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MathLinks Contest 2008

  1. Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously. Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$. 
  2. Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square. 
  3. We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n - 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called nice, if $ f_i(x_i) = f_i(x_{i + 1})$, for each $ i = 1,2,\dots,n - 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i = 1}^{n - 1} |A_i|}.\]
  4. Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1} + p_{2} + \ldots + p_{k} = 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)$, $ \left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)$, $ \ldots$, $ \left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)$ of $ k$-tuples of finite sets satisfying the following two properties
    • for every $ i$ and every $ j \neq j^{\prime}$, $ A_{i,j}\cap A_{i,j^{\prime}} = \emptyset$, and 
    • for every $ i\neq i^{\prime}$ there exist $ j\neq j^{\prime}$ for which $ A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset$.
    Prove that \[ \sum_{b = 1}^{m}{\prod_{a = 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1.\]
  5. For a prime $ p$ an a positive integer $ n$, denote by $ \nu_p(n)$ the exponent of $ p$ in the prime factorization of $ n!$. Given a positive integer $ d$ and a finite set $ \{p_1,p_2,\ldots, p_k\}$ of primes, show that there are infinitely many positive integers $ n$ such that $ \nu_{p_i}(n) \equiv 0 \pmod d$, for all $ 1\leq i \leq k$. 
  6. Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$. Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points. 
  7. Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that \[ \sum_{k = 0}^{p - 1}{\binom{2k}{k + d}}\equiv r \pmod{p},\] where $ r \equiv p-d \pmod 3$, $ r\in\{-1,0,1\}$. 
  8. Prove that for positive integers $ x,y,z$ the number $ x^2 + y^2 + z^2$ is not divisible by $ 3(xy + yz + zx)$. 
  9. Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$ \[ \frac ab + \frac bc + \frac ca - 3 \geq k \left( \frac a{b + c} + \frac b{c + a} + \frac c{a + b} - \frac 32 \right).\]
  10. Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB+BC+CA + DE < AD + AE + BD+BE + CD+CE .\]
  11. Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n+1}\}$, formed with nonnegative integers, for which $ a_1=a_{2n+1}=0$ and $ |a_k -a_{k+1}|=1$, for all $ k\in\{1,2,\ldots,2n\}$. 
  12. Let $ a,b,c$ be positive real numbers such that $ ab+bc+ca=3$. Prove that \[ \frac 1{1+a^2(b+c)} + \frac 1{1+b^2(c+a)} + \frac 1 {1+c^2(a+b) } \leq \frac 3 {1+2abc} .\]
  13. Find all real polynomials $ g(x)$ of degree at most $ n - 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) = x^n + nx^{n - 1} + \frac {n(n - 1)}2 x^{n - 2} + g(x)$ are real. 
  14. Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$. (If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$.) 
  15. If $ a\geq b\geq c\geq d > 0$ such that $ abcd=1$, then prove that \[ \frac 1{1+a} + \frac 1{1+b} + \frac 1{1+c} \geq \frac {3}{1+\sqrt[3]{abc}}.\]
  16. Let $ \{x_n\}_{n\geq 1}$ be a sequences, given by $ x_1 = 1$, $ x_2 = 2$ and \[ x_{n + 2} = \frac { x_{n + 1}^2 + 3 }{x_n}.\] Prove that $ x_{2008}$ is the sum of two perfect squares. 
  17. Find all functions $ f,g: \mathbb Q \to \mathbb Q$ such that for all rational numbers $ x,y$ we have \[ f(f(x) + g(y) ) = g(f(x)) + y.\]
  18. Let $ \Omega$ be the circumcircle of triangle $ ABC$. Let $ D$ be the point at which the incircle of $ ABC$ touches its side $ BC$. Let $ M$ be the point on $ \Omega$ such that the line $ AM$ is parallel to $ BC$. Also, let $ P$ be the point at which the circle tangent to the segments $ AB$ and $ AC$ and to the circle $ \Omega$ touches $ \Omega$. Prove that the points $ P$, $ D$, $ M$ are collinear. 
  19. Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}
  20. Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points. 
  21. Let $ n$ be a positive integer, and let $ M = \{1,2,\ldots, 2n\}$. Find the minimal positive integer $ m$, such that no matter how we choose the subsets $ A_i \subset M$, $ 1\leq i\leq m$, with the properties
    • $ |A_i-A_j|\geq 1$, for all $ i\neq j$, (2) $ \bigcup_{i=1}^m A_i = M$,
    • we can always find two subsets $ A_k$ and $ A_l$ such that $ A_k \cup A_l = M$ (here $ |X|$ represents the number of elements in the set $ X$.)

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Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,3,Amsterdam,5,Ấn Độ,2,An Giang,23,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,52,Bắc Giang,50,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,48,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,38,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,14,Bình Định,45,Bình Dương,23,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,6,Buôn Ma Thuột,1,BxMO,13,Cà Mau,14,Cần Thơ,14,Canada,40,Cao Bằng,7,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,353,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,618,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,56,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,1769,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,52,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,17,ELMO,19,EMC,9,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: MathLinks Contest 2008
MathLinks Contest 2008
MOlympiad
https://www.molympiad.net/2020/08/mathlinks-contest-2008.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2020/08/mathlinks-contest-2008.html
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