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 $v-u$.
Find all the ordered integers $(a, b, c)$ which satisfy $a+b+c=450$ and $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}=2c$.
$n$ straight lines are drawn on a plane such in such a way that no two of them are parallel and no three of them meet at one point. Show that the number of regions in which these lines divide the plane is $\frac{(n)(n+1)}{2}+1$.
Show
$$\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
Find the value of the following expression: $$\binom{2020}{0}-\binom{2020}{2}+\binom{2020}{4}-\cdots+\binom{2020}{2020}$$
Given integers $a$, $b$, $n$. Show that there exist integers $x$, $y$, such that $$(a^2+b^2)^n = x^2 + y^2$$.
The points $(0,0)$, $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Find $c$ if $a$, $b$, and $c$ and positive integers which satisfy $c = (a+bi)^3 - 107i$
Let $z_1$, $z_2$, $z_3$ be complex numbers with nonzero imaginary parts such that $|z_1| = |z_2| = |z_3|$. Show that if $z_1+z_2z_3$, $z_2+z_1z_3$, $z_3+z_1z_2$ are real, then $z_1z_2z_3 = 1$.
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 $a$, $b$, so that $(x-1)^2$ divides $ax^4+bx^3+1$.
Show that for each integer $n$ the polynomial $(\cos\theta+x\sin\theta)^n-\cos n\theta-x\sin n\theta$ is divisible by $x^2+1$
Show that
$$\sum_{i=0}^n \binom{n}{i}^2 = \binom{2n}{n}$$.
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.
Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.