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2017 Romanian Master of Mathematics Shortlist
3
RMM 2017 Shortlist
Algebra
A1
A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:
(1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ ,
(2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ ,
(3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A$, and
(4) If a and b are distinct elements of $A-\{e\}$, then $a^3 * b = b^3 * a^2$, where $x^k = x * x^{k-1}$ for all integers $k \ge 2$ and all $x$ in $A$.
Determine the largest cardinality $A$ may have.

proposed by Bojan Basic, Serbia
parmenides51
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Combinatorics
C1
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
GGPiku
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C2
Fix an integer $n \ge 2$ and let $A$ be an $n\times n$ array with $n$ cells cut out so that exactly one cell is removed out of every row and every column. A stick is a $1\times k$ or $k\times 1$ subarray of $A$, where $k$ is a suitable positive integer.
(a) Determine the minimal number of sticks $A$ can be dissected into.
(b) Show that the number of ways to dissect $A$ into a minimal number of sticks does not exceed $100^n$.

proposed by Palmer Mebane and Nikolai Beluhov

a few comments
parmenides51
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Geometry
G1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
GGPiku
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G2
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.

proposed by Luis Eduardo Garcia Hernandez, Mexico
parmenides51
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G3
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.
GGPiku
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Number Theory
N1
For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system.
Show that there is an integer $k$, with no $9$ in it's decimal representation, such that:
$$S(2^{24^{2017}}k)=S(k)$$
freghyy
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N2
Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$.

proposed by Bojan Basic, Serbia
parmenides51
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a