## $hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 Hanoi Open Mathematics Competition (HOMC) was first organized in 2004 by the Hanoi Mathematical Society for Junior (Grade 8) and Senior (Grade 10) students. As the original regulation of HOMC, all questions, problems, and contestant’s presentation should be presented in English. From 2013 Hanoi Department of Education and Training became the co- organizer and promoted the competition as nationwide with nearly 1,000 contestants participated every year from 50 cities across the country. From 2018, with the support of the Hanoi People’s Committee, the competition was opened to international teams to participate by invitation only. The 15th Hanoi Open Mathematics Competition 2018 (HOMC), an annual event held by the Hanoi Mathematical Society, will see the participation of international contestants for the first time, announced the Hanoi Department of Education and Training on March 23. 87 international contestants from 10 countries, including China, Thailand, Myanmar, Indonesia, Poland, Hungary, and others will attend the competition. ### Junior Individual 1. Let$x$and$y$be real numbers satisfying the conditions$x + y = 4$and$xy = 3$. Compute the value of$(x - y)^2$. 2. Let$f(x)$be a polynomial such that$2f(x) + f(2 - x) = 5 + x$for any real number$x$. Find the value of$f(0) + f(2)$. 3. There are$3$unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings. 4. Find the number of distinct real roots of the following equation $$x^2 +\frac{9x^2}{(x + 3)^2} = 40.$$ 5. Let$ABC$be an acute triangle with$AB = 3$and$AC = 4$. Suppose that$AH$,$AO$and$AM$are the altitude, the bisector and the median derived from$A$, respectively. Calculate the length of$BC$if$HO = 3 MO$. 6. Nam spent$20$dollars for$20$stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost$3$dollars,$1.5$dollars and$0.5$dollar respectively. How many dollars did Nam spend for books? 7. Suppose that$ABCDE$is a convex pentagon with$\angle A = 90^\circ$,$\angle B = 105^\circ$,$\angle C = 90^\circ$and$AB = 2$,$BC = CD = DE =\sqrt2$. If the length of$AE$is$\sqrt{a }- b$where$a$,$b$are integers, what is the value of$a + b$? 8. Let$k$be a positive integer such that $$1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}.$$ Find the remainder when$k$is divided by$7$. 9. There are three polygons and the area of each one is$3$. They are drawn inside a square of area$6$. Find the greatest value of$m$such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than$m$. 10. Let$T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$where$x,y,z$are real numbers such that$1 \leq x,y,z \leq 4$and$x-y+z=4$. Find the smallest value of$10T$. 11. Find all pairs of nonnegative integers$(x, y)$for which$(xy + 2)^2 = x^2 + y^2 $. 12. Let$ABCD$be a rectangle with$45^\circ < \angle ADB < 60^\circ$. The diagonals$AC$and$ BD$intersect at$O$. A line passing through$O$and perpendicular to$BD$meets$AD$and$CD$at$M$and$N$respectively. Let$K$be a point on side$BC$such that$MK \parallel AC$. Show that$\angle MKN = 90^\circ$. 13. A competition room of HOMC has$m \times n$students where$m, n$are integers larger than$2$. Their seats are arranged in$m$rows and$n$columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally$27$handshakes. Find the number of students in the room. 14. Let$P(x)$be a polynomial with degree$2017$such that $$P(k) =\frac{k}{k + 1},\,\forall k = 0, 1, 2, ..., 2017.$$ Calculate$P(2018)$. 