A KS5 proof collection

Here are a collection of statements to prove, to help you to practise writing out clear mathematical proofs.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Here are a collection of statements for you to try to prove clearly and concisely. Once you have had a go, follow the links to see how other students solved each one.

Perfectly Square

We are first given that: $$x_1 = 2^2 + 3^2 + 6^2$$ $$x_2 = 3^2 + 4^2 + 12^2$$ $$x_3 = 4^2 + 5^2 + 20^2$$ Then show that $x_n$ is always a perfect square.

Always Perfect

Prove that if you add 1 to the product of four consecutive whole numbers the answer is ALWAYS a perfect square.

Unit Interval

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product.

That is, if $0< x< 1$ and $0< y< 1$ then prove

$$x+y< 1+xy$$.

Sums of Squares

$2(5^2 + 3^2) = 2(25 + 9) = 68 = 64 + 4 = 8^2 + 2^2$

$2(7^2 + 4^2) = 2(49 + 16) = 130 = 121 + 9 = 11^2 + 3^2$

Prove that if you double the sum of two squares you get the sum of two squares.

Mod 3

Prove that if $a^2+b^2$ is a multiple of $3$ then both $a$ and $b$ are multiples of $3$.

IFFY Triangles

  1. Let $ABC$ be a triangle with angles $A$, $B$ and $C$ where $A \le B \le C$ and $A$, $B$ and $C$ form an arithmetic progression. Prove that $B=60^{\circ}$.

     
  2. $XYZ$ is a triangle with angles $X$, $Y$ and $Z$ and we have $Y = 60^{\circ}$.

    Prove that $X$, $Y$ and $Z$ form an arithmetic progression.

     
  3. $PQR$ is a triangle with length $PQ$ equal to length $QR$.

    Prove that angles $P$ and $R$ are equal.

     
  4. $KLM$ is a triangle with equal angles $K$ and $M$.

    Prove that the lengths $KL$ and $LM$ are equal.