## $hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 # ĐẶT MUA TẠP CHÍ / PURCHASE JOURNALS ### Algebra 1. Find all triples$(f,g,h)of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbersx$and$y$. (We say a function$F$is injective if$F(a)\neq F(b)$for any distinct real numbers$a$and$b$.) 2. Prove that for all positive reals$a,b,c$, $\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}.$ 3. Find all$f:\mathbb{R}\to\mathbb{R}$such that for all$x,y\in\mathbb{R}$,$f(x)+f(y) = f(x+y)$and$f(x^{2013}) = f(x)^{2013}$. 4. Positive reals$a$,$b$, and$c$obey $$\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}.$$ Prove that $\sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}.$ 5. Let$a,b,c$be positive reals satisfying$a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that$a^a b^b c^c \ge 1$. 6. Let$a, b, c$be positive reals such that$a+b+c=3$. Prove that $18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15.$ 7. Consider a function$f: \mathbb Z \to \mathbb Z$such that for every integer$n \ge 0$, there are at most$0.001n^2$pairs of integers$(x,y)$for which$f(x+y) \neq f(x)+f(y)$and$\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer$n \ge 0$, there are more than$n$integers$a$such that$f(a) \neq a \cdot f(1)$and$\lvert a \rvert \le n$? 8. Let$a, b, c$be positive reals with$a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that $\frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}.$ 9. Let$a, b, c$be positive reals, and let $$\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P.$$ Prove that $\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).$ ### Combinatorics 1. Let$n\ge2$be a positive integer. The numbers$1,2,..., n^2$are consecutively placed into squares of an$n\times n$, so the first row contains$1,2,...,n$from left to right, the second row contains$n+1,n+2,...,2n$from left to right, and so on. The magic square value of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of$\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.) 2. Let$n$be a fixed positive integer. Initially,$n$1's are written on a blackboard. Every minute, David picks two numbers$x$and$y$written on the blackboard, erases them, and writes the number$(x+y)^4$on the blackboard. Show that after$n-1$minutes, the number written on the blackboard is at least$2^{\frac{4n^2-4}{3}}$. 3. Let$a_1,a_2,...,a_9$be nine real numbers, not necessarily distinct, with average$m$. Let$A$denote the number of triples$1 \le i < j < k \le 9$for which$a_i + a_j + a_k \ge 3m$. What is the minimum possible value of$A$? 4. Let$n$be a positive integer. The numbers$\{1, 2, ..., n^2\}$are placed in an$n \times n$grid, each exactly once. The grid is said to be Muirhead-able if the sum of the entries in each column is the same, but for every$1 \le i,k \le n-1$, the sum of the first$k$entries in column$i$is at least the sum of the first$k$entries in column$i+1$. For which$n$can one construct a Muirhead-able array such that the entries in each column are decreasing? 5. There is a$2012\times 2012$grid with rows numbered$1,2,\dots 2012$and columns numbered$1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!) a) Show that there exist$2012^2$unique integers$a_{i,j}$where$i,j \in [1,2012]$such that for all$x,y\in [1,2012]$, the sum $\sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j}$ is equal to the sum of the thicknesses of all the napkins that cover the grid square in row$x$and column$y$. b) Show that if we use at most$500,000$napkins, at least half of the$a_{i,j}$will be$0$. 6. A$4\times4$grid has its 16 cells colored arbitrarily in three colors. A swap is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. 7. A$2^{2014} + 1$by$2^{2014} + 1$grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer$n$greater than$2$, there do not exist pairwise distinct black squares$s_1$,$s_2$, \dots,$s_n$such that$s_i$and$s_{i+1}$share an edge for$i=1,2, \dots, n$(here$s_{n+1}=s_1$). What is the maximum possible number of filled black squares? 8. There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that$A$gives$B$a coin and$B$gives$A$a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer$n$, there exists a person who will give away coins during the$n$th minute. What is the smallest number of coins that could be at the party? 