1. In the diagram, the circle has radius $\sqrt 7$ and and centre $O.$ Points $A$, $B$ and $C$ are on the circle. If $\angle BOC=120^\circ$ and $AC = AB + 1,$ determine the length of $AB.$
2. Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set?
3. Determine all solutions to the system of equations $\begin{cases}x^2 + y^2 + x + y &= 12 \\ xy + x + y &= 3\end{cases}$
4. Alphonse and Beryl play a game starting with a blank blackboard. Alphonse goes first and the two players alternate turns. On Alphonse's first turn, he writes the integer $10^{2011}$ on the blackboard. On each subsequent turn, each player can do exactly one of the following two things
• Replace any number $x$ that is currently on the blackboard with two integers a and b greater than $1$ such that $x = ab,$ or
• Erase one or two copies of a number $y$ that appears at least twice on the blackboard.
1. Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.
2. In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD$. The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ$, for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$
3. One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc,$ where each of $a, b$ and $c$ is a digit between $0$ and $9,$ inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the markers will be given a unique two-digit identification number $xy,$ with each of $x$ and $y$ a digit between $0$ and $9,$ inclusive. Marker $xy$ will be able to mark any contest with an identification number of the form $xyA$ or $xAy$ or $Axy,$ for any digit $A.$ What is the minimum possible number of markers to be hired to ensure that all contests will be marked?
4. Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$
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