## $hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 # ĐẶT MUA TẠP CHÍ / PURCHASE JOURNALS ### Algebra 1. Let$a_1,a_2,\ldots a_n,k$, and$M$be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If$M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$has no positive roots. 2. Let$q$be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: • In the first line, Gugu writes down every number of the form$a-b$, where$a$and$b$are two (not necessarily distinct) numbers on his napkin. • In the second line, Gugu writes down every number of the form$qab$, where$a$and$b$are two (not necessarily distinct) numbers from the first line. • In the third line, Gugu writes down every number of the form$a^2+b^2-c^2-d^2$, where$a, b, c, d$are four (not necessarily distinct) numbers from the first line. Determine all values of$q$such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line. 1. Let$S$be a finite set, and let$\mathcal{A}$be the set of all functions from$S$to$S$. Let$f$be an element of$\mathcal{A}$, and let$T=f(S)$be the image of$S$under$f$. Suppose that$f\circ g\circ f\ne g\circ f\circ g$for every$g$in$\mathcal{A}$with$g\ne f$. Show that$f(T)=T$. 2. A sequence of real numbers$a_1,a_2,\ldots$satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant$M$such that$|a_n|\leq M$for all positive integers$n$. 3. An integer$n \geq 3$is given. We call an$n$-tuple of real numbers$(x_1, x_2, \dots, x_n)$Shiny if for each permutation$y_1, y_2, \dots, y_n$of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$Find the largest constant$K = K(n)$such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$holds for every Shiny$n$-tuple$(x_1, x_2, \dots, x_n)$. 4. Let$\mathbb{R}$be the set of real numbers. Determine all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that, for any real numbers$x$and$y$, $f(f(x)f(y)) + f(x+y) = f(xy).$ 5. Let$a_0,a_1,a_2,\ldots$be a sequence of integers and$b_0,b_1,b_2,\ldots$be a sequence of positive integers such that$a_0=0,a_1=1$, and $a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if b_{n-1}=1} \\ a_nb_n-a_{n-1} & \text{if b_{n-1}>1} \end{cases}\qquad\text{for }n=1,2,\ldots.$for$n=1,2,\ldots.$Prove that at least one of the two numbers$a_{2017}$and$a_{2018}$must be greater than or equal to$2017$. 6. A function$f:\mathbb{R} \to \mathbb{R}$has the following property $$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$Prove that$f(x)+y \leq f(y)+x$whenever$x>y$. ### Combinatorics 1. A rectangle$\mathcal{R}$with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of$\mathcal{R}$are either all odd or all even. 2. Let$n$be a positive integer. Define a chameleon to be any sequence of$3n$letters, with exactly$n$occurrences of each of the letters$a, b,$and$c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon$X$, there exists a chameleon$Y$such that$X$cannot be changed to$Y$using fewer than$3n^2/2$swaps. 3. Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: • Choose any number of the form$2^j$, where$j$is a non-negative integer, and put it into an empty cell. • Choose two (not necessarily adjacent) cells with the same number in them; denote that number by$2^j$. Replace the number in one of the cells with$2^{j+1}$and erase the number in the other cell. At the end of the game, one cell contains$2^n$, where$n$is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of$n$. 4. An integer$N \ge 2$is given. A collection of$N(N + 1)$soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove$N(N - 1)$players from this row leaving a new row of$2N$players in which the following$N$conditions hold: • ($1$) no one stands between the two tallest players, • ($2$) no one stands between the third and fourth tallest players, •$\;\;\vdots$• ($N$) no one stands between the two shortest players. Show that this is always possible. 5. A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point,$A_0,$and the hunter's starting point,$B_0$are the same. After$n-1$rounds of the game, the rabbit is at point$A_{n-1}$and the hunter is at point$B_{n-1}.$In the$n^{\text{th}}$round of the game, three things occur in order: • The rabbit moves invisibly to a point$A_n$such that the distance between$A_{n-1}$and$A_n$is exactly$1.$• A tracking device reports a point$P_n$to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between$P_n$and$A_n$is at most$1.$• The hunter moves visibly to a point$B_n$such that the distance between$B_{n-1}$and$B_n$is exactly$1.$Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after$10^9$rounds, she can ensure that the distance between her and the rabbit is at most$100?$1. Let$n > 1$be a given integer. An$n \times n \times n$cube is composed of$n^3$unit cubes. Each unit cube is painted with one colour. For each$n \times n \times 1$box consisting of$n^2$unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get$3n$sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of$n$, the maximal possible number of colours that are present. 