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[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2016

  1. Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB$, $AC$ in $D$, $E$, then $ADE$ is equilateral.
  2. Let $m$, $n$ are positive intergers and $x$, $y$, $z$ positive real numbers such that $0 \leq x,y,z \leq 1$. Let $m+n=p$. Prove that $$0 \leq x^p+y^p+z^p-x^m y^n-y^m z^n-z^m x^n \leq 1.$$
  3. Let $M$ be the set of natural numbers $k$ for which there exists a natural number $n$ such that $$3^n \equiv k\pmod n.$$ Prove that $M$ has infinitely many elements.
  4. Let $ABC$ be an acute triangle with $AB<AC$ and $D$, $E$, $F$ be the contact points of the incircle $(I)$ with $BC$, $AC$, $AB$. Let $M$, $N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.
  5. Let $n\ge 2$. Each $1\times 1$ square of a $n\times n$ board is colored in black or white such that every black square has at least $3$ white neighbors (Two squares are neighbors if they have a common side). What is the maximum number of black squares?
  6. Triangle $\triangle{ABC}$, $O$ is circumcenter of $(ABC)$, $OA=R$ the $A$-excircle intersect $(AB)$, $(BC)$, $(CA)$ at points $F$, $D$, $E$. If the $A$-excircle has radius $R$, prove that $OD\perp EF$
  7. Let $a,b,c>0$ and $abc\ge 1$. Prove that $$\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$$
  8. Let $n$ be an integer greater than $2$ and consider the set \begin{align*} A = \{2^n-1,3^n-1,\dots,(n-1)^n-1\}. \end{align*}Given that $n$ does not divide any element of $A$, prove that $n$ is a square-free number. Does it necessarily follow that $n$ is a prime?
  9. We have a $4\times 4$ board. All $1\times 1$ squares are white.A move is changing colours of all squares of a $1\times 3$ rectangle from black to white and from white to black. It is possible to make all the $1\times 1$ squares black after several moves?
  10. Let $n$ be a positive integer and consider the system
    \begin{align*} S(n):\begin{cases} x^2+ny^2=z^2\\ nx^2+y^2=t^2 \end{cases}, \end{align*}where $x,y,z$, and $t$ are naturals. If $M_1=\{n\in\mathbb N:$ system $S(n)$ has infinitely many solutions$\}$, and $M_1=\{n\in\mathbb N:$ system $S(n)$ has no solutions$\}$, prove that $7 \in M_1$ and $10 \in M_2$. sets $M_1$ and $M_2$ are infinite.
  11. Let $x,y$ are real numbers different from $0$ such that $x^3+y^3+3x^2y^2=x^3y^3$. Find all possible values of $E=\dfrac{1}{x}+\dfrac{1}{y}$
  12. Let $ABCD$ is cyclic quadrilateral, $AC\cap BD=X$, $AA'\perp BD$, $A'\in BD$, $CC'\perp BD$, $C'\in BD$, $BB'\perp AC$, $B'\in AC$, $DD'\perp AC$, $D'\in AC$. Prove that
    a) Perpendiculars from midpoints of the sides to the opposite sides are concurrent. The point is called Mathot Point
    b) $A'$, $B'$, $C'$, $D'$ are concyclic
    c) If $O'$ is circumcenter of $(A'B'C')$, prove that $O'$ is midpoint of the line that connects the orthocenter of triangle $XAB$ and $XCD$
    d) $O'$ is the Mathot Point
  13. In each $1\times 1$ square of a $n\times n$ board we write $n^2$ numbers with sum $S$. A move is choosing a $2\times 2$ square and adding $1$ to three numbers(from three different $1\times 1$ squares). We say that a number n is good if we can make all the numbers on the board equal by applying a successive number of moves and it not depends of $S$.
    a) Show that $6$ is not good
    b) Show that $4$ and $1024$ are good
  14. The altitudes $AA_1$, $BB_1$, $CC_1$ of $\triangle{ABC}$ intersect at $H$. $O$ is the circumcenter of $\triangle{ABC}$. Let $A_2$ be the reflection of $A$ wrt $B_1C_1$. Prove that
    a) $O$, $A_2$, $B_1$, $C$ are all on a circle
    b) $O$, $H$, $A_1$, $A_2$ are all on a circle
  15. Given three colors and a rectangle m × n dice unit, we want to color each segment constituting one side of a square drive with one of three colors so that each square unit have two sides of one color and two sides another color. How many colorings we have?
  16. Let $a,b,c$ be real numbers such that:$a\ge b\ge 1\ge c\ge 0$ and $a+b+c=3$. Prove that
    a) $2\le ab +bc+ca\le 3$.
    b) $\dfrac{24}{a^3+b^3+c^3}+\dfrac{25}{ab+bc+ca}\ge 14$.
  17. Let $ABCD$ be a cyclic quadrilateral.$E$ is the midpoint of $(AC)$ and $F$ is the midpoint of $(BD)$, $G$=$AB\cap CD$ and $H$=$AD\cap BC$.
    a) Prove that the intersections of the angle bisector of $\angle{AHB}$ and the sides $AB$ and $CD$ and the intersections of the angle bisector of $\angle{AGD}$ with $BC$ and $AD$ are the verticles of a rhombus
    b) Prove that the center of this rhombus lies on $EF$.

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Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,2,Amsterdam,5,Ấn Độ,1,An Giang,21,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,51,Bắc Giang,49,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,46,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,37,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,13,Bình Định,43,Bình Dương,21,Bình Phước,38,Bình Thuận,34,Birch,1,Booklet,11,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,5,Buôn Ma Thuột,1,BxMO,12,Cà Mau,13,Cần Thơ,14,Canada,39,Cao Bằng,6,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,347,Chu Tuấn Anh,1,Chuyên Đề,124,Chuyên Sư Phạm,31,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,610,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,25,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,54,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi HSG,1637,Đề Thi JMO,1,Điện Biên,8,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,4,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,49,Đồng Tháp,51,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,16,ELMO,19,EMC,8,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. 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MOlympiad: [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2016
[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2016
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