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[Solutions] Canadian Mathematical Olympiad Qualification Repechage 2015

  1. Find all integer solutions to the equation $$7x^2y^2 + 4x^2 = 77y^2 + 1260$$
  2. A polynomial $f(x)$ with integer coefficients is said to be tri-divisible if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.
  3. Let $N$ be a $3$-digit number with three distinct non-zero digits. We say that $N$ is mediocre if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.
  4. Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X$, $Y$ be points on $AB$, $AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A$, $H$, $X$, $Y$ lie on a common circle.
  5. Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$
  6. Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D$, $E$, $F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$.
  7. A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that
    • If there is a $0$ in position $i$ then there is a $1$ in position $i + x$,
    • If there is a $1$ in position $j$ then there is a $2$ in position $j + y$,
    • If there is a $2$ in position $k$ then there is a $0$ in position $k + z$.
      For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?
    1. A magical castle has $n$ identical rooms, each of which contains $k$ doors arranged in a line. In room $i, 1 \leq i \leq n - 1$ there is one door that will take you to room $i + 1$, and in room $n$ there is one door that takes you out of the castle. All other doors take you back to room $1$. When you go through a door and enter a room, you are unable to tell what room you are entering and you are unable to see which doors you have gone through before. You begin by standing in room $1$ and know the values of $n$ and $k$. Determine for which values of $n$ and $k$ there exists a strategy that is guaranteed to get you out of the castle and explain the strategy. For such values of $n$ and $k$, exhibit such a strategy and prove that it will work.

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    MOlympiad: [Solutions] Canadian Mathematical Olympiad Qualification Repechage 2015
    [Solutions] Canadian Mathematical Olympiad Qualification Repechage 2015
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