2. For any finite sets$X$and$Y$os positive integers, denote by$f_X(k)$the$k^{\text{th}}$smallest positive integer not in$X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let$A$be a set of$a>0$positive integers and let$B$be a set of$b>0$positive integers. Prove that if$A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears b times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears a times}}.$$ 3. Let$n$be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point$cconsists of all lattice points within the axis-aligned(2n+1) \times (2n+1)$square entered at$c$, apart from$c$itself. We call a butterfly lonely, crowded, or comfortable, depending on whether the number of butterflies in its neighborhood$N$is respectively less than, greater than, or equal to half of the number of lattice points in$N$. Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state. ### Geometry 1. Let$ABCDE$be a convex pentagon such that$AB=BC=CD$,$\angle{EAB}=\angle{BCD}$, and$\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from$E$to$BC$and the line segments$AC$and$BD$are concurrent. 2. Let$R$and$S$be different points on a circle$\Omega$such that$RS$is not a diameter. Let$\ell$be the tangent line to$\Omega$at$R$. Point$T$is such that$S$is the midpoint of the line segment$RT$. Point$J$is chosen on the shorter arc$RS$of$\Omega$so that the circumcircle$\Gamma$of triangle$JST$intersects$\ell$at two distinct points. Let$A$be the common point of$\Gamma$and$\ell$that is closer to$R$. Line$AJ$meets$\Omega$again at$K$. Prove that the line$KT$is tangent to$\Gamma$. 3. Let$O$be the circumcenter of an acute triangle$ABC$. Line$OA$intersects the altitudes of$ABC$through$B$and$C$at$P$and$Q$, respectively. The altitudes meet at$H$. Prove that the circumcenter of triangle$PQH$lies on a median of triangle$ABC$. 4. In triangle$ABC$, let$\omega$be the excircle opposite to$A$. Let$D, E$and$F$be the points where$\omega$is tangent to$BC, CA$, and$AB$, respectively. The circle$AEF$intersects line$BC$at$P$and$Q$. Let$M$be the midpoint of$AD$. Prove that the circle$MPQ$is tangent to$\omega$. 5. Let$ABCC_1B_1A_1$be a convex hexagon such that$AB=BC$, and suppose that the line segments$AA_1, BB_1$, and$CC_1$have the same perpendicular bisector. Let the diagonals$AC_1$and$A_1C$meet at$D$, and denote by$\omega$the circle$ABC$. Let$\omega$intersect the circle$A_1BC_1$again at$E \neq B$. Prove that the lines$BB_1$and$DE$intersect on$\omega$. 6. Let$n\ge3$be an integer. Two regular$n$-gons$\mathcal{A}$and$\mathcal{B}$are given in the plane. Prove that the vertices of$\mathcal{A}$that lie inside$\mathcal{B}$or on its boundary are consecutive. (That is, prove that there exists a line separating those vertices of$\mathcal{A}$that lie inside$\mathcal{B}$or on its boundary from the other vertices of$\mathcal{A}$.) 7. A convex quadrilateral$ABCD$has an inscribed circle with center$I$. Let$I_a, I_b, I_c$and$I_d$be the incenters of the triangles$DAB, ABC, BCD$and$CDA$, respectively. Suppose that the common external tangents of the circles$AI_bI_d$and$CI_bI_d$meet at$X$, and the common external tangents of the circles$BI_aI_c$and$DI_aI_c$meet at$Y$. Prove that$\angle{XIY}=90^{\circ}$. 8. There are$2017$mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn. Find all possible numbers of tangent segments when Luciano stops drawing. ### Number Theory 1. For each integer$a_0 > 1$, define the sequence$a_0, a_1, a_2, \ldots$for$n \geq 0$as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases}$$Determine all values of$a_0$such that there exists a number$A$such that$a_n = A$for infinitely many values of$n$. 2. Let$ p \geq 2$be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index$i$in the set$\{1,2,\ldots, p-1 \}$that was not chosen before by either of the two players and then chooses an element$a_i$from the set$\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i.$$ The goal of Eduardo is to make$M$divisible by$p$, and the goal of Fernando is to prevent this. 3. Determiner all integers$ n\geq 2$having the following property: for any integers$a_1,a_2,\ldots, a_n$whose sum is not divisible by$n$, there exists an index$1 \leq i \leq n$such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$is divisible by$n$. Here, we let$a_i=a_{i-n}$when$i >n$. 4. Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer$m$, we say that a positive integer$t$is$m-$tastic if there exists a number$c\in \{1,2,3,\ldots ,2017\}$such that$\dfrac{10^t-1}{c\cdot m}$is short, and such that$\dfrac{10^k-1}{c\cdot m}$is not short for any$1\le k<t$. Let$S(m)$be the the set of$m-$tastic numbers. Consider$S(m)$for$m=1,2,\ldots{}.