20. Five logicians each make a statement, as follows:
Mr P: Of these five statements, an odd number are true. Ms Q: Both statements made by women are true. Mr R: My first name is Robert and Mr P's statement is true. Ms S: Exactly one statement made by a man is true. Mr T: Neither statement made by a woman is true.
How many of the five statements can be simultaneously true?
A none
B 1 only
C 2 only
D 3 only
E 4 only F none or 1 only G 1 or 2 only H 2 or 3 only

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Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The following statements are attached to the urns.
Urn P This urn contains one or four balls.
Urn Q This urn contains two or four balls.
Urn R This urn contains more than two balls and fewer than five balls.
Urn S This urn contains one or two balls.
Urn T This urn contains fewer than three balls.
Exactly one of the urns has a true statement attached to it.
Which urn is it?

Consider the statement:
() A whole number $n$ is prime if it is 1 less or 5 less than a multiple of 6 .
How many counterexamples to () are there in the range $0 < n < 50$ ?

Triangles $A B C$ and $X Y Z$ have the same area.
Which of these extra conditions, taken independently, would imply that they are congruent?
(1) $A B = X Y$ and $B C = Y Z$
(2) $A B = X Y$ and $\angle A B C = \angle X Y Z$
(3) $\angle A B C = \angle X Y Z$ and $\angle B C A = \angle Y Z X$

In this question, $a , b$, and $c$ are positive integers.
The following is an attempted proof of the false statement:
If $a$ divides $b c$, then $a$ divides $b$ or $a$ divides $c$.
['$a$ divides $b c$' means '$a$ is a factor of $b c$']
Which line contains the error in this proof?
1. The statement is equivalent to if $a$ does not divide $b$ and $a$ does not divide $c$ then $a$ does not divide $b c$'.
2. Suppose $a$ does not divide $b$ and $a$ does not divide $c$. Then the remainder when dividing $b$ by $a$ is $r$, where $0 < r < a$, and the remainder when dividing $c$ by $a$ is $s$, where $0 < s < a$.
3. So $b = a x + r$ and $c = a y + s$ for some integers $x$ and $y$.
4. Thus $b c = a ( a x y + x s + y r ) + r s$.
5. So the remainder when dividing $b c$ by $a$ is $r s$.
6. Since $r > 0$ and $s > 0$, it follows that $r s > 0$.
7. Hence $a$ does not divide $b c$.

Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.
Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively.
This is the side elevation of the object.
This is the front elevation of the object.
This is the plan view of the object.
How many cubes were used to construct the object?

The diagram shows a square-based pyramid with base $P Q R S$ and vertex $O$. All the edges of the pyramid are of length 20 metres.
Find the shortest distance, in metres, along the outer surface of the pyramid from $P$ to the midpoint of $O R$.
A $10 \sqrt { 5 - 2 \sqrt { 3 } }$ B $10 \sqrt { 3 }$ C $10 \sqrt { 5 }$ D $10 \sqrt { 7 }$ E $10 \sqrt { 5 + 2 \sqrt { 3 } }$

Consider the following three statements:
$1 \quad 10 p ^ { 2 } + 1$ and $10 p ^ { 2 } - 1$ are both prime when $p$ is an odd prime.
2 Every prime greater than 5 is of the form $6 n + 1$ for some integer $n$.
3 No multiple of 7 greater than 7 is prime.
The result $91 = 7 \times 13$ can be used to provide a counterexample to which of the above statements?
A none of them
B 1 only
C 2 only
D 3 only
E 1 and 2 only
F 1 and 3 only
G 2 and 3 only
H 1, 2 and 3

Consider the following attempt to prove this true theorem:
Theorem: $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$ has no solutions with $a , b$ and $c$ positive integers.
Attempted proof:
Suppose that there are positive integers $a , b$ and $c$ such that $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$.
I We have $a ^ { 3 } = c ^ { 3 } - b ^ { 3 }$.
II $\quad$ Hence $a ^ { 3 } = ( c - b ) \left( c ^ { 2 } + c b + b ^ { 2 } \right)$.
III It follows that $a = c - b$ and $a ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$, since $a \leqslant a ^ { 2 }$ and $c - b \leqslant c ^ { 2 } + c b + b ^ { 2 }$.
IV Eliminating $a$, we have $( c - b ) ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
V Multiplying out, we have $c ^ { 2 } - 2 c b + b ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
VI Hence $3 c b = 0$ so one of $b$ and $c$ is zero.
But this is a contradiction to the original assumption that all of $a , b$ and $c$ are positive. It follows that the equation has no solutions.
Comment on this proof by choosing one of the following options:
A The proof is correct
B The proof is incorrect and the first mistake occurs on line I.
C The proof is incorrect and the first mistake occurs on line II.
D The proof is incorrect and the first mistake occurs on line III.
E The proof is incorrect and the first mistake occurs on line IV.
F The proof is incorrect and the first mistake occurs on line V.
G The proof is incorrect and the first mistake occurs on line VI.

