Algebra, Algebraic Techniques
This focuses on the specific methods and skills used to manipulate and simplify algebraic expressions, such as factoring, expanding, collecting like terms, working with fractions, and simplifying radicals. Questions require proficient application of these techniques.
Short Multiplication Formulas / Algebraic Identities Telescoping Sums Roots / Radicals Multiplication by the Conjugate-
Relationships Between the Roots of Quadratic Equations
Given a quadratic equation `ax^2+bx+c=0` whose solutions are `x_{1,2}={-b+-sqrt{b^2-4ac}}/{2a}`.
A: Show that Vieta's formulas hold: `x_1x_2=c/a` `x_1+x_2=-b/a,`.
B: Express the following in terms of a, b, c: `1/{x_1^2}+1/{x_2^2}` `1/x_1+1/x_2, ` `x_1^3+x_2^3,` `x_1^2+x_2^2`.
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Factoring and Using the Formula
An interesting formula is `x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)`.
A: Use it to factor the expression `a^n-b^n`.
B: Factor the expression `a^n+b^n` for any odd integer n.
C: Prove that if `2^n-1` is prime, then n is also prime.
D: Prove that if `2^n+1` is prime, then n is necessarily a power of 2, which is equivalent to `n=2^m`
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Polynomial with Integer Coefficients
Let `p(x)` be a polynomial with integer coefficients such that `p(-2006) < p(2006)=2005`. Prove that `p(-2006)<=-2007`.
Sources:
- Grossman Math Olympiad, 2006 Question 6
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LADY BELINDA'S GARDEN
Lady Belinda is an enthusiastic gardener. In the illustration she is depicted in the act of worrying out a pleasant little problem which I will relate. One of her gardens is oblong in shape, enclosed by a high holly hedge, and she is turning it into a rosary for the cultivation of some of her choicest roses. She wants to devote exactly half of the area of the garden to the flowers, in one large bed, and the other half to be a path going all round it of equal breadth throughout. Such a garden is shown in the diagram at the foot of the picture. How is she to mark out the garden under these simple conditions? She has only a tape, the length of the garden, to do it with, and, as the holly hedge is so thick and dense, she must make all her measurements inside. Lady Belinda did not know the exact dimensions of the garden, and, as it was not necessary for her to know, I also give no dimensions. It is quite a simple task no matter what the size or proportions of the garden may be. Yet how many lady gardeners would know just how to proceed? The tape may be quite plain—that is, it need not be a graduated measure.
Sources:Topics:Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Algebraic Techniques Algebra -> Equations- Amusements in Mathematics, Henry Ernest Dudeney Question 195
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Question
Calculate without a calculator using the most efficient method:
\( { (2001 \cdot 2021+100) \cdot (1991 \cdot 2031+400) \over 2011^4}\)
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Question
Answer without using pen and paper (and without a calculator, of course
):Which number is larger: `12345678^2` or `12345677*12345679` ?
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Question
Prove that the difference of squares of two consecutive odd numbers is divisible by `8`.
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Question
Prove that for every prime number `p>3 ` the following holds: `p^2-1` is divisible by `6`.
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Question
Given a natural number `A`. When it is increased by `1`, its square increases by `1001`. Find `A`.
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Question
Is the following number prime?
`4^9Â + 6^10Â + 3^20`