$What is the maximum number of elements in$S(m)$? 5. Find all pairs$(p,q)$of prime numbers which$p>q$and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$is an integer. 6. Find the smallest positive integer$n$or show no such$n$exists, with the following property: there are infinitely many distinct$n$-tuples of positive rational numbers$(a_1, a_2, \ldots, a_n)$such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$are integers. 7. An ordered pair$(x, y)$of integers is a primitive point if the greatest common divisor of$x$and$y$is$1$. Given a finite set$S$of primitive points, prove that there exist a positive integer$n$and integers$a_0, a_1, \ldots , a_n$such that, for each$(x, y)$in$S$, we have $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ 8. Let$p$be an odd prime number and$\mathbb{Z}_{>0}$be the set of positive integers. Suppose that a function$f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$satisfies the following properties: •$f(1,1)=0$. •$f(a,b)+f(b,a)=1$for any pair of relatively prime positive integers$a,b$not both equal to 1; •$f(a+b,b)=f(a,b)$for any pair of relatively prime positive integers$(a,b)$. Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$ ##$hide=mobile$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,76,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,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,513,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,2227,Đề Thi JMO,1,DHBB,30,Điện Biên,14,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,43,Hà Nội,254,Hà Tĩnh,95,Hà Trung Kiên,1,Hải Dương,68,Hải Phòng,57,Hậu Giang,13,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 10,125,HSG 10 2010-2011,4,HSG 10 2011-2012,7,HSG 10 2012-2013,7,HSG 10 2013-2014,7,HSG 10 2014-2015,6,HSG 10 2015-2016,2,HSG 10 2016-2017,7,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 2022-2023,11,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 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,136,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 2017-2018,6,HSG 11 2018-2019,8,HSG 11 2019-2020,4,HSG 11 2020-2021,7,HSG 11 2021-2022,2,HSG 11 2022-2023,6,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,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 12,653,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 2021-2022,35,HSG 12 2022-2023,41,HSG 12 2023-2024,11,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,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,4,HSG 12 Hà Nội,17,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,15,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,13,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,10,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,12,HSG 12 Thanh Hóa,17,HSG 12 Thừa Thiên Huế,19,HSG 12 Tiền Giang,3,HSG 12 TPHCM,13,HSG 12 Tuyên Quang,2,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,20,HSG 12 Yên Bái,6,HSG 9,568,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,42,HSG 9 2012-2013,42,HSG 9 2013-2014,35,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,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,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,8,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,82,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,42,Hương Sơn,2,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,29,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,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,73,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,60,Ninh Thuận,26,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,134,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,121,Olympic 12,52,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,343,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,59,Putnam,27,Quảng Bình,63,Quảng Nam,55,Quảng Ngãi,49,Quảng Ninh,59,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,Thạch Hà,1,Thái Bình,44,Thái Nguyên,60,Thái Vân,2,Thanh Hóa,68,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,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,153,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,542,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 2020-2021,29,TST 2021-2022,38,TST 2022-2023,42,TST 2023-2024,22,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,7,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,6,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,13,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,85,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,57,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: [Shortlists & Solutions] International Mathematical Olympiad 2017
[Shortlists & Solutions] International Mathematical Olympiad 2017