## $hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 # ĐẶT MUA TẠP CHÍ / PURCHASE JOURNALS ### Algebra 1. Let$\mathbb{Z}$be the set of integers. Determine all functions$f: \mathbb{Z} \rightarrow \mathbb{Z}$such that, for all integers$a$and$b$, $$f(2a)+2f(b)=f(f(a+b)).$$ 2. Let$u_1, u_2, \dots, u_{2019}$be real numbers satisfying $u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.$Let$a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$and$b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that $a b \leqslant-\frac{1}{2019}.$ 3. Let$n \geqslant 3$be a positive integer and let$\left(a_{1}, a_{2}, \ldots, a_{n}\right)$be a strictly increasing sequence of$n$positive real numbers with sum equal to 2. Let$X$be a subset of$\{1,2, \ldots, n\}$such that the value of $\left|1-\sum_{i \in X} a_{i}\right|$is minimised. Prove that there exists a strictly increasing sequence of$n$positive real numbers$\left(b_{1}, b_{2}, \ldots, b_{n}\right)$with sum equal to 2 such that $\sum_{i \in X} b_{i}=1.$ 4. Let$n\geqslant 2$be a positive integer and$a_1,a_2, \ldots ,a_n$be real numbers such that $a_1+a_2+\dots+a_n=0.$Define the set$A$by $A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}$Prove that, if$A$is not empty, then $\sum_{(i, j) \in A} a_{i} a_{j}<0.$ 5. Let$x_1, x_2, \dots, x_n$be different real numbers. Prove that $\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.$ 6. A polynomial$P(x, y, z)$in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial$F(t)$in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$ 7. Let$\mathbb Z$be the set of integers. We consider functions$f :\mathbb Z\to\mathbb Z$satisfying $f\left(f(x+y)+y\right)=f\left(f(x)+y\right)$for all integers$x$and$y$. For such a function, we say that an integer$v$is f-rare if the set $X_v=\{x\in\mathbb Z:f(x)=v\}$is finite and nonempty. a) Prove that there exists such a function$f$for which there is an$f$-rare integer. b) Prove that no such function$f$can have more than one$f$-rare integer. ### Combinatorics 1. The infinite sequence$a_0,a _1, a_2, \dots$of (not necessarily distinct) integers has the following properties:$0\le a_i \le i$for all integers$i\ge 0$, and $\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k$for all integers$k\ge 0$. Prove that all integers$N\ge 0$occur in the sequence (that is, for all$N\ge 0$, there exists$i\ge 0$with$a_i=N$). 2. You are given a set of$n$blocks, each weighing at least$1$; their total weight is$2n$. Prove that for every real number$r$with$0 \leq r \leq 2n-2$you can choose a subset of the blocks whose total weight is at least$r$but at most$r + 2$. 3. The Bank of Bath issues coins with an$H$on one side and a$T$on the other. Harry has$n$of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly$k>0$coins showing$H$, then he turns over the$k$th coin from the left; otherwise, all coins show$T$and he stops. For example, if$n=3$the process starting with the configuration$THT$would be$THT \to HHT  \to HTT \to TTT$, which stops after three operations. a) Show that, for each initial configuration, Harry stops after a finite number of operations. b) For each initial configuration$C$, let$L(C)$be the number of operations before Harry stops. For example,$L(THT) = 3$and$L(TTT) = 0$. Determine the average value of$L(C)$over all$2^n$possible initial configurations$C$. 4. On a flat plane in Camelot, King Arthur builds a labyrinth$\mathfrak{L}$consisting of$n$walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let$k(\mathfrak{L})$be the largest number$k$such that, no matter how Merlin paints the labyrinth$\mathfrak{L},$Morgana can always place at least$k$knights such that no two of them can ever meet. For each$n,$what are all possible values for$k(\mathfrak{L}),$where$\mathfrak{L}$is a labyrinth with$n$walls? 5. A social network has$2019$users, some pairs of whom are friends. Whenever user$A$is friends with user$B$, user$B$is also friends with user$A$. Events of the following kind may happen repeatedly, one at a time: Three users$A$,$B$, and$C$such that$A$is friends with both$B$and$C$, but$B$and$C$are not friends, change their friendship statuses such that$B$and$C$are now friends, but$A$is no longer friends with$B$, and no longer friends with$C$. All other friendship statuses are unchanged. Initially,$1010$users have$1009$friends each, and$1009$users have$1010$friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. 6. Let$n>1$be an integer. Suppose we are given$2n$points in the plane such that no three of them are collinear. The points are to be labelled$A_1, A_2, \dots , A_{2n}$in some order. We then consider the$2n$angles $$\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2.$$ We measure each angle in the way that gives the smallest positive value (i.e. between$0^{\circ}$and$180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting$2n$angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group. 7. There are 60 empty boxes$B_1,\ldots,B_{60}$in a row on a table and an unlimited supply of pebbles. Given a positive integer$n$, Alice and Bob play the following game. In the first round, Alice takes$n$pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps a) Bob chooses an integer$k$with$1\leq k\leq 59$and splits the boxes into the two groups$B_1,\ldots,B_k$and$B_{k+1},\ldots,B_{60}$. b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest$n$such that Alice can prevent Bob from winning. 8. Alice has a map of Wonderland, a country consisting of$n \geq 2$towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most$4n$questions. 9. For any two different real numbers$x$and$y$, we define$D(x,y)$to be the unique integer$d$satisfying$2^d\le |x-y| < 2^{d+1}$. Given a set of reals$\mathcal F$, and an element$x\in \mathcal F$, we say that the scales of$x$in$\mathcal F$are the values of$D(x,y)$for$y\in\mathcal F$with$x\neq y$. Let$k$be a given positive integer. Suppose that each member$x$of$\mathcal F$has at most$k$different scales in$\mathcal F$(note that these scales may depend on$x$). What is the maximum possible size of$\mathcal F$? ### Geometry 1. Let$ABC$be a triangle. Circle$\Gamma$passes through$A$, meets segments$AB$and$AC$again at points$D$and$E$respectively, and intersects segment$BC$at$F$and$G$such that$F$lies between$B$and$G$. The tangent to circle$BDF$at$F$and the tangent to circle$CEG$at$G$meet at point$T$. Suppose that points$A$and$T$are distinct. Prove that line$AT$is parallel to$BC$. 2. Let$ABC$be an acute-angled triangle and let$D, E$, and$F$be the feet of altitudes from$A, B$, and$C$to sides$BC, CA$, and$AB$, respectively. Denote by$\omega_B$and$\omega_C$the incircles of triangles$BDF$and$CDE$, and let these circles be tangent to segments$DF$and$DE$at$M$and$N$, respectively. Let line$MN$meet circles$\omega_B$and$\omega_C$again at$P \ne M$and$Q \ne N$, respectively. Prove that$MP = NQ$. 3. In triangle$ABC$, point$A_1$lies on side$BC$and point$B_1$lies on side$AC$. Let$P$and$Q$be points on segments$AA_1$and$BB_1$, respectively, such that$PQ$is parallel to$AB$. Let$P_1$be a point on line$PB_1$, such that$B_1$lies strictly between$P$and$P_1$, and$\angle PP_1C=\angle BAC$. Similarly, let$Q_1$be the point on line$QA_1$, such that$A_1$lies strictly between$Q$and$Q_1$, and$\angle CQ_1Q=\angle CBA$. Prove that points$P,Q,P_1$, and$Q_1$are concyclic. 4. Let$P$be a point inside triangle$ABC$. Let$AP$meet$BC$at$A_1$, let$BP$meet$CA$at$B_1$, and let$CP$meet$AB$at$C_1$. Let$A_2$be the point such that$A_1$is the midpoint of$PA_2$, let$B_2$be the point such that$B_1$is the midpoint of$PB_2$, and let$C_2$be the point such that$C_1$is the midpoint of$PC_2$. Prove that points$A_2, B_2$, and$C_2$cannot all lie strictly inside the circumcircle of triangle$ABC$. 5. Let$ABCDE$be a convex pentagon with$CD= DE$and$\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point$P$is located in the interior of the pentagon such that$AP =AE$and$BP= BC$. Prove that$P$lies on the diagonal$CE$if and only if area$(BCD)$+ area$(ADE)$= area$(ABD)$+ area$(ABP)$. 6. Let$I$be the incentre of acute-angled triangle$ABC$. Let the incircle meet$BC, CA$, and$AB$at$D, E$, and$F,$respectively. Let line$EF$intersect the circumcircle of the triangle at$P$and$Q$, such that$F$lies between$E$and$P$. Prove that$\angle DPA + \angle AQD =\angle QIP$. 7. Let$I$be the incentre of acute triangle$ABC$with$AB\neq AC$. The incircle$\omega$of$ABC$is tangent to sides$BC, CA$, and$AB$at$D, E,$and$F$, respectively. The line through$D$perpendicular to$EF$meets$\omega$at$R$. Line$AR$meets$\omega$again at$P$. The circumcircles of triangle$PCE$and$PBF$meet again at$Q$. Prove that lines$DI$and$PQ$meet on the line through$A$perpendicular to$AI$. 8. Let$\mathcal L$be the set of all lines in the plane and let$f$be a function that assigns to each line$\ell\in\mathcal L$a point$f(\ell)$on$f(\ell)$. Suppose that for any point$X$, and for any three lines$\ell_1,\ell_2,\ell_3$passing through$X$, the points$f(\ell_1),f(\ell_2),f(\ell_3)$, and$X$lie on a circle. Prove that there is a unique point$P$such that$f(\ell)=P$for any line$\ell$passing through$P$. ### Number Theory 1. Find all pairs$(k,n)$of positive integers such that $k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}).$ 2. Find all triples$(a, b, c)$of positive integers such that$a^3 + b^3 + c^3 = (abc)^2$. 3. We say that a set$S$of integers is rootiful if, for any positive integer$n$and any$a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial$a_0+a_1x+\cdots+a_nx^n$are also in$S$. Find all rootiful sets of integers that contain all numbers of the form$2^a - 2^b$for positive integers$a$and$b$. 4. Find all functions$f:\mathbb Z_{>0}\to \mathbb Z_{>0}$such that$a+f(b)$divides$a^2+bf(a)$for all positive integers$a$and$b$with$a+b>2019$. 5. Let$a$be a positive integer. We say that a positive integer$b$is$a$-good if$\tbinom{an}{b}-1$is divisible by$an+1$for all positive integers$n$with$an \geq b$. Suppose$b$is a positive integer such that$b$is$a$-good, but$b+2$is not$a$-good. Prove that$b+1$is prime. 6. Let$H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$and let$n$be a positive integer. Prove that there exists a constant$C$such that, if$A\subseteq \{1,2,\dots, n\}$satisfies$|A| \ge C\sqrt{n}$, then there exist$a,b\in A$such that$a-b\in H$. (Here$\mathbb Z_{>0}$is the set of positive integers, and$\lfloor z\rfloor$denotes the greatest integer less than or equal to$z$.) 7. Prove that there is a constant$c>0$and infinitely many positive integers$n$with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than$cn\log(n)$pairwise coprime$n$th powers. 8. Let$a$and$b$be two positive integers. Prove that the integer $a^2+\left\lceil\frac{4a^2}b\right\rceil$ is not a square. (Here$\lceil z\rceil$denotes the least integer greater than or equal to$z$.) ##$hide=mobile$type=ticker$c=36$cols=2$l=0$sr=random$b=0

Name

Abel,5,Albania,2,AMM,2,Amsterdam,4,An Giang,45,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Lan,1,Bà Rịa Vũng Tàu,77,Bắc Bộ,2,Bắc Giang,62,Bắc Kạn,4,Bạc Liêu,18,Bắc Ninh,53,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,41,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,72,Benelux,16,Bình Định,65,Bình Dương,38,Bình Phước,52,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,22,Cần Thơ,27,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,515,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,30,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,50,Đa Thức,2,Đại Số,20,Đắk Lắk,76,Đắk Nông,15,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2249,Đề Thi JMO,1,DHBB,30,Điện Biên,15,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,64,Đồng Tháp,63,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,30,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,40,Gia Viễn,2,Giải Tích Hàm,1,Giới hạn,2,Goldbach,1,Hà Giang,5,Hà Lan,1,Hà Nam,45,Hà Nội,255,Hà Tĩnh,91,Hà Trung Kiên,1,Hải Dương,70,Hải Phòng,57,Hậu Giang,14,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,33,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,126,HSG 10 2010-2011,4,HSG 10 2011-2012,7,HSG 10 2012-2013,8,HSG 10 2013-2014,7,HSG 10 2014-2015,6,HSG 10 2015-2016,2,HSG 10 2016-2017,8,HSG 10 2017-2018,4,HSG 10 2018-2019,4,HSG 10 2019-2020,7,HSG 10 2020-2021,3,HSG 10 2021-2022,4,HSG 10 2022-2023,11,HSG 10 2023-2024,1,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,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 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 PTNK,10,HSG 10 Quảng Nam,1,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,9,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,135,HSG 11 2009-2010,1,HSG 11 2010-2011,6,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 2017-2018,7,HSG 11 2018-2019,8,HSG 11 2019-2020,5,HSG 11 2020-2021,8,HSG 11 2021-2022,4,HSG 11 2022-2023,7,HSG 11 2023-2024,1,HSG 11 An Giang,2,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,3,HSG 11 Bắc Ninh,2,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,12,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 12,668,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 2017-2018,55,HSG 12 2018-2019,43,HSG 12 2019-2020,43,HSG 12 2020-2021,52,HSG 12 2021-2022,35,HSG 12 2022-2023,42,HSG 12 2023-2024,23,HSG 12 2023-2041,1,HSG 12 An Giang,8,HSG 12 Bà Rịa Vũng Tàu,13,HSG 12 Bắc Giang,18,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,HSG 12 Bình Định,17,HSG 12 Bình Dương,8,HSG 12 Bình Phước,9,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,7,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,11,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,14,HSG 12 Hà Nam,5,HSG 12 Hà Nội,17,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,16,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,4,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,10,HSG 12 Khánh Hòa,4,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,3,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,13,HSG 12 Ninh Bình,12,HSG 12 Ninh Thuận,7,HSG 12 Phú Thọ,18,HSG 12 Phú Yên,13,HSG 12 Quảng Bình,14,HSG 12 Quảng Nam,11,HSG 12 Quảng Ngãi,6,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,10,HSG 12 Sóc Trăng,4,HSG 12 Sơn La,5,HSG 12 Tây Ninh,6,HSG 12 Thái Bình,11,HSG 12 Thái Nguyên,13,HSG 12 Thanh Hóa,17,HSG 12 Thừa Thiên Huế,19,HSG 12 Tiền Giang,3,HSG 12 TPHCM,13,HSG 12 Tuyên Quang,3,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,20,HSG 12 Yên Bái,6,HSG 9,573,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,42,HSG 9 2012-2013,41,HSG 9 2013-2014,35,HSG 9 2014-2015,41,HSG 9 2015-2016,38,HSG 9 2016-2017,42,HSG 9 2017-2018,45,HSG 9 2018-2019,41,HSG 9 2019-2020,18,HSG 9 2020-2021,50,HSG 9 2021-2022,53,HSG 9 2022-2023,55,HSG 9 2023-2024,15,HSG 9 An Giang,9,HSG 9 Bà Rịa Vũng Tàu,8,HSG 9 Bắc Giang,14,HSG 9 Bắc Kạn,1,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,12,HSG 9 Bến Tre,9,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,5,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,15,HSG 9 Hà Tĩnh,13,HSG 9 Hải Dương,16,HSG 9 Hải Phòng,8,HSG 9 Hậu Giang,6,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,17,HSG 9 Quảng Trị,10,HSG 9 Sóc Trăng,9,HSG 9 Sơn La,5,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,11,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,12,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,12,HSG 9 Yên Bái,5,HSG Cấp Trường,80,HSG Quốc Gia,113,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,43,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,58,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,30,KHTN,64,Kiên Giang,74,Kon Tum,24,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,12,Lâm Đồng,47,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,37,Langlands,1,Lào Cai,35,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,52,Lớp 10 Chuyên,709,Lớp 10 Không Chuyên,355,Lớp 11,1,Lục Ngạn,1,Lượng giác,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 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KHTN,8,Olympic Sinh Viên,78,Olympic Tháng 4,12,Olympic Toán,344,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,42,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,64,Putnam,27,Quảng Bình,64,Quảng Nam,57,Quảng Ngãi,49,Quảng Ninh,60,Quảng Trị,42,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,36,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,37,Thái Bình,45,Thái Nguyên,61,Thái Vân,2,Thanh Hóa,69,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,56,Tiền Giang,30,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,158,Trà Vinh,10,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,39,Trại Hè Hùng Vương,30,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,23,Trường Hè,10,Trường Thu,1,Trường Xuân,3,TST,544,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 2019-2020,34,TST 2020-2021,30,TST 2021-2022,38,TST 2022-2023,42,TST 2023-2024,23,TST An Giang,8,TST Bà Rịa Vũng Tàu,11,TST Bắc Giang,5,TST Bắc Ninh,11,TST Bến Tre,10,TST Bình Định,5,TST Bình Dương,7,TST Bình Phước,9,TST Bình Thuận,9,TST Cà Mau,7,TST Cần Thơ,6,TST Cao Bằng,2,TST Đà Nẵng,8,TST Đắk Lắk,12,TST Đắk Nông,2,TST Điện Biên,2,TST Đồng Nai,13,TST Đồng Tháp,12,TST Gia Lai,4,TST Hà Nam,8,TST Hà Nội,12,TST Hà Tĩnh,15,TST Hải Dương,11,TST Hải Phòng,13,TST Hậu Giang,1,TST Hòa Bình,4,TST Hưng Yên,10,TST Khánh Hòa,8,TST Kiên Giang,11,TST Kon Tum,6,TST Lâm Đồng,12,TST Lạng Sơn,3,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,13,TST Phú Yên,5,TST PTNK,15,TST Quảng Bình,12,TST Quảng Nam,7,TST Quảng Ngãi,8,TST Quảng Ninh,9,TST Quảng Trị,10,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,6,TST TPHCM,14,TST Trà Vinh,1,TST Tuyên Quang,1,TST Vĩnh Long,7,TST Vĩnh Phúc,7,TST Yên Bái,8,Tuyên Quang,14,Tuyển Sinh,4,Tuyển Sinh 10,1064,Tuyển Sinh 10 An Giang,18,Tuyển Sinh 10 Bà Rịa Vũng Tàu,22,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,34,Tuyển Sinh 10 Bình Định,19,Tuyển Sinh 10 Bình Dương,12,Tuyển Sinh 10 Bình Phước,21,Tuyển Sinh 10 Bình Thuận,15,Tuyển Sinh 10 Cà Mau,5,Tuyển Sinh 10 Cần Thơ,10,Tuyển Sinh 10 Cao Bằng,2,Tuyển Sinh 10 Chuyên SPHN,19,Tuyển Sinh 10 Đà Nẵng,18,Tuyển Sinh 10 Đại Học Vinh,13,Tuyển Sinh 10 Đắk Lắk,21,Tuyển Sinh 10 Đắk Nông,7,Tuyển Sinh 10 Điện Biên,5,Tuyển Sinh 10 Đồng Nai,18,Tuyển Sinh 10 Đồng Tháp,23,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,19,Tuyển Sinh 10 Hải Dương,17,Tuyển Sinh 10 Hải Phòng,15,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,21,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,10,Tuyển Sinh 10 Long An,18,Tuyển Sinh 10 Nam Định,21,Tuyển Sinh 10 Nghệ An,23,Tuyển Sinh 10 Ninh Bình,20,Tuyển Sinh 10 Ninh Thuận,10,Tuyển Sinh 10 Phú Thọ,18,Tuyển Sinh 10 Phú Yên,12,Tuyển Sinh 10 PTNK,37,Tuyển Sinh 10 Quảng Bình,12,Tuyển Sinh 10 Quảng Nam,15,Tuyển Sinh 10 Quảng Ngãi,13,Tuyển Sinh 10 Quảng Ninh,12,Tuyển Sinh 10 Quảng Trị,7,Tuyển Sinh 10 Sóc Trăng,17,Tuyển Sinh 10 Sơn La,5,Tuyển Sinh 10 Tây Ninh,15,Tuyển Sinh 10 Thái Bình,17,Tuyển Sinh 10 Thái Nguyên,18,Tuyển Sinh 10 Thanh Hóa,27,Tuyển Sinh 10 Thừa Thiên Huế,24,Tuyển Sinh 10 Tiền Giang,14,Tuyển Sinh 10 TPHCM,23,Tuyển Sinh 10 Trà Vinh,6,Tuyển Sinh 10 Tuyên Quang,3,Tuyển Sinh 10 Vĩnh Long,12,Tuyển Sinh 10 Vĩnh Phúc,22,Tuyển Sinh 2008-2009,1,Tuyển Sinh 2009-2010,1,Tuyển Sinh 2010-2011,6,Tuyển Sinh 2011-2012,20,Tuyển Sinh 2012-2013,65,Tuyển Sinh 2013-2014,77,Tuyển Sinh 2013-2044,1,Tuyển Sinh 2014-2015,81,Tuyển Sinh 2015-2016,64,Tuyển Sinh 2016-2017,72,Tuyển Sinh 2017-2018,126,Tuyển Sinh 2018-2019,61,Tuyển Sinh 2019-2020,90,Tuyển Sinh 2020-2021,59,Tuyển Sinh 2021-202,1,Tuyển Sinh 2021-2022,69,Tuyển Sinh 2022-2023,113,Tuyển Sinh 2023-2024,49,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,6,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,32,Vĩnh Long,41,Vĩnh Phúc,86,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,58,VNTST,25,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: [Shortlists & Solutions] International Mathematical Olympiad 2019
[Shortlists & Solutions] International Mathematical Olympiad 2019