Here are some problems to keep you occupied! Some of the questions are easy to solve whilst others require more time and thought. The ones marked with an asterisk are especially difficult and will required techniques/knowledge beyond A level. The goal is to have fun and persevere, so enjoy!
Q1. Find all polynomials \(P(x)\) such that $$P(x) P(2x^2-1)\quad = \quad P(x^2)P(2x-1) $$
Q2. Let \(F_n\), \(F_{n+1}\) and \(F_{n+2}\) denote any three consecutive Fibonacci numbers show that \(a\), \(b\) form the two smallest values of a Pythagorean triple with \begin{equation}a = F_{n}F_{n+3} \quad \mathrm{and} \quad b = F_{n+1}F_{n+2} \end{equation} Hint: \(F_{n+2} = F_{n+1} +F_{n}\) and \(c^2 = a^2 + b^2\)
Q3. Show that the last digit of a squared integer can only be one of the following: 0, 1, 4, 5, 6, 9.
Q4*. Prove that \begin{equation}3(a + b + 1)^2 + 1 \geq 3ab \end{equation}
Q5. Solve the difference equation $$a_{n+1} \left( 1 + ( 1 + a_{n}^2 )^{\frac{1}{2}} \right) = a_n \quad \mathrm{with} \quad a_0 = 1.$$
Also find
$$ \lim_{n \to \infty} 2^n a_n $$
Also find the limit2nan asn tends to infinity (see if you can do this rigorously by proving existence and convergence).
Q6. Let \(a, b , c\) be integers such that the polynomial \(a x^2 + b x + c\) can be factored into a pair of brackets (using integers only). Show how the later factorisation can be obtained from the factorisation of \(x^2 + b x + ac\) . (Factorising the later equation with integers is more efficient from a trail and error point of view since there are fewer possibilities to test).
Q7.
- Prove that \((2n + 1)^3 - (2n - 1)^3\) is the sum of three prefect squares.
- Prove that \((2n + 1)^3 - 2\) is the sum of \(3n - 1\) perfect squares where each square is greater than one.
Q8.* Let \(a, b, c\) be positive real numbers such that \(abc = 1\).
Prove that $$\begin{equation}(a - 1 + b^{-1} )(b - 1 + c^{-1} )(c - 1 + a^{-1} ) \leq 1.\end{equation} $$
Q9. Is it possible to place 8 Queens on a chess board so that none are attacking each other?
Q10. Prove that: $$\arctan(\frac{1}{F_{2n}}) = \arctan(\frac{1}{F_{2n+1}}) + \arctan(\frac{1}{F_{2n+2}})$$
where \(F_{2n}\) is the \(2n\)th Fibonacci number.
Q11. Find all solutions to the following coupled equations:
$$\begin{eqnarray}
x+ y + z &= 3 \\
x^2 + y^2 + z^2 &= 3 \\
x^3 + y^3 + z^3 &= 3
\end{eqnarray}$$