1. The front row of a movie theatre contains $45$ seats.
a) If $42$ people are sitting in the front row, prove that there are $10$ consecutive seats that are all occupied.
b) Show that this conclusion doesn’t necessarily hold if only $41$ people are sitting in the front row.
2. Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.
3. We say that $(a,b,c)$ form a fantastic triplet if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.
4. Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*}S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\ S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\ S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.
5. Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies $a_{i+1} = a_i +d(a_i),$ for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are inﬁnitely many terms in the sequence divisible by $3^{2011}$.
6. Determine whether there exist two real numbers $a$ and $b$ such that both $(x-a)^3+ (x-b)^2+x$ and $(x-b)^3 + (x-a)^2 +x$ contain only real roots.
7. Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players $\{\mathit{A}, \mathit{B}, \mathit{C}\}$ is said to be cyclic if $\mathit{A}$ wins against $\mathit{B}$, $\mathit{B}$ wins against $\mathit{C}$ and $\mathit{C}$ wins against $\mathit{A}$. a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible. b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?
8. Suppose circles $\mathit{W}_1$ and $\mathit{W}_2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.
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