^2+ cx + d = 0$ has solutions $x = a + 2$ and $x = b + 2$, what is the value of d?