Proof & Number Theory

9 methods · 22 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

The full chapter (9 methods · 22 worked questions) comes with Platform Access. See pricing

STEP scaffolds force a result early precisely so you can deploy it later. Treat any proved lemma, bound or identity as a reusable instrument: re-substitute specific values, specialise variables, or apply it to a new tuple to clear the next part almost mechanically. If a later part looks intractable, ask which earlier result it is begging you to plug in.

Trigger: the words "hence", "deduce", "use the result above", or a part that mirrors the previous one with different inputs.

Instances:
(i) feed a special value into a just-proved inequality to chain to a stronger bound.
(ii) apply a proved identity to a cleverly chosen new tuple (e.g. a substitution $d + e = c$ , or averaged terms $b_{i}$ ).
(iii) reuse a closed form recursively (build the odd case from the even case).

Linked questions (3)

STEP 2 2008 · Q3

Find the coordinates of the turning points of the curve $y = 27 x^{3} - 27 x^{2} + 4$ . Sketch the curve and deduce that $x^{2} (1 - x) ⩽ 4/27$ for all $x ⩾ 0$ . Given that each of the numbers $a$ , $b$ and $c$ lies between $0$ and $1$ , prove by contradiction that at least one of the numbers $b c (1 - a)$ , $c a (1 - b)$ and $ab (1 - c)$ is less than or equal to $4/27$ .
Given that each of the numbers $p$ and $q$ lies between $0$ and $1$ , prove that at least one of the numbers $p (1 - q)$ and $q (1 - p)$ is less than or equal to $1/4$ .

STEP 3 2009 · Q4

For any given (suitable) function

f

, the Laplace transform of

f

is the function

F

defined by

F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t (s > 0) .

Show that the Laplace transform of $e^{- b t} f (t)$ , where $b > 0$ , is $F (s + b)$ .
Show that the Laplace transform of $f (a t)$ , where $a > 0$ , is $a^{- 1} F (\frac{s}{a})$ .
Show that the Laplace transform of $f^{'} (t)$ is $s F (s) - f (0)$ .
In the case $f (t) = sin t$ , show that $F (s) = \frac{1}{s ^{2} + 1}$ .

Using only these four results, find the Laplace transform of

e^{- pt} cos q t

, where

p > 0

and

q > 0

STEP 2 2013 · Q7

Write down a solution of the equation $x^{2} - 2 y^{2} = 1, (*)$ for which $x$ and $y$ are non-negative integers. Show that, if $x = p$ , $y = q$ is a solution of ( $*$ ), then so also is $x = 3 p + 4 q$ , $y = 2 p + 3 q$ . Hence find two solutions of $(*)$ for which $x$ is a positive odd integer and $y$ is a positive even integer.
Show that, if $x$ is an odd integer and $y$ is an even integer, $(*)$ can be written in the form $n^{2} = \frac{1}{2} m (m + 1),$ where $m$ and $n$ are integers.
The positive integers $a$ , $b$ and $c$ satisfy $b^{3} = c^{4} - a^{2},$ where $b$ is a prime number. Express $a$ and $c^{2}$ in terms of $b$ in the two cases that arise. Find a solution of $a^{2} + b^{3} = c^{4}$ , where $a$ , $b$ and $c$ are positive integers but $b$ is not prime.