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Результаты Romanian Masters of Mathematics 2018

1 USA 93 First + Trophy
2 HUN 88 Second
3 RUS 88 Second
4 UKR 85 Third
5 KOR 83
6 POL 78
7 SRB 77
8 UNK 77
9 ITA 72
10 BGR 67
11 ROU 64
12 IDN 60
13 HRV 56
14 PER 52
15 BRA 47
16 SVN 47
17 FRA 45
18 BLR 44



Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent.

Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$

Ann and Bob play a game on an infinite checkered plane making moves in turn; Ann makes the first move. A move consists in orienting any unit grid-segment that has not been oriented before. If at some stage some oriented segments form an oriented cycle, Bob wins. Does Bob have a strategy that guarantees him to win?

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

Let $n$ be a positive integer and fix $2n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment).
The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $ABCD$ is a convex quadrangle oriented clockwise. Determine the number of good configurations.

Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ traces $\Omega$, circle $XYZ$ is tangent to two fixed circles.


@темы: Новости

2018-02-26 в 23:48 

О, суровые румынские математики)))

2018-02-26 в 23:58 

Четвертая с НОДом кажется из ряда относительно стандартных задач, правда я на свойства НОДа опыта в решении не имею.

2018-02-27 в 06:35 

Груша Вильямс, по ссылке в тексте сообщения есть файлы с решениями.


Не решается алгебра/высшая математика?.. ПОМОЖЕМ!