15. Find all pairs of prime numbers$(p,q)$such that for each pair$(p,q)$, there is a positive integer$m$satisfying $$\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}.$$ ### Junior Team 1. Let$a, b$, and$c$be distinct positive integers such that$a + 2b + 3c < 12$. Which of the following inequalities must be true? •$a + b + c < 7$•$a- b + c < 4$•$b + c- a < 3$•$a + b- c <5 $•$5a + 3b + c < 27$2. Let$ABCD$be a rectangle with$\angle ABD = 15^\circ$,$BD = 6cm$. Compute the area of the rectangle. 3. Consider all triples$(x,y,p)$of positive integers, where$p$is a prime number, such that $$4x^2 + 8y^2 + (2x-3y)p-12xy = 0.$$ Which below number is a perfect square number for every such triple$(x,y, p)$? 4. How many triangles are there for which the perimeters are equal to$30cm$and the lengths of sides are integers in centimeters? 5. Find all$3$-digit numbers$\overline{abc}$($a,b \ne 0$) such that$\overline{bcd} \times a = \overline{1a4d}$for some integer$d$from$1$to$9$6. In the below figure, there is a regular hexagon and three squares whose sides are equal to$4cm$. Let$M,N$, and$P$be the centers of the squares. The perimeter of the triangle$MNP$can be written in the form$a + b\sqrt3$(cm), where$a, b$are integers. Compute the value of$a + b$. 7. For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have? 8. Let$a,b, c$be real numbers with$a+b+c = 2018$. Suppose$x, y$, and$z$are the distinct positive real numbers which are satisfied$a = x^2 - yz - 2018, b = y^2 - zx - 2018$, and$c = z^2 - xy - 2018$. Compute the value of the following expression $$P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$$ 9. Each of the thirty squares in the diagram below contains a number$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to$S$. 10. The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of$3$levels. For the first and second levels, each has$10$doorways among which$3$doorways are located at the front,$3$at the back,$2$on the right side and$2$on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements. • The top level is painted with only one color. • In the second level, the$3$doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors. • For the first level, we apply the same rules as for the second level. a) In how many ways the first level can be painted?. b) In how many ways the whole tower model can be painted? ### Senior Individual 1. How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). 2. What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side$3$? 3. How many integers$nare there those satisfy the following inequality$$n^4 - n^3 - 3n^2 - 3^n - 17 < 0$$ 4. Let \begin{align}a &= (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3), \\ b &= (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3).\end{align} Which set does he differencea - b$belongs to?. 5. The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? 6. Write down all real numbers$(x, y)$satisfying two conditions$x^{2018} + y^2 = 2$and$x^2 + y^{2018} = 2$. 7. Let$\{u_n\}_ {n\ge 1}$be given sequence satisfying the conditions:$u_1 = 0$,$u_2 = 1$,$u_{n+1} = u_{n-1} + 2n - 1$for$n \ge 2$. a) Calculate$u_5$. b) Calculate$u_{100} + u_{101}$. 8. Let$P$be a point inside the square$ABCD$such that$\angle PAC = \angle PCD = 17^\circ$. Calculate$\angle APB$? 9. How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point. 10. There are$100$school students from two clubs$A$and$B$standing in circle. Among them$62$students stand next to at least one student from club$A$, and$54$students stand next to at least one student from club$B$. a) How many students stand side-by-side with one friend from club$A$and one friend from club$B$? b) What is the number of students from club$A$? 