[Shortlists & Solutions] Eric Larsen Math Olympiad 2012


  1. Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0$, $y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\]
  2. Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\]
  3. Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$.
  4. Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
  5. Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.)
  6. Let $a,b,c\ge0$. Show that $$(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\\ \le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$$
  7. Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
  8. Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $$f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$$ for all $x,y\in\mathbb{Q}$.
  9. Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
  10. Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.


  1. Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise. Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class.
  2. Determine whether it's possible to cover a $K_{2012}$ with
    a) 1000 $K_{1006}$'s;
    b) 1000 $K_{1006,1006}$'s.
  3. Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.
  4. A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
  5. Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other.
  6. Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
  7. Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$.
  8. Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a tasteful tiling is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).
    a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.
    b) Prove that such a tasteful tiling is unique.
  9. For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$?.


  1. In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
  2. In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$.
    a) Prove $SX,TY, AD$ are concurrent at a point $Z$.
    b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$.
  3. $ABC$ is a triangle with incenter $I$. The foot of the perpendicular from $I$ to $BC$ is $D$, and the foot of the perpendicular from $I$ to $AD$ is $P$. Prove that $\angle BPD = \angle DPC$.
  4. Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
  5. Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
  6. In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear.
  7. Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$.

Number Theory 

  1. Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. 
  2. For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
    a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
    b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$. 
  3.  Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.)
  4. Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$.
  5. Prove that if $a$ and $b$ are positive integers and $ab>1$, then
    \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.
  6. A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$).
  7. Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.
  8. Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
  9. Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares?
MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệ
Chúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...


