## $hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 # ĐẶT MUA TẠP CHÍ / PURCHASE JOURNALS ### National Math Olympiad (Second Round) 2010 1. Let$a,b$be two positive integers and$a>b$. We know that$\gcd(a-b,ab+1)=1$and$\gcd(a+b,ab-1)=1$. Prove that$(a-b)^2+(ab+1)^2$is not a perfect square. 2. There are$n$points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these$n$points and area of them is 1,is not greater than$\frac23(n^2-n)$. 3. Circles$W_1$,$W_2$meet at$D$and$P$.$A$and$B$are on$W_1$,$W_2$respectively, such that$AB$is tangent to$W_1$and$W_2$. Suppose$D$is closer than$P$to the line$AB$.$AD$meet circle$W_2$for second time at$C$. Let$M$be the midpoint of$BC$. Prove that $$\angle{DPM}=\angle{BDC}.$$ 4. Let$P(x)=ax^3+bx^2+cx+d$be a polynomial with real coefficients such that $\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}$ Prove that$P(x)$do not have a real root in$[-1,1]$. 5. In triangle$ABC$we havev$\angle A=\frac{\pi}{3}$. Construct$E$and$F$on continue of$AB$and$AC$respectively such that$BE=CF=BC$. Suppose that$EF$meets circumcircle of$\triangle ACE$in$K$($K\not \equiv E$). Prove that$K$is on the bisector of$\angle A$. 6. A school has$n$students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that$\left(n-1\right)^2$. ### National Math Olympiad (Third Round) 2010 1. Suppose that polynomial$p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$does not have a real root. what is the maximum number of coefficients to be$-1$? 2.$a,b,c$are positive real numbers. prove the following inequality $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c}\right)^2$$ 3. Prove that for each natural number$n$there exist a polynomial with degree$2n+1$with coefficients in$\mathbb{Q}[x]$such that it has exactly$2$complex zeros and it's irreducible in$\mathbb{Q}[x]$. 4. For each polynomial$p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$we define it's derivative as this and we show it by$p'(x)$$p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1$ a) For each two polynomials$p(x)$and$q(x)$prove that $(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)$ b) Suppose that$p(x)$is a polynomial with degree$n$and$x_1,x_2,...,x_n$are it's zeros. prove that:(3 points) $\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}$ c)$p(x)$is a monic polynomial with degree$n$and$z_1,z_2,...,z_n$are it's zeros such that $|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1$ Prove that$p'(x)$has at least one zero in the disc with length one with the center$z_1$in complex plane. (disc with length one with the center$z_1$in complex plane:$D=\{z \in \mathbb C: |z-z_1|\le1\}$) 5. Let$x,y,z$be positive real numbers such that$xy+yz+zx=1$. Prove that $$3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$$ 6. Suppose that$a=3^{100}$and$b=5454$. how many$z$s in$[1,3^{99})$exist such that for every$c$that$\gcd(c,3)=1$, two equations$x^z\equiv c$and$x^b\equiv c$(mod$a$) have the same number of answers? 7.$R$is a ring such that$xy=yx$for every$x,y\in R$and if$ab=0$then$a=0$or$b=0$. if for every Ideal$I\subset R$there exist$x_1,x_2,..,x_n$in$R$($n$is not constant) such that$I=(x_1,x_2,...,x_n)$, prove that every element in$R$that is not$0$and it's not a unit, is the product of finite irreducible elements. 8. If$p$is a prime number, what is the product of elements like$g$such that$1\le g\le p^2$and$g$is a primitive root modulo$p$but it's not a primitive root modulo$p^2$, modulo$p^2$? 9. Suppose that$\sigma_k:\mathbb N \longrightarrow \mathbb R$is a function such that$\sigma_k(n)=\sum_{d|n}d^k$.$\rho_k:\mathbb N \longrightarrow \mathbb R$is a function such that$\rho_k\ast \sigma_k=\delta$. find a formula for$\rho_k$. 10. Prove that if$p$is a prime number such that$p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with$p^2$elements. 11.$g$and$n$are natural numbers such that$gcd(g^2-g,n)=1$and$A=\{g^i|i \in \mathbb N\}$and$B=\{x\equiv (n)|x\in A\}$(by$x\equiv (n)$we mean a number from the set$\{0,1,...,n-1\}$which is congruent with$x$modulo$n$). if for$0\le i\le g-1a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$. Prove that$g-1|\sum_{i=0}^{g-1}ia_i$. (the symbol$||$means the number of elements of the set) 12. In a triangle$ABC$,$O$is the circumcenter and$I$is the incenter.