11. Find all positive integers$k$such that there exists a positive integer$n$, for which$2^n + 11$is divisible by$2^k - 1$. 12. Let$ABC$be an acute triangle with$AB < AC$, and let$BE$and$CF$be the altitudes. Let the median$AM$intersect$BE$at point$P$, and let line$CP$intersect$AB$at point$D$. Prove that$DE \parallel BC$, and$AC$is tangent to the circumcircle of$\vartriangle DEF$. 13. For a positive integer$n$, let$S(n), P(n)$denote the sum and the product of all the digits of$n$respectively. a) Find all values of n such that$n = P(n)$. b) Determine all values of n such that$n = S(n) + P(n)$. 14. Let$a,b, c$denote the real numbers such that$1 \le a, b, c\le 2$. Consider $$T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}.$$ Determine the largest possible value of$T$. 15. There are$n$distinct straight lines on a plane such that every line intersects exactly$12$others. Determine all the possible values of$n$. ### Senior Team 1. If$x$and$y$are positive real numbers such that $$(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018.$$ What is the minimum possible value of$x + y$?. 2. In triangle$ABC$,$\angle BAC = 60^\circ$,$AB = 3a$and$AC = 4a(a > 0)$. Let$M$be point on the segment$AB$such that$AM =\frac13 AB, N$be point on the side$AC$such that$AN =\frac12AC$. Let$I$be midpoint of$MN$. Determine the length of$BI$. 3. The lines$\ell_1$and \ell_2 are parallel. The points$A_1,A_2, ...,A_7$are on$\ell_1$and the points$B_1,B_2,...,B_8$are on$\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure). What is the greatest number of intersection points? 4. A pyramid of non-negative integers is constructed as follows • The first row consists of only$0$, • The second row consists of$1$and$1$, • The$n^{th}$(for$n > 2$) is an array of$n$integers among which the left most and right most elements are equal to$n - 1$and the interior numbers are equal to the sum of two adjacent numbers from the$(n - 1)^{th}$row. Let$S_n$be the sum of numbers in row$n^{th}$. Determine the remainder when dividing$S_{2018}$by$2018$. 1. Let$f$be a polynomial such that, for all real number$x$, $$f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019.$$ Compute$f(2018)$. 2. Three students$A$,$B$and$C$are traveling from a location on the National Highway No.$5$on direction to Hanoi for participating the HOMC$2018$. At beginning,$A$takes$B$on the motocycle, and at the same time$C$rides the bicycle. After one hour and a half,$B$switches to a bicycle and immediately continues the trip to Hanoi, while$A$returns to pick up$C$. Upon meeting,$C$continues the travel on the motocycle to Hanoi with$A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is$50$km per hour and of the both bicycles are$10$km per hour. Find the distance from the starting point to Hanoi. 3. Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of$2$. What is the maximal possible number written on the blackboard? 4. Let$ABCD$be rhombus, with$\angle ABC = 80^\circ$: Let$E$be midpoint of$BC$and$F$be perpendicular projection of$A$onto$DE$. Find the measure of$\angle DFC$in degree. 