The digits of a 3-digit integer are reversed to form a new integer of greater value. The product of this new integer and the original integer is 91,567. What is the new integer?
If $(x^2 + 3x + 6)(x^2 + ax + b) = x^4 + mx^2 + n$ for integers $a, b, m$ and $n$, what is the product of $m$ and $n$?
The function $f (n) = a ⋅ n! + b$, where a and b are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
Find the largest of three prime divisors of $13^4+16^5-172^2$.
During the annual frog jumping contest at the county fair, the height of the frog's jump, in feet, is given by $$f(x)= -
\frac{1}{3}x^2+\frac{4}{3}x\:$$ What was the maximum height reached by the frog?
Let $x, y,$ and $z$ be some real numbers such that: $x+2y-z=6$ and $x-y+2z=3$. Find the minimal value of $x^2 + y^2 + z^2$.
Let $x$ be a negative real number. Find the maximum value of $y=x+\frac{4}{x} +2007$.
Let real numbers $a$ and $b$ satisfy $a^2 + ab + b^2 = 1$. Find the range of $a^2 - ab + b^2$.
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 $\delta$ be the roots of $x^2+qx+1$. Show $$(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=q^2-p^2$$
Let $a, b, c$ be distinct real numbers. Show that there is a real number $x$ such that $$x^2 +2(a+b+c)x+3(ab+bc+ca)$$ is negative.
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 real roots.
Solve the equation in real numbers $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
Find $a$ and $b$ so that $(x-1)^2$ divides $ax^4 + bx^3+1$.
Find all pairs of real numbers $a, b$, such that the polynomial $$p(x)=(a+b)x^5 + abx^2 +1$$ is divisible by $x^2 - 3x+2$.
Find the real root of the polynomial $p(x)=8x^3 -3x^2 -3x -1$.
Let $n$ be a positive integer, and for $1\le k\le n$, let $S_k$ be the sum of the products of $1, \frac{1}{2}, \cdots, \frac{1}{n}$, taken $k$ a time ($k^{th}$ elementary symmetric polynomial). Find $S_1 + S_2 + \cdots +S_n$.
If the product of two roots of polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ is $- 32$. Find the value of $k$.
Simplify $\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$.
Simplify $\sqrt{5\sqrt{3}+6\sqrt{2}}$.
Solve $x^2 +6x - 4\sqrt{5}=0$.
Let $P(x)$ be a monic polynomial of degree 3. (Monic here means that the coefficient of $x^3$ is 1.) Suppose that the remainder when $P(x)$ is divided by $x^2 - 5x+6$ equals 2 times the remainder when $P(x)$ is divided by $x^2 - 5x + 4$. If $P(0) = 100$, what is $P(5)$?
Let $P(x)$ be a polynomial with degree 2008 and leading coeffi\u000ecient 1 such that $P(0) = 2007, P(1) = 2006, P(2) = 2005, \cdots, P(2007) = 0$. Determine the value of $P(2008)$. You may use factorials in your answer.
Compute the value of $\sqrt{1+1995\sqrt{4+1995\times 1999}}$.