$X$is the reflection of$I$to$O$.$A_1$is foot of the perpendicular from$X$to$BC$.$B_1$and$C_1$are defined similarly. prove that$AA_1$,$BB_1$and$CC_1$are concurrent. 13. In a quadrilateral$ABCD$,$E$and$F$are on$BC$and$AD$respectively such that the area of triangles$AED$and$BCF$is$\dfrac{4}{7}$of the area of$ABCD$.$R$is the intersection point of digonals of$ABCD$.$\dfrac{AR}{RC}=\dfrac{3}{5}$and$\dfrac{BR}{RD}=\frac{5}{6}$. a) in what ratio does$EF$cut the digonals? b) find$\dfrac{AF}{FD}$. 14. In a quadrilateral$ABCD$digonals are perpendicular to each other. let$S$be the intersection of digonals.$K$,$L$,$M$and$N$are reflections of$S$to$AB$,$BC$,$CD$and$DA$.$BN$cuts the circumcircle of$SKN$in$E$and$BM$cuts the circumcircle of$SLM$in$F$. Prove that$EFLK$is concyclic. 15. In a triangle$ABC$,$I$is the incenter.$BI$and$CI$cut the circumcircle of$ABC$at$E$and$F$respectively.$M$is the midpoint of$EF$.$C$is a circle with diameter$EF$.$IM$cuts$C$at two points$L$and$K$and the arc$BC$of circumcircle of$ABC$(not containing$A$) at$D$. Prove that $$\frac{DL}{IL}=\frac{DK}{IK}$$ 16. In a triangle$ABC$,$I$is the incenter.$D$is the reflection of$A$to$I$. the incircle is tangent to$BC$at point$E$.$DE$cuts$IG$at$P$($G$is centroid).$M$is the midpoint of$BC$. Prove that a)$AP||DM$. b)$AP=2DM$. 17. In a triangle$ABC$,$\angle C=45$.$AD$is the altitude of the triangle.$X$is on$AD$such that$\angle XBC=90-\angle B$($X$is in the triangle).$AD$and$CX$cut the circumcircle of$ABC$in$M$and$N$respectively. if tangent to circumcircle of$ABC$at$M$cuts$AN$at$P$, prove that$P$,$B$and$O$are collinear. 18. Suppose that$\mathcal F\subseteq X^{(k)}$and$|X|=n$. we know that for every three distinct elements of$\mathcal F$like$A$,$B$,$C$, at most one of$A\cap B$,$B\cap C$and$C\cap A$is$\phi$. for$k\le \frac{n}{2}$. Prove that a)$|\mathcal F|\le \max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$. b) find all cases of equality in a) for$k\le \frac{n}{3}$. 19. Suppose that$\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$and$|X|=n$. we know that$\mathcal F$is a sperner family and it's also$H_k$. Prove that $$\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$$ 20. Suppose that$\mathcal F\subseteq p(X)$and$|X|=n$. we know that for every$A_i,A_j\in \mathcal F$that$A_i\supseteq A_j$we have$3\le |A_i|-|A_j|$. Prove that $$|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$$ 21. Suppose that$\mathcal F\subseteq X^{(K)}$and$|X|=n$. we know that for every three distinct elements of$\mathcal F$like$A,B$and$C$we have$A\cap B \not\subset C$. a) Prove that $|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1$ b) if elements of$\mathcal F$do not necessarily have$k$elements, with the above conditions show that $|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2$ 22. Suppose that$\mathcal F\subseteq p(X)$and$|X|=n$. prove that if$|\mathcal F|>\sum_{i=0}^{k-1}\dbinom{n}{i}$then there exist$Y\subseteq X$with$|Y|=k$such that$p(Y)=\mathcal F\cap Y$that$\mathcal F\cap Y=\{F\cap Y:F\in \mathcal F\}$23. Suppose that$X$is a set with$n$elements and$\mathcal F\subseteq X^{(k)}$and$X_1,X_2,...,X_s$is a partition of$X$. We know that for every$A,B\in \mathcal F$and every$1\le j\le s$,$E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$shows that none of$E,F$contains the other one. Prove that $|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}$ 24.$P(x,y)$is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of$x$and$y$in it. we denote by$Q(x,y)$sum of monomials with the most degree in$P(x,y)$. (for example if$P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$then$Q(x,y)=3x^4y-2x^2y^3$.) Suppose that there are real numbers$x_1$,$y_1$,$x_2$and$y_2$such that 25.$Q(x_1,y_1)>0$,$Q(x_2,y_2)<0$. Prove that the set$\{(x,y)|P(x,y)=0\}$is not bounded. (We call a set$S$of plane bounded if there exist positive number$M$such that the distance of elements of$S$from the origin is less than$M$.) 26.$a$,$b$and$c$are natural numbers. we have a$(2a+1)\times (2b+1)\times (2c+1)$cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) Prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. 27. Set$A$containing$n$points in plane is given. a$copy$of$A$is a set of points that is made by using transformation, rotation, homogeneity or their combination on elements of$A$. we want to put$ncopies$of$A$in plane, such that every two copies have exactly one point in common and every three of them have no common elements. a) prove that if no$4$points of$A$make a parallelogram, you can do this only using transformation. ($A$doesn't have a parallelogram with angle$0$and a parallelogram that it's two non-adjacent vertices are one!) b) prove that you can always do this by using a combination of all these things. 28. Suppose that$S$is a figure in the plane such that it's border doesn't contain any lattice points. suppose that$x,y$are two lattice points with the distance$1$(we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of$S$such that$x,y$always go on lattice points ( you can rotate or reverse copies of$S$). prove that the area of$S$is equal to lattice points inside it. 29.$n$is a natural number and$x_1,x_2,...$is a sequence of numbers$1$and$-1$with these properties: it is periodic and its least period number is$2^n-1$. (it means that for every natural number$j$we have$x_{j+2^n-1}=x_j$and$2^n-1$is the least number with this property.) There exist distinct integers$0\le t_1<t_2<...<t_k<n$such that for every natural number$j$we have $x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}$ Prove that for every natural number$s$that$s<2^n-1$we have $\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1$ 30. We call a$12$-gon in plane good whenever: first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points$(0,1)$,$(1,0)$,$(-1,0)$and$(0,-1)$. find the faces of the massivest polyhedral that it's image on every three plane$xy$,$yz$and$zx$is a good$12$-gon. (it's obvios that centers of these three$12$-gons are the origin of coordinates for three dimensions.) 31.$S$is a set with$n$elements and$P(S)$is the set of all subsets of$S$and$f : P(S) \rightarrow \mathbb N$is a function with these properties: for every subset$A$of$S$we have$f(A)=f(S-A)$. for every two subsets of$S$like$A$and$B$we have$\max(f(A),f(B))\ge f(A\cup B)$. Prove that number of natural numbers like$x$such that there exists$A\subseteq S$and$f(A)=x$is less than$n$. Prove that there are infinitely many natural numbers of the form$n^2+1$such that they don't have any divisor of the form$k^2+1$except$1$and themselves. 32. Prove that the group of orientation-preserving symmetries of the cube is isomorphic to$S_4$(the group of permutations of$\{1,2,3,4\}$). 33. Prove the third sylow theorem: suppose that$G$is a group and$|G|=p^em$which$p$is a prime number and$(p,m)=1$. Suppose that$a$is the number of$p$-sylow subgroups of$G$($H<G$that$|H|=p^e$). Prove that$a|m$and$p|a-1$. (Hint: you can use this: every two$p$-sylow subgroups are conjugate. 34. Suppose that$G<S_n$is a subgroup of permutations of$\{1,...,n\}$with this property that for every$e\neq g\in G$there exist exactly one$k\in \{1,...,n\}$such that$g.k=k$. prove that there exist one$k\in \{1,...,n\}$such that for every$g\in G$we have$g.k=k$. 35. a) Prove that every discrete subgroup of$(\mathbb R^2,+)$is in one of these forms i-$\{0\}$. ii-$\{mv|m\in \mathbb Z\}$for a vector$v$in$\mathbb R^2$. iii-$\{mv+nw|m,n\in \mathbb Z\}$for the linearly independent vectors$v$and$w$in$\mathbb R^2$.(lattice$L$) b) prove that every finite group of symmetries that fixes the origin and the lattice$L$is in one of these forms:$\mathcal C_i$or$\mathcal D_i$that$i=1,2,3,4,6$($\mathcal C_i$is the cyclic group of order$i$and$\mathcal D_i$is the dyhedral group of order$i$). 