5. Let$ABC$be acute, non-isosceles triangle, inscribed in the circle$(O)$. Let$D$be perpendicular projection of$A$onto$BC$, and$E$,$F$be perpendicular projections of$D$onto$CA$,$AB$respectively. a) Prove that$AO \perp EF$. b) The line$AO$intersects$DE,DF$at$I,J$respectively. Prove that$\vartriangle DIJ$and$\vartriangle ABC$are similar. c) Prove that circumcenter of$\vartriangle DIJ$is equidistant from$B$and$C$6. The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of$3$levels. For the first and second levels, each has$10$doorways among which$3$doorways are located at the front,$3$at the back,$2$on the right side and$2$on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements • The top level is painted with only one color. • The$3$doorways at the front on the second level are painted with the same color. • The$3$doorways at the front on the first level are painted with the same color. Each of the remaining$14$doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. a) How many ways are there to paint the first level? b) How many ways are there to paint the entire tower model? ##$hide=mobile$type=ticker$c=36$cols=2$l=0$sr=random$b=0

Name

Abel,5,Albania,2,AMM,2,Amsterdam,4,An Giang,43,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Lan,1,Bà Rịa Vũng Tàu,75,Bắc Bộ,2,Bắc Giang,60,Bắc Kạn,4,Bạc Liêu,17,Bắc Ninh,59,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,41,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,70,Benelux,16,Bình Định,64,Bình Dương,37,Bình Phước,48,Bình Thuận,42,Birch,1,BMO,41,Booklet,12,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,British,16,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,2,BxMO,15,Cà Mau,21,Cần Thơ,25,Canada,40,Cao Bằng,12,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,492,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,29,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,675,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,28,Đà Nẵng,49,Đa Thức,2,Đại Số,20,Đắk Lắk,74,Đắk Nông,14,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2192,Đề Thi JMO,1,DHBB,28,Điện Biên,13,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,5,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đồng Nai,63,Đồng Tháp,62,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,28,E-Book,31,EGMO,30,ELMO,19,EMC,11,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,30,Gauss,1,GDTX,3,Geometry,14,GGTH,30,Gia Lai,39,Gia Viễn,2,Giải Tích Hàm,1,Giới hạn,2,Goldbach,1,Hà Giang,5,Hà Lan,1,Hà Nam,43,Hà Nội,249,Hà Tĩnh,93,Hà Trung Kiên,1,Hải Dương,66,Hải Phòng,56,Hậu Giang,13,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,32,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,124,HSG 10 2010-2011,4,HSG 10 2011-2012,7,HSG 10 2012-2013,6,HSG 10 20122-2023,2,HSG 10 2013-2014,5,HSG 10 2014-2015,6,HSG 10 2015-2016,2,HSG 10 2016-2017,6,HSG 10 2017-2018,4,HSG 10 2018-2019,4,HSG 10 2019-2020,8,HSG 10 2020-2021,2,HSG 10 2021-2022,3,HSG 10 2022-2023,9,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,HSG 10 Bắc Ninh,3,HSG 10 Bình Định,1,HSG 10 Bình Dương,1,HSG 10 Bình Thuận,4,HSG 10 Chuyên SPHN,5,HSG 10 Đắk Lắk,2,HSG 10 Đồng Nai,4,HSG 10 Gia Lai,2,HSG 10 Hà Nam,4,HSG 10 Hà Tĩnh,15,HSG 10 Hải Dương,10,HSG 10 KHTN,9,HSG 10 Kon Tum,1,HSG 10 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 PTNK,5,HSG 10 Quảng Nam,1,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,9,HSG 10 Trà Vinh,5,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,132,HSG 11 2010-2011,5,HSG 11 2011-2012,10,HSG 11 2012-2013,9,HSG 11 2013-2014,7,HSG 11 2014-2015,10,HSG 11 2015-2016,6,HSG 11 2016-2017,8,HSG 11 2017-2018,6,HSG 11 2018-2019,8,HSG 11 2019-2020,4,HSG 11 2020-2021,7,HSG 11 2021-2022,2,HSG 11 2022-2023,5,HSG 11 An Giang,2,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,2,HSG 11 Bắc Ninh,4,HSG 11 Bình