A KS5 proof collection

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.