Here are six 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.

For What?

Prove that if the integer $n$ is divisible by $4$ then it can be written as the difference 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 $3$.

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.

For What?

Prove that if the integer $n$ is divisible by $4$ then it can be written as the difference 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 $3$.