Junior Science Team Exercises, 2024-2025, Assignment 1

• Question 1 - 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`

  Topics:

  Algebra -> Equations Algebra -> Algebraic Techniques -> Roots / Radicals

• Question 2 - Partitioned Roots

  Find the sum `1/{sqrt1+sqrt2}+1/{sqrt2+sqrt3}+...+1/{sqrt99+sqrt100}`.

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  Topics:

  Algebra -> Algebraic Techniques -> Telescoping Sums Algebra -> Algebraic Techniques -> Roots / Radicals

• Question 3 - 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`.

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations

• Question 4 - 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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  Topics:

  Algebra -> Algebraic Techniques Number Theory -> Prime Numbers -> Prime Factorization Proof and Example -> Proof by Contradiction

• Question 5 - Finite Division

  Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2 `.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Proof and Example -> Proof by Contradiction Algebra -> Equations -> Diophantine Equations

• Question 6 - Integer Expression

  Find all integers n for which the expression `{(n+2)^4}/{n-1}` is defined and an integer.

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  Topics:

  Number Theory Algebra -> Equations