Junior Science Team Exercises, 2024-2025
Assignment 1-
Question from sources: Assignment 1(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`
Sources: -
Question from sources: Assignment 1(2) - 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 -
Question from sources: Assignment 1(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`.
Sources: -
Question from sources: Assignment 1(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`
Sources: -
Question from sources: Assignment 1(5) - Finite Division
Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2 `.
Sources: -
Question from sources: Assignment 1(6) - Integer Expression
Find all integers n for which the expression `{(n+2)^4}/{n-1}` is defined and an integer.
Sources: