Problems

1. Imagine that you are writing down integers in increasing order, starting from 1, until you have written 1000 digits, at which point you stop (even possibly in the middle of a number). How many times have you used the digit 7?
2. In how many zeros does the number 365! terminate, when written in base 10?

1. Consider the sequence \(\{a_n\}\) of positive real numbers defined by \(a_1 = 1\), \(a_{n+1} = a_n + 2\). Prove by induction or otherwise that \(a_n < 2\) and \(a_{n+1} > a_n\) for all \(n \geq 1\).
2. Prove by induction that 19 divides \(2 \cdot 5^{2n+1} + 2^n \cdot 3^{n+2}\) for all integers \(n \geq 0\).

The distant island of Amphibia is populated by speaking frogs and toads. They spend much of their time in little groups, making statements to themselves. Toads always tell the truth and frogs always lie. In each of the following four scenes from Amphibian life decide which characters mentioned are frogs and which are toads, explaining your reasoning carefully:

1. \(A\): `Both myself and \(B\) are frogs.'
2. \(C\): `At least one of \(D\) and myself is a frog.'
3. \(E\): `Both \(G\) and \(H\) are toads.' \\ \(G\): `That is true.' \\ \(H\): `No, that is not true.'
4. \(I\) and \(J\) talking about \(I\), \(J\) and \(K\): \\ \(I\): `All of us are frogs.' \\ \(J\): `Exactly one of us is a toad.'

A quartic polynomial \(f(x)\) with real coefficients is such that the equation \(f(x) = 0\) has exactly three distinct roots, which are all real. Show that just one of these roots is also a root of \(f'(x) = 0\). If \(f(x) = x^4 + 2x^3 - 3x^2 - 4x + a\) (where \(a\) is a constant) satisfies these conditions, show that there is only one possible value for \(a\), and find it.

Show that every odd square leaves remainder 1 when divided by 8, and that every even square leaves remainder 0 or 4. Deduce that a number of the form \(8n + 7\), where \(n\) is a positive integer, cannot be expressed as a sum of three squares.

Consider the \(2 \times 2\) complex matrices $$A = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$ List all the matrices which may be obtained from \(A\) and \(B\) by matrix multiplication, and show that they form a non-commutative group \(G\) of order 8. [You may assume the associativity of matrix multiplication.] By considering the elements in \(G\) whose square is the identity, or otherwise, determine whether \(G\) is isomorphic to the group of symmetries of a square.

Let \(p_r, q_r\) (\(r = 1, 2, \ldots\)) be two sequences such that \(p_r = q_{r+1} - q_r\) for all \(r \geq 1\). Evaluate \(\sum_{r=1}^N p_r\). Hence or otherwise evaluate

1. \(\sum_{r=1}^N \frac{1}{4r^2 - 1}\),
2. \(\sum_{r=1}^N \frac{1}{r(r+1)(r+2)}\),
3. \(\sum_{r=1}^N \sin r\theta\) [hint: first multiply by \(\sin \frac{1}{2}\theta\)].

1. Let \(a, b, c\) be real numbers with \(a > 0\). Prove that \(ax^2 + 2bx + c \geq 0\) for all \(x\) if and only if \(ac \geq b^2\).
2. Let \(n\) be a positive integer and let \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\) be real numbers. By considering \(\sum_{i=1}^n (a_i x - b_i)^2\), prove that $$\left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right) \geq \left(\sum_{i=1}^n a_i b_i\right)^2.$$
3. Now let \(c_1, c_2, \ldots, c_n\) be non-negative real numbers. Prove that $$\left(\sum_{i=1}^n c_i\right)^2 \geq \sum_{i=1}^n c_i^2 \geq \frac{1}{n}\left(\sum_{i=1}^n c_i\right)^2.$$

1. Prove that the angle subtended by a chord of a circle at any point on the circumference on the same side of the chord as the centre is half that subtended by the chord at the centre, and is equal to the angle between the chord and the tangent to the circle at either end of the chord; i.e. \(\angle AOB = 2\angle ACB = 2\angle DAB\).
2. A circle \(C\) has \(QR\) as a diameter, and \(P\) is any point inside \(C\) and not on \(QR\). \(PQ\), \(PR\) meet \(C\) in \(Q'\), \(R'\), and \(C'\) is the circle through \(PQ'R'\) (see figure). \(O'\) is the centre of \(C'\). Prove that

A room has a square horizontal ceiling of side \(a\), and vertical walls of height \(h\). A spider is located at distance \(h\) below the ceiling at the intersection of two walls, moving along the walls and ceiling it moves to a point on the intersection of the other two walls, also at distance \(h\) below the ceiling. Find the length of its shortest path for all possible values of \(h/a\).