Định,12,HSG 11 Bình Dương,3,HSG 11 Bình Thuận,1,HSG 11 Cà Mau,1,HSG 11 Đà Nẵng,9,HSG 11 Đồng Nai,1,HSG 11 Hà Nam,2,HSG 11 Hà Tĩnh,12,HSG 11 Hải Phòng,1,HSG 11 Kiên Giang,4,HSG 11 Lạng Sơn,11,HSG 11 Nghệ An,6,HSG 11 Ninh Bình,2,HSG 11 Quảng Bình,9,HSG 11 Quảng Nam,1,HSG 11 Quảng Ngãi,9,HSG 11 Quảng Trị,3,HSG 11 Sóc Trăng,1,HSG 11 Thái Nguyên,8,HSG 11 Thanh Hóa,3,HSG 11 Trà Vinh,1,HSG 11 Tuyên Quang,1,HSG 11 Vĩnh Long,3,HSG 11 Vĩnh Phúc,11,HSG 12,641,HSG 12 2009-2010,2,HSG 12 2010-2011,39,HSG 12 2011-2012,44,HSG 12 2012-2013,58,HSG 12 2013-2014,53,HSG 12 2014-2015,44,HSG 12 2015-2016,37,HSG 12 2016-2017,46,HSG 12 2017-2018,56,HSG 12 2018-2019,43,HSG 12 2019-2020,43,HSG 12 2020-2021,52,HSG 12 2021-2022,34,HSG 12 2022-2023,41,HSG 12 An Giang,8,HSG 12 Bà Rịa Vũng Tàu,12,HSG 12 Bắc Giang,17,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,HSG 12 Bình Định,16,HSG 12 Bình Dương,8,HSG 12 Bình Phước,8,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,7,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,10,HSG 12 Đà Nẵng,3,HSG 12 Đắk Lắk,21,HSG 12 Đắk Nông,1,HSG 12 Điện Biên,3,HSG 12 Đồng Nai,20,HSG 12 Đồng Tháp,18,HSG 12 Gia Lai,13,HSG 12 Hà Nam,4,HSG 12 Hà Nội,16,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,14,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,4,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,9,HSG 12 Khánh Hòa,3,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,2,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,11,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,17,HSG 12 Long An,18,HSG 12 Nam Định,7,HSG 12 Nghệ An,12,HSG 12 Ninh Bình,11,HSG 12 Ninh Thuận,7,HSG 12 Phú Thọ,17,HSG 12 Phú Yên,13,HSG 12 Quảng Bình,13,HSG 12 Quảng Nam,9,HSG 12 Quảng Ngãi,6,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,9,HSG 12 Sóc Trăng,4,HSG 12 Sơn La,5,HSG 12 Tây Ninh,6,HSG 12 Thái Bình,11,HSG 12 Thái Nguyên,12,HSG 12 Thanh Hóa,16,HSG 12 Thừa Thiên Huế,18,HSG 12 Tiền Giang,3,HSG 12 TPHCM,13,HSG 12 Tuyên Quang,2,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,19,HSG 12 Yên Bái,6,HSG 9,570,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,42,HSG 9 2012-2013,42,HSG 9 2013-2014,36,HSG 9 2014-2015,41,HSG 9 2015-2016,39,HSG 9 2016-2017,43,HSG 9 2017-2018,45,HSG 9 2018-2019,43,HSG 9 2019-2020,18,HSG 9 2020-2021,51,HSG 9 2021-2022,55,HSG 9 2022-2023,55,HSG 9 An Giang,9,HSG 9 Bà Rịa Vũng Tàu,8,HSG 9 Bắc Giang,13,HSG 9 Bắc Kạn,1,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,13,HSG 9 Bến Tre,10,HSG 9 Bình Định,11,HSG 9 Bình Dương,7,HSG 9 Bình Phước,13,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,2,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,2,HSG 9 Đà Nẵng,11,HSG 9 Đắk Lắk,12,HSG 9 Đắk Nông,3,HSG 9 Điện Biên,4,HSG 9 Đồng Nai,8,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,9,HSG 9 Hà Giang,4,HSG 9 Hà Nam,10,HSG 9 Hà Nội,14,HSG 9 Hà Tĩnh,17,HSG 9 Hải Dương,15,HSG 9 Hải Phòng,8,HSG 9 Hậu Giang,5,HSG 9 Hòa Bình,4,HSG 9 Hưng Yên,11,HSG 9 Khánh Hòa,6,HSG 9 Kiên Giang,16,HSG 9 Kon Tum,9,HSG 9 Lai Châu,2,HSG 9 Lâm Đồng,14,HSG 9 Lạng Sơn,10,HSG 9 Lào Cai,4,HSG 9 Long An,10,HSG 9 Nam Định,9,HSG 9 Nghệ An,21,HSG 9 Ninh Bình,14,HSG 9 Ninh Thuận,4,HSG 9 Phú Thọ,13,HSG 9 Phú Yên,9,HSG 9 Quảng Bình,14,HSG 9 Quảng Nam,12,HSG 9 Quảng Ngãi,13,HSG 9 Quảng Ninh,16,HSG 9 Quảng Trị,10,HSG 9 Sóc Trăng,9,HSG 9 Sơn La,5,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,10,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,11,HSG 9 Thừa Thiên Huế,9,HSG 9 Tiền Giang,7,HSG 9 