Practice (6)

3063

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?

3081

$(x^2 + 3x + 6)(x^2 + ax + b) = x^4 + mx^2 + n$ for integers

$a, b, m$ and

$n$ , what is the product of

$m$ and

$n$ ?

3099

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

$f (1)$ ?

3119

Find the largest of three prime divisors of $13^4+16^5-172^2$ .

3123

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?

3124

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

$x^2 + y^2 + z^2$ .

3125

Let

$x$ be a negative real number. Find the maximum value of

$y=x+\frac{4}{x} +2007$ .

3126

Let real numbers

$a$ and

$b$ satisfy

$a^2 + ab + b^2 = 1$ . Find the range of

$a^2 - ab + b^2$ .

3203

The roots of

$x^2 + ax + b+1$ are positive integers. Show that

$a^2+b^2$ is not a prime number.

3204

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

$(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=q^2-p^2$

3205

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.

3206

Solve the equation $x^4 -97x^3+2012x^2-97x+1=0$ .

3207

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.

3208

Solve the equation in real numbers

$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$

3211

Find

$a$ and

$b$ so that

$(x-1)^2$ divides

$ax^4 + bx^3+1$ .

3212

Find all pairs of real numbers

$a, b$ , such that the polynomial

$p(x)=(a+b)x^5 + abx^2 +1$ is divisible by

$x^2 - 3x+2$ .

3213

Find the real root of the polynomial

$p(x)=8x^3 -3x^2 -3x -1$ .

3214

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

$S_1 + S_2 + \cdots +S_n$ .

3215

If the product of two roots of polynomial

$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ is

$- 32$ . Find the value of

$k$ .

3239

Simplify

$\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$ .

3257

Simplify

$\sqrt{5\sqrt{3}+6\sqrt{2}}$ .

3261

Solve

$x^2 +6x - 4\sqrt{5}=0$ .

3265

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

$P(5)$ ?

3270

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.

3295

Compute the value of

$\sqrt{1+1995\sqrt{4+1995\times 1999}}$ .

PolynomialAndEquation Root VietaTheorem SpecialEquation InfiniteRepitition FunctionEquation PlaneGeometry NumberTheoryBasic Trigonometry LawOfCosines Inequality SpecialValueMethod SubstitutionMethod

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