Algebra
Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.
Algebraic Techniques Equations Inequalities Word Problems Sequences-
Quadratic Equations
Solve the equations:
a: `({x^2+6}/{x^2-4})^2=({5x}/{4-x^2})^2`
b: `7(x+1/x)-2(x^2+1/x^2)=9`
c: `sqrt{x+2sqrt{x-1}}-sqrt{x-2sqrt{x-1}}=2`
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Partitioned Roots
Find the sum `1/{sqrt1+sqrt2}+1/{sqrt2+sqrt3}+...+1/{sqrt99+sqrt100}`.
Sources:Topics:Algebra -> Algebraic Techniques -> Telescoping Sums Algebra -> Algebraic Techniques -> Roots / Radicals -
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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Finite Division
Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2`.
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Finite Division
Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2 `.
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Enlarging a Rectangle
A rectangle is given in the plane. Is it possible that after each side of the rectangle is increased by 1 cm, the area increases by 1 square meter? Provide an example or prove that it is impossible.
(If the rectangle is 1x5, it becomes 2x6 and no side can be 0)
Sources:- Grossman Math Olympiad, 2017, Juniors Question 1
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Integer Expression
Find all integers n for which the expression `{(n+2)^4}/{n-1}` is defined and an integer.
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Integer Coefficients?
Given real numbers a, b, c such that for every integer x, the number `ax^2+bx+c` is an integer. Does this necessarily imply that a, b, c are all integers? Prove it, or provide a counterexample.
Sources:Topics:Algebra Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample- Grossman Math Olympiad, 2017, Juniors Question 2
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The Confrontation
Ehud and Benjamin are participating in a public debate. Each one, in turn, presents a question to their opponent. Ehud is chosen to be the first to present a question. A "trick question" is a question to which the opponent has no answer. A contestant who manages to ask a trick question immediately wins the debate. The probability of each of the two contestants finding a trick question (in their turn) is exactly 1/2. It is also known that there is no dependence between the questions. What is the probability of Ehud winning the debate?