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[Solutions] European Mathematical Cup 2016

Junior Division

  1. A grasshopper is jumping along the number line. Initially it is situated at zero. In $k$-th step, the length of his jump is $k$.
    a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to the left...). Will it visit on every integer at least once?
    b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left, then four steps to the right...). Will it visit every integer at least once?
  2. Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$.
  3. Prove that for all positive integers $n$ there exist $n$ distinct, positive rational numbers with sum of their squares equal to $n$.
  4. We will call a pair of positive integers $(n, k)$ with $k > 1$ a $lovely$ $couple$ if there exists a table $nxn$ consisting of ones and zeros with following properties: in every row there are exactly $k$ ones; for each two rows there is exactly one column such that on both intersections of that column with the mentioned rows, number one is written. Solve the following subproblems
    a) Let $d \neq 1$ be a divisor of $n$. Determine all remainders that $d$ can give when divided by $6$.
    b) Prove that there exist infinitely many lovely couples.

Senior Division

  1. Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum $$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$(with $1 \le r \le s \le 2016$) is a composite number, but:
    a) $\gcd (a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$;
    b) $\gcd (a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $\gcd (a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$?
    Note. $\gcd (x, y)$ denotes the greatest common divisor of $x$, $y$.
  2. For two positive integers $a$ and $b$, Ivica and Marica play the following game: Given two piles of $a$ and $b$ cookies, on each turn a player takes $2n$ cookies from one of the piles, of which he eats $n$ and puts $n$ of them on the other pile. Number $n$ is arbitrary in every move. Players take turns alternatively, with Ivica going first. The player who cannot make a move, loses. Assuming both players play perfectly, determine all pairs of numbers $(a, b)$ for which Marica has a winning strategy.
  3. Determine all functions $f:\mathbb R\to\mathbb R$ such that equality $$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$holds for all real numbers $x$, $y$.
  4. Let $C_{1}$, $C_{2}$ be circles intersecting in $X$, $Y$ . Let $A$, $D$ be points on $C_{1}$ and $B$, $C$ on $C_2$ such that $A$, $X$, $C$ are collinear and $D$, $X$, $B$ are collinear. The tangent to circle $C_{1}$ at $D$ intersects $BC$ and the tangent to $C_{2}$ at $B$ in $P$, $R$ respectively. The tangent to $C_2$ at $C$ intersects $AD$ and tangent to $C_1$ at $A$, in $Q$, $S$ respectively. Let $W$ be the intersection of $AD$ with the tangent to $C_{2}$ at $B$ and $Z$ the intersection of $BC$ with the tangent to $C_1$ at $A$. Prove that the circumcircles of triangles $YWZ$, $RSY$ and $PQY$ have two points in common, or are tangent in the same point.

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MOlympiad: [Solutions] European Mathematical Cup 2016
[Solutions] European Mathematical Cup 2016
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