The positive real numbers $a \times 10 ^ { - 3 } , b \times 10 ^ { - 2 }$ and $c \times 10 ^ { - 1 }$ are each in standard form, and
$$\left( a \times 10 ^ { - 3 } \right) + \left( b \times 10 ^ { - 2 } \right) = \left( c \times 10 ^ { - 1 } \right)$$
Which of the following statements (I, II, III, IV) must be true?
$$\begin{array} { l l } \text { I } & a > 9 \\ \text { II } & b > 9 \\ \text { III } & a < c \\ \text { IV } & b < c \end{array}$$
A I only
B II only
C I and II only
D I and III only
E I and IV only
F II and III only
G II and IV only
H I, II, III and IV

Consider the following statement:
(*) If $f ( x )$ is an integer for every integer $x$, then $f ^ { \prime } ( x )$ is an integer for every integer $x$.
Which one of the following is a counterexample to (*)?
A $f ( x ) = \frac { x ^ { 3 } + x + 1 } { 4 }$
B $f ( x ) = \frac { x ^ { 4 } + x ^ { 2 } + x } { 2 }$
C $f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x } { 2 }$
D $f ( x ) = \frac { x ^ { 4 } + 2 x ^ { 3 } + x ^ { 2 } } { 4 }$

A set $S$ of whole numbers is called stapled if and only if for every whole number $a$ which is in $S$ there exists a prime factor of $a$ which divides at least one other number in $S$.
Let $T$ be a set of whole numbers. Which of the following is true if and only if $T$ is not stapled?
A For every number $a$ which is in $T$, there is no prime factor of $a$ which divides every other number in $T$.
B For every number $a$ which is in $T$, there is no prime factor of $a$ which divides at least one other number in $T$.
C For every number $a$ which is in $T$, there is a prime factor of $a$ which does not divide any other number in $T$.
D For every number $a$ which is in $T$, there is a prime factor of $a$ which does not divide at least one other number in $T$.
E There exists a number $a$ which is in $T$ such that there is no prime factor of $a$ which divides every other number in $T$.
F There exists a number $a$ which is in $T$ such that there is no prime factor of $a$ which divides at least one other number in $T$.
G There exists a number $a$ which is in $T$ such that there is a prime factor of $a$ which does not divide any other number in $T$.
H There exists a number $a$ which is in $T$ such that there is a prime factor of $a$ which does not divide at least one other number in $T$.

Consider the following statement:
A car journey consists of two parts. In the first part, the average speed is $u \mathrm {~km} / \mathrm { h }$. In the second part, the average speed is $v \mathrm {~km} / \mathrm { h }$. Hence the average speed for the whole journey is $\frac { 1 } { 2 } ( u + v ) \mathrm { km } / \mathrm { h }$.
Which of the following examples of car journeys provide(s) a counterexample to the statement?
I In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for 100 km . In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for 100 km .
II In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for one hour. In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for one hour.
III In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for 80 km . In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for 100 km .

The two diagonals of the quadrilateral $Q$ are perpendicular.
Consider the following statements:
I One of the diagonals of $Q$ is a line of symmetry of $Q$.
II The midpoints of the sides of $Q$ are the vertices of a square.
Which of these statements is/are necessarily true for the quadrilateral $Q$ ?

Which one of the following functions provides a counterexample to the statement: if $\mathrm { f } ^ { \prime } ( x ) > 0$ for all real $x$, then $\mathrm { f } ( x ) > 0$ for all real $x$.

The diagram shows an example of a mountain profile.
[Figure]
This consists of upstrokes which go upwards from left to right, and downstrokes which go downwards from left to right. The example shown has six upstrokes and six downstrokes. The horizontal line at the bottom is known as sea level.
A mountain profile of order $n$ consists of $n$ upstrokes and $n$ downstrokes, with the condition that the profile begins and ends at sea level and never goes below sea level (although it might reach sea level at any point). So the example shown is a mountain profile of order 6.
Mountain profiles can be coded by using U to indicate an upstroke and D to indicate a downstroke. The example shown has the code UDUUUDUDDUDD. A sequence of U's and D's obtained from a mountain profile in this way is known as a valid code.
Which of the following statements is/are true?
I If a valid code is written in reverse order, the result is always a valid code.
II If each $U$ in a valid code is replaced by $D$ and each $D$ by $U$, the result is always a valid code.
III If U is added at the beginning of a valid code and D is added at the end of the code, the result is always a valid code.

