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[Solutions] Canadian Mathematical Olympiad Qualification Repechage 2013

  1. Determine all real solutions to the following equation \[2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.\]
  2. In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$. 
  3. A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property. 
  4. Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that both of these events occur
    • Each person receives exactly one gift;
    • No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$, then $B$ did not give her gift to $A$).
    1. For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$. 
    2. Let $x, y, z$ be real numbers that are greater than or equal to $0$ and less than or equal to $\dfrac{1}{2}$.
      a) Determine the minimum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this minimum is obtained.
      b) Determine the maximum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this maximum is obtained.
    3. Consider the following layouts of nine triangles with the letters $A, B, C, D, E, F, G, H, I$ in its interior. [asy]
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 /* end of picture */[/asy]A sequence of letters, each letter chosen from $ A, B, C, D, E, F, G, H, I$ is said to be triangle-friendly if the first and last letter of the sequence is $C$, and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after $C$ must be either $A, B$, or $D$. For example, $CBF BC$ is triangle-friendly, but $CBF GH$ and $CBBHC$ are not.
      a) Determine the number of triangle-friendly sequences with $2012$ letters.
      b) Determine the number of triangle-friendly sequences with exactly $2013$ letters.
    4. Let $\triangle ABC$ be an acute-angled triangle with orthocentre $H$ and circumcentre $O$. Let $R$ be the radius of the circumcircle. Let $A'$ be the point on $AO$ (extended if necessary) for which $HA'\perp AO$. Let $B'$ be the point on $BO$ (extended if necessary) for which $HB'\perp BO$. Let $C'$ be the point on $CO$ (extended if necessary) for which $HC'\perp CO$. Prove that $HA'+HB'+HC'<2R$. (The orthocentre of a triangle is the intersection of the three altitudes of the triangle. The circumcircle of a triangle is the circle passing through the triangle’s three vertices. The circummcentre is the centre of the circumcircle)

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    MOlympiad: [Solutions] Canadian Mathematical Olympiad Qualification Repechage 2013
    [Solutions] Canadian Mathematical Olympiad Qualification Repechage 2013
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