36. Suppose that$p$is a prime number. find that smallest$n$such that there exists a non-abelian group$G$with$|G|=p^n$. SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries. ### Iran Team Selection Test 2011 1. In acute triangle$ABC$angle$B$is greater than$C$. Let$M$is midpoint of$BC$.$D$and$E$are the feet of the altitude from$C$and$B$respectively.$K$and$L$are midpoint of$ME$and$MD$respectively. If$KL$intersect the line through$A$parallel to$BC$in$T$, prove that$TA=TM$. 2. Find all natural numbers$n$greater than$2$such that there exist$n$natural numbers$a_{1},a_{2},\ldots,a_{n}$such that they are not all equal, and the sequence$a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$forms an arithmetic progression with nonzero common difference. 3. There are$n$points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals$F$such that for every element of$F$like$A$there is almost one other element of$F$like$B$such that$A \subseteq B$(in this case we call$A$is sub-interval of$B$). We call an interval maximal if it is not a sub-interval of any other interval. If$m$is the number of maximal elements of$F$and$a$is number of non-maximal elements of$F,$prove that$n\geq m+\frac a2.$4. Define a finite set$A$to be 'good' if it satisfies the following conditions • For every three disjoint element of$A,$like$a,b,c$we have$\gcd(a,b,c)=1;$• For every two distinct$b,c\in A,$there exists an$a\in A,$distinct from$b,c$such that$bc$is divisible by$a.$Find all good sets. 5. Find all surjective functions$f: \mathbb R \to \mathbb R$such that for every$x,y\in \mathbb R,$we have $f(x+f(x)+2f(y))=f(2x)+f(2y).$ 6. The circle$\omega$with center$O$has given. From an arbitrary point$T$outside of$\omega$draw tangents$TB$and$TC$to it.$K$and$H$are on$TB$and$TC$respectively. a)$B'$and$C'$are the second intersection point of$OB$and$OC$with$\omega$respectively.$K'$and$H'$are on angle bisectors of$\angle BCO$and$\angle CBO$respectively such that$KK' \bot BC$and$HH'\bot BC$. Prove that$K,H',B'$are collinear if and only if$H,K',C'$are collinear. b) Consider there exist two circle in$TBC$such that they are tangent two each other at$J$and both of them are tangent to$\omega$.and one of them is tangent to$TB$at$K$and other one is tangent to$TC$at$H$. Prove that two quadrilateral$BKJI$and$CHJI$are cyclic ($I$is incenter of triangle$OBC$). 7. Find the locus of points$P$in an equilateral triangle$ABC$for which the square root of the distance of$P$to one of the sides is equal to the sum of the square root of the distance of$P$to the two other sides. 8. Let$p$be a prime and$k$a positive integer such that$k \le p$. We know that$f(x)$is a polynomial in$\mathbb Z[x]$such that for all$x \in \mathbb{Z}$we have$p^k | f(x)$. a) Prove that there exist polynomials$A_0(x),\ldots,A_k(x)$all in$\mathbb Z[x]$such that $f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),$ b) Find a counter example for each prime$p$and each$k > p$. 9. We have$n$points in the plane such that they are not all collinear. We call a line$\ell$a 'good' line if we can divide those$n$points in two sets$A,B$such that the sum of the distances of all points in$A$to$\ell$is equal to the sum of the distances of all points in$B$to$\ell$. Prove that there exist infinitely many points in the plane such that for each of them we have at least$n+1$good lines passing through them. 10. Find the least value of$k$such that for all$a,b,c,d \in \mathbb{R}$the inequality $\sqrt{(a^2+1)(b^2+1)(c^2+1)} \\ +\sqrt{(b^2+1)(c^2+1)(d^2+1)} \\ +\sqrt{(c^2+1)(d^2+1)(a^2+1)} \\ +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ge 2( ab+bc+cd+da+ac+bd)-k$ holds. 11. Let$ABC$be a triangle and$A',B',C'$be the midpoints of$BC,CA,AB$respectively. Let$P$and$P'$be points in plane such that$PA=P'A'$,$PB=P'B'$,$PC=P'C'$. Prove that all$PP'$pass through a fixed point. 12. Suppose that$f : \mathbb{N} \rightarrow \mathbb{N}$is a function for which the expression$af(a)+bf(b)+2ab$for all$a,b \in \mathbb{N}$is always a perfect square. Prove that$f(a)=a$for all$a \in \mathbb{N}$. ### DOWNLOAD SOLUTIONS ##$hide=mobile$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 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,
ltr
item