TPHCM,11,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,6,HSG 9 Vĩnh Long,12,HSG 9 Vĩnh Phúc,11,HSG 9 Yên Bái,5,HSG Cấp Trường,80,HSG Quốc Gia,111,HSG Quốc Tế,16,HSG11 2021-2022,3,HSG11 2022-2023,1,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,41,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,57,IMT,2,IMU,2,India - Ấn Độ,47,Inequality,13,InMC,1,International,349,Iran,13,Jakob,1,JBMO,41,Jewish,1,Journal,30,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,29,KHTN,62,Kiên Giang,73,Kon Tum,24,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,12,Lâm Đồng,46,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,36,Langlands,1,Lào Cai,34,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Hồng Phong,5,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Viết Hải,1,Lê Việt Hưng,2,Leibniz,1,Long An,51,Lớp 10 Chuyên,678,Lớp 10 Không Chuyên,349,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lưu Lý Tưởng,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,13,Menelaus,1,Metropolises,4,Mexico,1,MIC,1,Michael Atiyah,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,MYM,25,MYTS,4,Nam Định,45,Nam Phi,1,National,276,Nesbitt,1,Newton,4,Nghệ An,71,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 Minh Hà,1,Nguyễn Minh Tuấn,9,Nguyễn Nhất Huy,1,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,2,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Song Thiên Long,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,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,59,Ninh Thuận,26,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,130,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,117,Olympic 12,50,Olympic 23/3,2,Olympic 24/3,10,Olympic 24/3 Quảng Nam,10,Olympic 27/4,24,Olympic 30/4,60,Olympic KHTN,8,Olympic Sinh Viên,78,Olympic Tháng 4,12,Olympic Toán,337,Olympic Toán Sơ Cấp,3,Ôn Thi 10,2,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 Quang Đạt,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,32,Phú Yên,41,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,57,Putnam,27,Quảng Bình,59,Quảng Nam,54,Quảng Ngãi,47,Quảng Ninh,57,Quảng Trị,39,Quỹ Tích,1,Riemann,1,RMM,14,RMO,24,Romania,38,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,70,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia - Ả Rập Xê Út,9,Scholze,1,Serbia,17,Sharygin,28,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,28,Sóc Trăng,34,Sơn La,22,Spain,8,Star Education,1,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,17,Tập San,3,Tây Ban Nha,1,Tây Ninh,36,Thạch Hà,1,Thái Bình,43,Thái Nguyên,58,Thái Vân,2,Thanh Hóa,66,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,Thông Tin Toán Học,43,THPT Chuyên Lê Quý Đôn,1,THPT Chuyên Nguyễn Du,9,THPTQG,16,THTT,31,Thừa Thiên Huế,55,Tiền Giang,29,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,151,Trà Vinh,9,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,37,Trại Hè Hùng Vương,28,Trại Hè Phương Nam,7,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,12,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,Trường Đông,21,Trường Hè,8,Trường Thu,1,Trường Xuân,3,TST,521,TST 2008-2009,1,TST 2010-2011,22,TST 2011-2012,23,TST 2012-2013,32,TST 2013-2014,29,TST 2014-2015,27,TST 2015-2016,26,TST 2016-2017,41,TST 2017-2018,42,TST 2018-2019,30,TST 2019-2020,35,TST 2020-2021,30,TST 2021-2022,37,TST 2022-2023,42,TST An Giang,7,TST Bà Rịa Vũng Tàu,11,TST Bắc Giang,5,TST Bắc Ninh,11,TST Bến Tre,8,TST Bình Định,5,TST Bình Dương,6,TST Bình Phước,8,TST Bình Thuận,9,TST Cà Mau,6,TST Cần Thơ,5,TST