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[Shortlist] Junior Balkan Mathematical Olympiad 2008

Algebra

  1. If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and $$x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008.$$ Find the product $xyz$.
  2. Find all real numbers $ a,b,c,d$ such that \[\begin{cases}a + b + c + d &= 20, \\ ab + ac + ad + bc + bd + cd &= 150. \end{cases}.\]
  3. Let the real parameter $p$ be such that the system $$\begin{cases} p(x^2 - y^2) &= (p^2- 1)xy \\ |x - 1|+ |y| &= 1 \end{cases}$$ has at least three different real solutions. Find $p$ and solve the system for that $p$.
  4. Find all triples $(x,y,z)$ of real numbers that satisfy the system $$\begin{cases} x + y + z &= 2008 \\ x^2 + y^2 + z^2 &= 6024^2 \\ \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}&=\dfrac{1}{2008} \end{cases}$$
  5. Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $$\begin{cases} \dfrac{1}{x}+\dfrac{4}{y}+\dfrac{9}{z}=3 \\ x + y + z \le 12 \end{cases}.$$ If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $$1 + ab + bc + cd + da + ac + bd > a + b + c + d.$$
  6. Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality $$\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$$
  7. Show that, for all real positive numbers $x, y $ and $z$,  $$(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \Big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\Big)^2$$
  8. Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and find this number.

Combinatorics

  1. On a $5 \times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board, it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they share a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was good.
    a) Prove that if $n = 20$, then a good initial positioning exists.
    b) Prove that if $n = 21$, then a good initial positioning does not exist.
  2. Kostas and Helene have the following dialogue
    • Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products.
    • Helene: I think that I know the numbers you have in mind. They are all equal to $1$.
    • Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem.
    Can you decide if Kostas is right? (Explain your answer).
  3. Integers $1,2, ...,2n$ are arbitrarily assigned to boxes labeled with numbers $1, 2,..., 2n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2n$.
  4. Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

Geometry

  1. Two perpendicular chords of a circle, $AM$, $BN$ which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD$, $BC$ with $AD || BC$ and $C$, $D$ different from $N$, $M$. If $L$ is the point of intersection of $DN$, $MC$ and $T$ the point of intersection of $DC$, $KL$. Prove that $\angle KTC = \angle KNL$.
  2. For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$), a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$) in a way such that $$AM -BC = BN- AC = CP - AB.$$ Prove that the angles of triangle $MNP$ do not depend on the choice of $M$, $N$, $P$.
  3. The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD=AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.
  4. Let $ABC$ be a triangle $(BC < AB)$. The line $l$ passing trough the vertices $C$ and orthogonal to the angle bisector $BE$ of $\angle B$, meets $BE$ and the median $BD$ of the side $AC$ at points $F$ and $G$, respectively. Prove that segment $DF$ bisects the segment $EG$.
  5. Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)
  6. Let $ABC$ be a triangle with $\angle A<{{90}^{o}} $. Outside of a triangle we consider isosceles triangles $ABE$ and $ACZ$ with bases $AB$ and $AC$, respectively. If the midpoint $D$ of the side $BC$ is such that $DE \perp DZ$ and $EZ = 2ED$, prove that $\angle AEB = 2 \angle AZC$ .
  7. Let $ABC$ be an isosceles triangle with $AC = BC$. The point $D$ lies on the side $AB$ such that the semicircle with diameter $BD$ and center $O$ is tangent to the side $AC$ in the point $P$ and intersects the side $BC$ at the point $Q$. The radius $OP$ intersects the chord $DQ$ at the point $E$ such that $5PE = 3DE$. Find the ratio $\dfrac{AB}{BC}$ .
  8. The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \dfrac{xy}{2}$, show that $$\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right) \quad \text{or} \quad \left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right).$$
  9. Let $O$ be a point inside the parallelogram $ABCD$ such that $$\angle AOB + \angle COD = \angle BOC + \angle AOD.$$ Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$ and $\vartriangle DOA$.
  10. Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$, and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$, by $X$ the (other than $M$) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.
  11. Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $DE$ and $BC$. The perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

