Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
(Sophie Germain's Identity) Prove $a^4 + 4b^4 = (a^2+2b^2-2ab)(a^2+2b^2+2ab)$.
Solve this equation $$2\sqrt{2}x^2 + x -\sqrt{1-x ^2}-\sqrt{2}=0$$
Find the remainder when $x^{2017}$ is divided by $(x+1)^2$.
Let $f(x)=2016x - 2015$. Solve this equation $$\underbrace{f(f(f(\cdots f(x))))}_{2017\text{ iterations}}=f(x)$$
Let integers $u$ and $v$ be two integral roots to the quadratic equation $x^2 + bx+c=0$ where $b+c=298$. If $u < v$, find the smallest possible value of $v-u$.
The sum and product of two numbers are equal to $y$. For which values of $y$ are these two numbers real?
Let $m$ and $n$ be the roots of $P(x)=ax^2+bx+c$. Find the coefficients of the quadratic polynomial whose roots are $m^2-n$ and $n^2-m$.
The roots of $x^2+ax+b+1$ are positive integers. Show that $a^2+b^2$ is not a prime number.
Let $\alpha$ and $\beta$ be the roots of $x^2+px+1$, and let $\gamma$ and $\sigma$ be the roots of $x^2+qx+1$.
Show
$$(\alpha - \gamma)(\beta-\gamma)(\alpha+\sigma)(\beta+\sigma) = q^2 - p^2$$
Let $a$, $b$, $c$ be distinct real numbers. Show that there is a real number $x$ such that
$$x^2+2(a+b+c)x+3(ab+bc+ac)$$
is negative.
Consider the quadratic equation $ax^2-bx+c=0$ where $a$, $b$, $c$ are real numbers and $a \ne 0$. Find the values of $a$, $b$, $c$ such that $a$ and $b$ are the roots of the equation and $c$ is it's discriminant.
Let $b \ge 0$ be a real number. The product of the four real roots of the equations $x^2+2bx+c=0$ and $x^2+2cx+b=0$ is equal to $1$. Find the values of $b$ and $c$.
Solve the equation
$$x^4-97x^3+2012x^2-97x+1=0$$
Show that if $a$, $b$, $c$ are the lengths of the sides of a triangle, then the equation
$$b^2x^2+(b^2+c^2-a^2)x + c^2=0$$
does not have any real roots.
Solve the equation $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
Find all real numbers $m$ such that $$x^2+my^2-4my+6y-6x+2m+8 \ge 0$$ for every pair of real numbers $x$ and $y$.
Does there exist a polynomial $P(x)$ such that $P(1)=2015$ and $P(2015)=2016$?
Show that $a=1+\sqrt{2}$ is irrational using the following steps:
(a) Find a polynomial with integer coefficients that has $a$ as a root.
(b) Use the Rational Root Theorem to show $a$ is irrational.
Show that $\sqrt{2}+\sqrt{3}$ is irrational using the same steps.
Find all polynomials with integer coefficients $p(x)$ that satisfy the following identity
$$2p(2x)=p(3x)+p(x)$$.
Let $p(x) = x^3-3x+1$. Show that if a complex number $a$ is a root of $p(x)$, then $a^2-2$ is also a root.
For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial $x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$ are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $x_1$ and $x_2$ be the two roots of equation $x^2 − 3x + 2 = 0$. Find the following values without
computing $x_1$ and $x_2$ directly.
i) $x_1^4 + x_2^4$
ii) $x_1 - x_2$
(Note: for (i) above, how many different solutions can you find?)
Find the sum of all possible integer values of $a$ such that the equation $(a + 1)x^2-(a^2 + 1)x + (2a^2 − 6) = 0$ is solvable in integers.