Cao Bằng,2,TST Đà Nẵng,8,TST Đắk Lắk,11,TST Đắk Nông,2,TST Điện Biên,2,TST Đồng Nai,12,TST Đồng Tháp,12,TST Gia Lai,4,TST Hà Nam,7,TST Hà Nội,11,TST Hà Tĩnh,14,TST Hải Dương,11,TST Hải Phòng,13,TST Hậu Giang,1,TST Hòa Bình,3,TST Hưng Yên,9,TST Khánh Hòa,8,TST Kiên Giang,10,TST Kon Tum,6,TST Lâm Đồng,11,TST Lạng Sơn,2,TST Lào Cai,4,TST Long An,6,TST Nam Định,8,TST Nghệ An,7,TST Ninh Bình,11,TST Ninh Thuận,4,TST Phú Thọ,14,TST Phú Yên,5,TST PTNK,14,TST Quảng Bình,12,TST Quảng Nam,6,TST Quảng Ngãi,7,TST Quảng Ninh,8,TST Quảng Trị,9,TST Sóc Trăng,5,TST Sơn La,7,TST Thái Bình,6,TST Thái Nguyên,8,TST Thanh Hóa,9,TST Thừa Thiên Huế,4,TST Tiền Giang,5,TST TPHCM,14,TST Trà Vinh,1,TST Tuyên Quang,1,TST Vĩnh Long,6,TST Vĩnh Phúc,7,TST Yên Bái,8,Tuyên Quang,13,Tuyển Sinh,4,Tuyển Sinh 10,1027,Tuyển Sinh 10 An Giang,17,Tuyển Sinh 10 Bà Rịa Vũng Tàu,21,Tuyển Sinh 10 Bắc Giang,19,Tuyển Sinh 10 Bắc Kạn,3,Tuyển Sinh 10 Bạc Liêu,9,Tuyển Sinh 10 Bắc Ninh,15,Tuyển Sinh 10 Bến Tre,33,Tuyển Sinh 10 Bình Định,19,Tuyển Sinh 10 Bình Dương,12,Tuyển Sinh 10 Bình Phước,19,Tuyển Sinh 10 Bình Thuận,15,Tuyển Sinh 10 Cà Mau,5,Tuyển Sinh 10 Cần Thơ,9,Tuyển Sinh 10 Cao Bằng,2,Tuyển Sinh 10 Chuyên SPHN,19,Tuyển Sinh 10 Đà Nẵng,17,Tuyển Sinh 10 Đắk Lắk,20,Tuyển Sinh 10 Đắk Nông,6,Tuyển Sinh 10 Điện Biên,4,Tuyển Sinh 10 Đồng Nai,18,Tuyển Sinh 10 Đồng Tháp,22,Tuyển Sinh 10 Gia Lai,10,Tuyển Sinh 10 Hà Giang,1,Tuyển Sinh 10 Hà Nam,16,Tuyển Sinh 10 Hà Nội,80,Tuyển Sinh 10 Hà Tĩnh,18,Tuyển Sinh 10 Hải Dương,16,Tuyển Sinh 10 Hải Phòng,14,Tuyển Sinh 10 Hậu Giang,3,Tuyển Sinh 10 Hòa Bình,15,Tuyển Sinh 10 Hưng Yên,12,Tuyển Sinh 10 Khánh Hòa,12,Tuyển Sinh 10 KHTN,19,Tuyển Sinh 10 Kiên Giang,31,Tuyển Sinh 10 Kon Tum,6,Tuyển Sinh 10 Lai Châu,6,Tuyển Sinh 10 Lâm Đồng,10,Tuyển Sinh 10 Lạng Sơn,6,Tuyển Sinh 10 Lào Cai,9,Tuyển Sinh 10 Long An,17,Tuyển Sinh 10 Nam Định,21,Tuyển Sinh 10 Nghệ An,22,Tuyển Sinh 10 Ninh Bình,19,Tuyển Sinh 10 Ninh Thuận,10,Tuyển Sinh 10 Phú Thọ,17,Tuyển Sinh 10 Phú Yên,11,Tuyển Sinh 10 PTNK,36,Tuyển Sinh 10 Quảng Bình,11,Tuyển Sinh 10 Quảng Nam,15,Tuyển Sinh 10 Quảng Ngãi,12,Tuyển Sinh 10 Quảng Ninh,11,Tuyển Sinh 10 Quảng Trị,6,Tuyển Sinh 10 Sóc Trăng,15,Tuyển Sinh 10 Sơn La,5,Tuyển Sinh 10 Tây Ninh,14,Tuyển Sinh 10 Thái Bình,16,Tuyển Sinh 10 Thái Nguyên,16,Tuyển Sinh 10 Thanh Hóa,26,Tuyển Sinh 10 Thừa Thiên Huế,24,Tuyển Sinh 10 Tiền Giang,14,Tuyển Sinh 10 TPHCM,23,Tuyển Sinh 10 Tuyên Quang,3,Tuyển Sinh 10 Vĩnh Long,12,Tuyển Sinh 10 Vĩnh Phúc,21,Tuyển Sinh 2008-2009,1,Tuyển Sinh 2009-2010,1,Tuyển Sinh 2010-2011,6,Tuyển Sinh 2011-2012,20,Tuyển Sinh 2012-2013,63,Tuyển Sinh 2013-2014,78,Tuyển Sinh 2014-2015,79,Tuyển Sinh 2015-2016,61,Tuyển Sinh 2016-2017,72,Tuyển Sinh 2017-2018,126,Tuyển Sinh 2018-2019,60,Tuyển Sinh 2019-2020,90,Tuyển Sinh 2020-2021,59,Tuyển Sinh 2021-202,1,Tuyển Sinh 2021-2022,70,Tuyển Sinh 2022-2023,114,Tuyển Sinh 2023-2024,12,Tuyển Sinh Chuyên SPHCM,7,Tuyển Sinh Yên Bái,6,Tuyển Tập,45,Tuymaada,6,UK - Anh,16,Undergraduate,69,USA - Mỹ,62,USA TSTST,6,USAJMO,12,USATST,8,USEMO,4,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,5,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,31,Vĩnh Long,40,Vĩnh Phúc,83,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,55,VNTST,24,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Xác Suất,1,Yên Bái,25,Yên Thành,1,Zhautykov,14,Zhou Yuan Zhe,1,
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MOlympiad.NET: [Solutions] Hanoi Open Mathematics Competition 2018
[Solutions] Hanoi Open Mathematics Competition 2018