9. Let$f_0$be the function from$\mathbb{Z}^2$to$\{0,1\}$such that$f_0(0,0)=1$and$f_0(x,y)=0$otherwise. For each positive integer$m$, let$f_m(x,y)$be the remainder when $f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k)$ is divided by$2$. Finally, for each nonnegative integer$n$, let$a_n$denote the number of pairs$(x,y)$such that$f_n(x,y) = 1$. Find a closed form for$a_n$. 10. Let$N\ge2$be a fixed positive integer. There are$2N$people, numbered$1,2,...,2N$, participating in a tennis tournament. For any two positive integers$i,j$with$1\le i<j\le 2N$, player$i$has a higher skill level than player$j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among$N$courts, numbered$1,2,...,N$. During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for$i=2,3,...,N$, the winner of court$i$moves to court$i-1$and the loser of court$i$stays on court$i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court$N$. Find all positive integers$M$such that, regardless of the initial pairing, the players$2, 3, \ldots, N+1$all change courts immediately after the$M$th round. ### Geometry 1. Let$ABC$be a triangle with incenter$I$. Let$U$,$V$and$W$be the intersections of the angle bisectors of angles$A$,$B$, and$C$with the incircle, so that$V$lies between$B$and$I$, and similarly with$U$and$W$. Let$X$,$Y$, and$Z$be the points of tangency of the incircle of triangle$ABC$with$BC$,$AC$, and$AB$, respectively. Let triangle$UVW$be the David Yang triangle of$ABC$and let$XYZ$be the Scott Wu triangle of$ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if$ABC$is equilateral. 2. Let$ABC$be a scalene triangle with circumcircle$\Gamma$, and let$D$,$E$,$F$be the points where its incircle meets$BC$,$AC$,$AB$respectively. Let the circumcircles of$\triangle AEF$,$\triangle BFD$, and$\triangle CDE$meet$\Gamma$a second time at$X,Y,Z$respectively. Prove that the perpendiculars from$A,B,C$to$AX,BY,CZ$respectively are concurrent. 3. In$\triangle ABC$, a point$D$lies on line$BC$. The circumcircle of$ABD$meets$AC$at$F$(other than$A$), and the circumcircle of$ADC$meets$AB$at$E$(other than$A$). Prove that as$D$varies, the circumcircle of$AEF$always passes through a fixed point other than$A$, and that this point lies on the median from$A$to$BC$. 4. Triangle$ABC$is inscribed in circle$\omega$. A circle with chord$BC$intersects segments$AB$and$AC$again at$S$and$R$, respectively. Segments$BR$and$CS$meet at$L$, and rays$LR$and$LS$intersect$\omega$at$D$and$E$, respectively. The internal angle bisector of$\angle BDE$meets line$ER$at$K$. Prove that if$BE = BR$, then$\angle ELK = \tfrac{1}{2} \angle BCD$. 5. Let$\omega_1$and$\omega_2$be two orthogonal circles, and let the center of$\omega_1$be$O$. Diameter$AB$of$\omega_1$is selected so that$B$lies strictly inside$\omega_2$. The two circles tangent to$\omega_2$, passing through$O$and$A$, touch$\omega_2$at$F$and$G$. Prove that$FGOB$is cyclic. 6. Let$ABCDEF$be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define$X=AB\cap DE$,$Y=BC\cap EF$, and$Z=CD\cap FA$. Prove that $\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.$ 7. Let$ABC$be a triangle inscribed in circle$\omega$, and let the medians from$B$and$C$intersect$\omega$at$D$and$E$respectively. Let$O_1$be the center of the circle through$D$tangent to$AC$at$C$, and let$O_2$be the center of the circle through$E$tangent to$AB$at$B$. Prove that$O_1$,$O_2$, and the nine-point center of$ABC$are collinear. 8. Let$ABC$be a triangle, and let$D$,$A$,$B$,$E$be points on line$AB$, in that order, such that$AC=AD$and$BE=BC$. Let$\omega_1, \omega_2$be the circumcircles of$\triangle ABC$and$\triangle CDE$, respectively, which meet at a point$F \neq C$. If the tangent to$\omega_2$at$F$cuts$\omega_1$again at$G$, and the foot of the altitude from$G$to$FC$is$H$, prove that$\angle AGH=\angle BGH$. 9. Let$ABCD$be a cyclic quadrilateral inscribed in circle$\omega$whose diagonals meet at$F$. Lines$AB$and$CD$meet at$E$. Segment$EF$intersects$\omega$at$X$. Lines$BX$and$CD$meet at$M$, and lines$CX$and$AB$meet at$N$. Prove that$MN$and$BC$concur with the tangent to$\omega$at$X$. 10. Let$AB=AC$in$\triangle ABC$, and let$D$be a point on segment$AB$. The tangent at$D$to the circumcircle$\omega$of$BCD$hits$AC$at$E$. The other tangent from$E$to$\omega$touches it at$F$, and$G=BF \cap CD$,$H=AG \cap BC$. Prove that$BH=2HC$. 11. Let$\triangle ABC$be a nondegenerate isosceles triangle with$AB=AC$, and let$D, E, F$be the midpoints of$BC, CA, AB$respectively.$BE$intersects the circumcircle of$\triangle ABC$again at$G$, and$H$is the midpoint of minor arc$BC$.