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[Solutions] International Mathematical Olympiad 2018

  1. Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.
  2. Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.
  3. An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$. \[\begin{array}{ c@{\hspace{4pt}}c@{\hspace{4pt}} c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c } & & & 4 & & & \\ & & 2 & & 6 & & \\ & 5 & & 7 & & 1 & \\ 8 & & 3 & & 10 & & 9 \\ \end{array}\]Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?
  4. A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.
  5. Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.
  6. A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\]Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$.

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Ả-rập Xê-út,1,Abel,2,Albania,2,American Mathematical Monthly,2,AMM,1,Amsterdam,8,Ấn Độ,1,An Giang,16,Andrew Wiles,1,Anh,2,Áo,1,APMO,16,Arabia,1,Ba Lan,1,Bà Rịa Vũng Tàu,44,Bắc Bộ,23,Bắc Giang,40,Bạc Liêu,7,Bắc Ninh,34,Bắc Trung Bộ,8,Bài Toán Hay,3,Balkan,29,Baltic Way,29,BAMO,1,Bất Đẳng Thức,77,BDHSG,14,Bến Tre,21,Benelux,11,Bình Định,36,Bình Dương,18,Bình Phước,20,Bình Thuận,25,Birch,1,Bosnia Herzegovina,2,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Văn Tuyên,1,Bulgaria,5,BxMO,10,Cà Mau,12,Cần Thơ,12,Canada,63,Cao Bằng,5,Cao Quang Minh,1,Câu Chuyện Toán Học,30,Chọn Đội Tuyển,274,Chu Tuấn Anh,1,Chuyên Đề,104,Chuyên Sư Phạm,28,Collection,8,College Mathematics Journal,1,Concours,1,Cono Sur,1,Correspondence,1,Cosmin Poahata,1,CPS,4,Crux,2,Đà Nẵng,36,Đa Thức,2,Đại Số,31,Đắk Lắk,48,Đắk Nông,4,Đan Phượng,1,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1957,Đề Thi HSG,1108,Đề Thi JMO,1,Điện Biên,5,Định Lý,1,Định Lý Beaty,1,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,3,Đồng Nai,42,Đồng Tháp,40,Đức,1,E-Book,19,EGMO,12,ELMO,17,EMC,7,Estonian,5,Evan Chen,1,Fermat,3,Finland,4,G. Polya,3,Gặp Gỡ Toán Học,21,GDTX,3,Geometry,5,Gia Lai,20,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,21,Hà Nội,151,Hà Tĩnh,60,Hà Trung Kiên,1,Hải Dương,41,Hải Phòng,36,Hàn Quốc,4,Hậu Giang,3,Hilbert,1,Hình Học,49,HKUST,6,Hòa Bình,12,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,Hong Kong,1,HongKong,6,HSG 10,86,HSG 11,63,HSG 12,469,HSG 9,309,HSG Cấp Trường,64,HSG Quốc Gia,86,HSG Quốc Tế,13,Hứa Lâm Phong,1,Huế,30,Hùng Vương,25,Hưng Yên,24,Hy Lạp,1,IMC,23,IMO,40,India,37,Inequality,13,International,208,Iran,4,Jakob,1,JBMO,16,Journal,16,K2pi,1,Kazakhstan,1,Khánh Hòa,10,KHTN,46,Kiên Giang,26,Kon Tum,17,Kvant,2,Kỷ Yếu,37,Lai Châu,3,Lâm Đồng,20,Lạng Sơn,17,Langlands,1,Lào Cai,9,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,4,Lê Viết Hải,1,Lê Việt Hưng,1,Long An,33,Lớp 10,8,Lớp 10 Chuyên,342,Lớp 10 Không Chuyên,140,Lớp 11,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Macedonian,1,Malaysia,1,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today Magazine,1,MathProblems Journal,1,Mathscope,8,MEMO,9,Metropolises,3,Mexico,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,7,MYM,74,MYTS,1,Nam Định,26,Nam Phi,1,National,177,Nesbitt,1,Nghệ An,43,Ngô Bảo Châu,1,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,2,Nguyễn Đức Tấn,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,4,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,4,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,7,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,23,Nguyễn Văn Nho,1,Nguyễn Văn Quý,1,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,2,Nhóm Toán,3,Ninh Bình,36,Ninh Thuận,13,Nội Suy Lagrange,1,Nội Suy Newton,1,Nordic,18,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,87,Olympic 10/3,3,Olympic 11,79,Olympic 12,27,Olympic 24/3,6,Olympic 27/4,19,Olympic 30/4,56,Olympic KHTN,5,Olympic Sinh Viên,63,Olympic Toán,258,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,Philippine,1,Philippines,4,Phú Thọ,24,Phú Yên,21,Phùng Hồ Hải,1,Phương Trình Hàm,26,Phương Trình Pythagoras,1,Pi,1,Problems,1,PT-HPT,32,PTNK,37,Putnam,24,Quảng Bình,37,Quảng Nam,26,Quảng Ngãi,29,Quảng Ninh,32,Quảng Trị,17,Riemann,1,RMM,11,Romania,8,Romanian Mathematical Magazine,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,79,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi,2,Scholze,1,Serbia,17,Sharygin,19,Shortlists,35,Simon Singh,1,Singapore,1,Số học,38,Sóc Trăng,7,Sơn La,10,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,15,Tây Ban Nha,1,Tây Ninh,24,Thái Bình,33,Thái Nguyên,31,Thanh Hóa,46,THCS,2,Thổ Nhĩ Kỳ,4,Thomas J. Mildorf,1,THPTQG,11,THTT,7,Tiền Giang,16,Titu Andreescu,2,Tổ hợp,7,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,20,Toán Tuổi Thơ,2,TOT,1,TPHCM,99,Trà Vinh,5,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,32,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,8,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,1,Trần Quốc Luật,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,11,Trường Đông,16,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,44,Tuyên Quang,6,Tuyển sinh,10,Tuyển Tập,33,Tuymaada,1,Undergraduate,61,USA,28,USAJMO,1,USATST,5,Uzbekistan,1,Vasile Cîrtoaje,3,Viện Toán Học,1,Vietnam,2,Viktor Prasolov,1,VIMF,1,Vinh,23,Vĩnh Long,17,Vĩnh Phúc,55,Virginia Tech,1,VLTT,1,VMEO,4,VMF,8,VMO,38,VNTST,18,Võ Quốc Bá Cẩn,18,Võ Thành Văn,1,Vojtěch Jarník,5,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,15,Yên Định,1,Zhautykov,10,Zhou Yuan Zhe,1,
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Mathematical Olympiad Contests Collection: [Solutions] International Mathematical Olympiad 2018
[Solutions] International Mathematical Olympiad 2018
Mathematical Olympiad Contests Collection
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