Italian Mathematical Olympiad 2014

For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all pos...

  1. For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.
  2. Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that
    i) $M$ is the midpoint of $AB$;
    ii) $N$ is the midpoint of $AC$;
    iii) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$.
    Prove that $\angle APM = \angle PBA$.
  3. For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
    a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
    b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
  4. Let $\omega$ be a circle with center $A$ and radius $R$. On the circumference of $\omega$ four distinct points $B, C, G, H$ are taken in that order in such a way that $G$ lies on the extended $B$-median of the triangle $ABC$, and H lies on the extension of altitude of $ABC$ from $B$. Let $X$ be the intersection of the straight lines $AC$ and $GH$. Show that the segment $AX$ has length $2R$.
  5. Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 0$, is colored in such a way that each of the cell is white or black. A cell is called special if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.
    a) Prove that there are at least $2n + 1$ special boxes.
    b) Provide an example where there are at most $4n$ special cells.
    c) Determine, as a function of $n$, the minimum possible number of special cells.



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MATHEMATICAL OLYMPIAD PROBLEMS: Italian Mathematical Olympiad 2014
Italian Mathematical Olympiad 2014
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