Number Theory

  1. Find all the positive integers $x$ and $y$ that satisfy the equation $$x(x - y) = 8y - 7$$
  2. Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infinite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.
  3. Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.
  4. Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.
  5. Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)
  6. Let $f : N \to R$ be a function, satisfying the following condition:
    for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $$f(n) = f \Big(\frac{n}{p}\Big)-f(p).$$ If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $$f(2007^2) + f(2008^3) + f(2009^5)$$
  7. Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.
  8. Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares.
    a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers.
    b) Give an example of such a number.
  9. Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that $$\frac{4a + p}{b}+\frac{4b + p}{a} \quad \text{and} \quad \frac{a^2}{b}+\frac{b^2}{a}$$ are integers.
  10. Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.
  11. Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.
  12. Find all prime numbers $p,q,r$ such that $$\frac{p}{q}-\frac{4}{r+1}=1$$

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Name

Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,2,Amsterdam,5,Ấn Độ,1,An Giang,16,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,47,Bắc Giang,45,Bắc Kạn,1,Bạc Liêu,8,Bắc Ninh,43,Bắc Trung Bộ,8,Bài Toán Hay,5,Balkan,37,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,43,Benelux,13,Bình Định,39,Bình Dương,19,Bình Phước,37,Bình Thuận,30,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,12,Cần Thơ,13,Canada,39,Cao Bằng,6,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,308,Chu Tuấn Anh,1,Chuyên Đề,122,Chuyên Sư Phạm,30,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,603,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,25,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,51,Đắk Nông,5,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi HSG,1500,Đề Thi JMO,1,Điện Biên,7,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,46,Đồng Tháp,50,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,31,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. Polya,3,Gặp Gỡ Toán Học,25,Gauss,1,GDTX,3,Geometry,12,Gia Lai,24,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,25,Hà Nội,220,Hà Tĩnh,66,Hà Trung Kiên,1,Hải Dương,46,Hải Phòng,40,Hàn Quốc,5,Hậu Giang,4,Hậu Lộc,1,Hilbert,1,Hình Học,32,HKUST,6,Hòa Bình,12,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,7,HSG 10,91,HSG 11,78,HSG 12,523,HSG 9,373,HSG Cấp Trường,76,HSG Quốc Gia,97,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hùng Vương,2,Hưng Yên,28,Hương Sơn,1,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,24,IMO,51,India,45,Inequality,13,InMC,1,International,303,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,14,KHTN,49,Kiên Giang,61,Kim Liên,1,Kon Tum,17,Korea,5,Kvant,2,Kỷ Yếu,42,Lai Châu,4,Lâm Đồng,31,Lạng Sơn,18,Langlands,1,Lào Cai,11,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,1,Leibniz,1,Long An,41,Lớp 10,10,Lớp 10 Chuyên,430,Lớp 10 Không Chuyên,218,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lý Thánh Tông,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,10,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,9,MYM,74,MYTS,4,Nam Định,30,Nam Phi,1,National,249,Nesbitt,1,Newton,4,Nghệ An,48,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,8,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,3,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,38,Ninh Thuận,14,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,94,Olympic 10/3,3,Olympic 11,86,Olympic 12,28,Olympic 24/3,6,Olympic 27/4,19,Olympic 30/4,65,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,10,Olympic Toán,292,Olympic Toán Sơ Cấp,3,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,26,Phú Yên,24,Phùng Hồ Hải,1,Phương Trình Hàm,10,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,41,Putnam,25,Quảng Bình,39,Quảng Nam,28,Quảng Ngãi,31,Quảng Ninh,41,Quảng Trị,23,Riemann,1,RMM,12,RMO,24,Romania,36,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,68,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia,7,Scholze,1,Serbia,17,Sharygin,22,Shortlists,55,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,27,Sóc Trăng,27,Sơn La,11,Spain,8,Star Education,3,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,14,Tập San,4,Tây Ban Nha,1,Tây Ninh,25,Thạch Hà,1,Thái Bình,37,Thái Nguyên,33,Thái Vân,2,Thanh Hóa,54,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,7,Thừa Thiên Huế,34,Tiền Giang,18,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TP Hồ Chí Minh,112,Trà Vinh,5,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,33,Trại Hè Hùng Vương,24,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,9,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,12,Trường Đông,17,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,55,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Tập,44,Tuymaada,4,Undergraduate,64,USA,44,USAJMO,10,USATST,7,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,1,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,26,Vĩnh Long,18,Vĩnh Phúc,58,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,42,VNTST,20,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,25,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,16,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
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MOlympiad: [Shortlist] Junior Balkan Mathematical Olympiad 2008
[Shortlist] Junior Balkan Mathematical Olympiad 2008
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