Consider the following attempt to solve the equation $4 x \sqrt { 2 x - 1 } = 10 x - 5$ :
$$\begin{aligned} 4 x \sqrt { 2 x - 1 } & = 10 x - 5 \\ 4 x \sqrt { 2 x - 1 } & = 5 ( 2 x - 1 ) \\ 16 x ^ { 2 } ( 2 x - 1 ) & = 25 ( 2 x - 1 ) ^ { 2 } \\ 16 x ^ { 2 } & = 25 ( 2 x - 1 ) \\ 16 x ^ { 2 } - 50 x + 25 & = 0 \\ ( 8 x - 5 ) ( 2 x - 5 ) & = 0 \end{aligned}$$
The solutions of the original equation are $x = \frac { 5 } { 8 }$ and $x = \frac { 5 } { 2 }$.
Which one of the following is true?

Consider the following statement:
For any positive integer $N$ there is a positive integer $K$ such that $N ( K m + 1 ) - 1$ is not prime for any positive integer $m$.
Which one of the following is the negation of this statement?

The following is an attempted proof of the conjecture:
$$\text { if } \tan \theta > 0 , \text { then } \sin \theta + \cos \theta > 1$$
Suppose $\tan \theta > 0$, so in particular $\cos \theta \neq 0$.
$$\text { Since } \tan \theta = \frac { \sin \theta } { \cos \theta } \text {, then } \sin \theta \cos \theta = \tan \theta \cos ^ { 2 } \theta > 0 . $$
It follows that $1 + 2 \sin \theta \cos \theta > 1$.
Therefore $\sin ^ { 2 } \theta + 2 \sin \theta \cos \theta + \cos ^ { 2 } \theta > 1$,
which factorises to give $( \sin \theta + \cos \theta ) ^ { 2 } > 1$.
Therefore $\sin \theta + \cos \theta > 1$.
Which one of the following is the case?

Three real numbers $x , y$ and $z$ satisfy $x > y > z > 1$.
Which one of the following statements must be true?

$a , b$ and $c$ are real numbers.
Given that $a b = a c$, which of the following statements must be true?
I $\quad a = 0$
II $b = 0$ or $c = 0$
III $b = c$

Consider the following conjecture:
If $N$ is a positive integer that consists of the digit 1 followed by an odd number of 0 digits and then a final digit 1 , then $N$ is a prime number.
Here are three numbers:
I $\quad N = 101$ (which is a prime number)
II $\quad N = 1001$ (which equals $7 \times 11 \times 13$ )
III $N = 10001$ (which equals $73 \times 137$ )
Which of these provide(s) a counterexample to the conjecture?

Consider the following statement about the positive integers $a , b$ and $n$ :
(*): $a b$ is divisible by $n$
The condition 'either $a$ or $b$ is divisible by $n$ ' is:

A student attempts to solve the equation
$$\cos x + \sin x \tan x = 2 \sin x - 1$$
in the range $0 \leq x \leq 2 \pi$.
The student's attempt is as follows:
$\cos x + \sin x \tan x = 2 \sin x - 1$
So $\quad \cos x - \sin x + \sin x \tan x - \sin x = - 1$
So $\quad ( \sin x - \cos x ) ( \tan x - 1 ) = - 1$
So $\quad \sin x - \cos x = - 1$ or $\tan x - 1 = - 1$
So $\quad ( \sin x - \cos x ) ^ { 2 } = 1$ or $\tan x = 0$
So $\quad 2 \sin x \cos x = 0$ or tan $x = 0$
So $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } , 2 \pi$
Which of the following best describes this attempt?

For which one of the following statements can the fact that $12 ^ { 2 } + 16 ^ { 2 } = 20 ^ { 2 }$ be used to produce a counterexample?
A If $a , b$ and $c$ are positive integers which satisfy the equation $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$, and the three numbers have no common divisor, then two of them are odd and the other is even.
B The equation $a ^ { 4 } + b ^ { 2 } = c ^ { 2 }$ has no solutions for which $a , b$ and $c$ are positive integers.
C The equation $a ^ { 4 } + b ^ { 4 } = c ^ { 4 }$ has no solutions for which $a , b$ and $c$ are positive integers.
D If $a , b$ and $c$ are positive integers which satisfy the equation $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$, then one is the arithmetic mean of the other two.