$CF\cap DG=I, BI\cap AC=J$. Prove that$\angle BJH=\angle ADG$if and only if$\angle BID=\angle GBC$. 12. Let$ABC$be a nondegenerate acute triangle with circumcircle$\omega$and let its incircle$\gamma$touch$AB, AC, BC$at$X, Y, Z$respectively. Let$XY$hit arcs$AB, AC$of$\omega$at$M, N$respectively, and let$P \neq X, Q \neq Y$be the points on$\gamma$such that$MP=MX, NQ=NY$. If$I$is the center of$\gamma$, prove that$P, I, Q$are collinear if and only if$\angle BAC=90^\circ$. 13. In$\triangle ABC$,$AB<AC$.$D$and$P$are the feet of the internal and external angle bisectors of$\angle BAC$, respectively.$M$is the midpoint of segment$BC$, and$\omega$is the circumcircle of$\triangle APD$. Suppose$Q$is on the minor arc$AD$of$\omega$such that$MQ$is tangent to$\omega$.$QB$meets$\omega$again at$R$, and the line through$R$perpendicular to$BC$meets$PQ$at$S$. Prove$SD$is tangent to the circumcircle of$\triangle QDM$. 14. Let$O$be a point (in the plane) and$T$be an infinite set of points such that$|P_1P_2| \le 2012$for every two distinct points$P_1,P_2\in T$. Let$S(T)$be the set of points$Q$in the plane satisfying$|QP| \le 2013$for at least one point$P\in T$. Now let$L$be the set of lines containing exactly one point of$S(T)$. Call a line$\ell_0$passing through$O$bad if there does not exist a line$\ell\in L$parallel to (or coinciding with)$\ell_0$. a) Prove that$L$is nonempty. b) Prove that one can assign a line$\ell(i)$to each positive integer$i$so that for every bad line$\ell_0$passing through$O$, there exists a positive integer$n$with$\ell(n) = \ell_0$### Number Theory 1. Find all ordered triples of non-negative integers$(a,b,c)$such that$a^2+2b+c$,$b^2+2c+a$, and$c^2+2a+b$are all perfect squares. 2. For what polynomials$P(n)$with integer coefficients can a positive integer be assigned to every lattice point in$\mathbb{R}^3$so that for every integer$n \ge 1$, the sum of the$n^3$integers assigned to any$n \times n \times n$grid of lattice points is divisible by$P(n)$? 3. Define a beautiful number to be an integer of the form$a^n$, where$a\in\{3,4,5,6\}$and$n$is a positive integer. Prove that each integer greater than$2$can be expressed as the sum of pairwise distinct beautiful numbers. 4. Find all triples$(a,b,c)$of positive integers such that if$n$is not divisible by any prime less than$2014$, then$n+c$divides$a^n+b^n+n$. 5. Let$m_1,m_2,...,m_{2013} > 1$be 2013 pairwise relatively prime positive integers and$A_1,A_2,...,A_{2013}$be 2013 (possibly empty) sets with$A_i\subseteq \{1,2,...,m_i-1\}$for$i=1,2,...,2013$. Prove that there is a positive integer$N$such that $N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right)$ and for each$i = 1, 2, ..., 2013$, there does not exist$a \in A_i$such that$m_i$divides$N-a$. 6. Let$\mathbb N$denote the set of positive integers, and for a function$f$, let$f^k(n)$denote the function$f$applied$k$times. Call a function$f : \mathbb N \to \mathbb N$saturated if $f^{f^{f(n)}(n)}(n) = n$ for every positive integer$n$. Find all positive integers$m$for which the following holds: every saturated function$f$satisfies$f^{2014}(m) = m$. 7. Let$p$be a prime satisfying$p^2\mid 2^{p-1}-1$, and let$n$be a positive integer. Define $f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}.$ Find the largest positive integer$N$such that there exist polynomials$g(x)$,$h(x)$with integer coefficients and an integer$r$satisfying$f(x) = (x-r)^N g(x) + p \cdot h(x)$. 8. We define the Fibonacci sequence$\{F_n\}_{n\ge0}$by$F_0=0$,$F_1=1$, and for$n\ge2$,$F_n=F_{n-1}+F_{n-2}$; we define the Stirling number of the second kind$S(n,k)$as the number of ways to partition a set of$n\ge1$distinguishable elements into$k\ge1$indistinguishable nonempty subsets. 9. For every positive integer$n$, let$t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let$p\ge7$be a prime. Prove that $t_{n+p^{2p}-1} \equiv t_n \pmod{p}$ for all$n\ge1$. ##$hide=mobile$type=ticker$c=36$cols=2$l=0$sr=random$b=0

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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 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,61,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 11,122,Olympic 12,52,Olympic 23/3,2,Olympic 24/3,10,Olympic 24/3 Quảng Nam,10,Olympic 27/4,24,Olympic 30/4,61,Olympic 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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