Abel Albania AMM Amsterdam An Giang Andrew Wiles Anh APMO Austria (Áo) Ba Đình Ba Lan Bà Rịa Vũng Tàu Bắc Bộ Bắc Giang Bắc Kạn Bạc Liêu Bắc Ninh Bắc Trung Bộ Bài Toán Hay Balkan Baltic Way BAMO Bất Đẳng Thức Bến Tre Benelux Bình Định Bình Dương Bình Phước Bình Thuận Birch BMO Booklet Bosnia Herzegovina BoxMath Brazil British Bùi Đắc Hiên Bùi Thị Thiện Mỹ Bùi Văn Tuyên Bùi Xuân Diệu Bulgaria Buôn Ma Thuột BxMO Cà Mau Cần Thơ Canada Cao Bằng Cao Quang Minh Câu Chuyện Toán Học Caucasus CGMO China - Trung Quốc Chọn Đội Tuyển Chu Tuấn Anh Chuyên Đề Chuyên Sư Phạm Chuyên Trần Hưng Đạo Collection College Mathematic Concours Cono Sur Contest Correspondence Cosmin Poahata Crux Czech-Polish-Slovak Đà Nẵng Đa Thức Đại Số Đắk Lắk Đắk Nông Đan Phượng Danube Đào Thái Hiệp ĐBSCL Đề Thi Đề Thi HSG Đề Thi JMO Điện Biên Định Lý Định Lý Beaty Đỗ Hữu Đức Thịnh Do Thái Doãn Quang Tiến Đoàn Quỳnh Đoàn Văn Trung Đống Đa Đồng Nai Đồng Tháp Du Hiền Vinh Đức Dương Quỳnh Châu Duyên Hải Bắc Bộ E-Book EGMO ELMO EMC Epsilon Estonian Euler Evan Chen Fermat Finland Forum Of Geometry Furstenberg G. Polya Gặp Gỡ Toán Học Gauss GDTX Geometry Gia Lai Gia Viễn Giải Tích Hàm Giảng Võ Giới hạn Goldbach Hà Giang Hà Lan Hà Nam Hà Nội Hà Tĩnh Hà Trung Kiên Hải Dương Hải Phòng Hậu Giang Hậu Lộc Hilbert Hình Học HKUST Hòa Bình Hoài Nhơn Hoàng Bá Minh Hoàng Minh Quân Hodge Hojoo Lee HOMC HongKong HSG 10 HSG 10 Bắc Giang HSG 10 Thái Nguyên HSG 10 Vĩnh Phúc HSG 11 HSG 11 Bắc Giang HSG 11 Lạng Sơn HSG 11 Thái Nguyên HSG 11 Vĩnh Phúc HSG 12 HSG 12 2010-2011 HSG 12 2011-2012 HSG 12 2012-2013 HSG 12 2013-2014 HSG 12 2014-2015 HSG 12 2015-2016 HSG 12 2016-2017 HSG 12 2017-2018 HSG 12 2018-2019 HSG 12 2019-2020 HSG 12 2020-2021 HSG 12 2021-2022 HSG 12 Bắc Giang HSG 12 Bình Phước HSG 12 Đồng Tháp HSG 12 Lạng Sơn HSG 12 Long An HSG 12 Quảng Nam HSG 12 Quảng Ninh HSG 12 Thái Nguyên HSG 12 Vĩnh Phúc HSG 9 HSG 9 2010-2011 HSG 9 2011-2012 HSG 9 2012-2013 HSG 9 2013-2014 HSG 9 2014-2015 HSG 9 2015-2016 HSG 9 2016-2017 HSG 9 2017-2018 HSG 9 2018-2019 HSG 9 2019-2020 HSG 9 2020-2021 HSG 9 2021-202 HSG 9 2021-2022 HSG 9 Bắc Giang HSG 9 Bình Phước HSG 9 Đồng Tháp HSG 9 Lạng Sơn HSG 9 Long An HSG 9 Quảng Nam HSG 9 Quảng Ninh HSG 9 Vĩnh Phúc HSG Cấp Trường HSG Quốc Gia HSG Quốc Tế Hứa Lâm Phong Hứa Thuần Phỏng Hùng Vương Hưng Yên Hương Sơn Huỳnh Kim Linh Hy Lạp IMC IMO IMT India - Ấn Độ Inequality InMC International Iran Jakob JBMO Jewish Journal Junior K2pi Kazakhstan Khánh Hòa KHTN Kiên Giang Kim Liên Kon Tum Korea - Hàn Quốc Kvant Kỷ Yếu Lai Châu Lâm Đồng Lăng Hồng Nguyệt Anh Lạng Sơn Langlands Lào Cai Lê Hải Châu Lê Hải Khôi Lê Hoành Phò Lê Khánh Sỹ Lê Minh Cường Lê Phúc Lữ Lê Phương Lê Quý Đôn Lê Viết Hải Lê Việt Hưng Leibniz Long An Lớp 10 Lớp 10 Chuyên Lớp 10 Không Chuyên Lớp 11 Lục Ngạn Lượng giác Lương Tài Lưu Giang Nam Lý Thánh Tông Macedonian Malaysia Margulis Mark Levi Mathematical Excalibur Mathematical Reflections Mathematics Magazine Mathematics Today Mathley MathLinks MathProblems Journal Mathscope MathsVN MathVN MEMO Metropolises Mexico MIC Michael Guillen Mochizuki Moldova Moscow MYM MYTS Nam Định Nam Phi National Nesbitt Newton Nghệ An Ngô Bảo Châu Ngô Việt Hải Ngọc Huyền Nguyễn Anh Tuyến Nguyễn Bá Đang Nguyễn Đình Thi Nguyễn Đức Tấn Nguyễn Đức Thắng Nguyễn Duy Khương Nguyễn Duy Tùng Nguyễn Hữu Điển Nguyễn Mình Hà Nguyễn Minh Tuấn Nguyễn Nhất Huy Nguyễn Phan Tài Vương Nguyễn Phú Khánh Nguyễn Phúc Tăng Nguyễn Quản Bá Hồng Nguyễn Quang Sơn Nguyễn Tài Chung Nguyễn Tăng Vũ Nguyễn Tất Thu Nguyễn Thúc Vũ Hoàng Nguyễn Trung Tuấn Nguyễn Tuấn Anh Nguyễn Văn Huyện Nguyễn Văn Mậu Nguyễn Văn Nho Nguyễn Văn Quý Nguyễn Văn Thông Nguyễn Việt Anh Nguyễn Vũ Lương Nhật Bản Nhóm $\LaTeX$ Nhóm Toán Ninh Bình Ninh Thuận Nội Suy Lagrange Nội Suy Newton Nordic Olympiad Corner Olympiad Preliminary Olympic 10 Olympic 10/3 Olympic 11 Olympic 12 Olympic 24/3 Olympic 24/3 Quảng Nam Olympic 27/4 Olympic 30/4 Olympic KHTN Olympic Sinh Viên Olympic Tháng 4 Olympic Toán Olympic Toán Sơ Cấp PAMO Phạm Đình Đồng Phạm Đức Tài Phạm Huy Hoàng Pham Kim Hung Phạm Quốc Sang Phan Huy Khải Phan Quang Đạt Phan Thành Nam Pháp Philippines Phú Thọ Phú Yên Phùng Hồ Hải Phương Trình Hàm Phương Trình Pythagoras Pi Polish Problems PT-HPT PTNK Putnam Quảng Bình Quảng Nam Quảng Ngãi Quảng Ninh Quảng Trị Quỹ Tích Riemann RMM RMO Romania Romanian Mathematical Russia Sách Thường Thức Toán Sách Toán Sách Toán Cao Học Sách Toán THCS Saudi Arabia - Ả Rập Xê Út Scholze Serbia Sharygin Shortlists Simon Singh Singapore Số Học - Tổ Hợp Sóc Trăng Sơn La Spain Star Education Stars of Mathematics Swinnerton-Dyer Talent Search Tăng Hải Tuân Tạp Chí Tập San Tây Ban Nha Tây Ninh Thạch Hà Thái Bình Thái Nguyên Thái Vân Thanh Hóa THCS Thổ Nhĩ Kỳ Thomas J. Mildorf THPT Chuyên Lê Quý Đôn THPTQG THTT Thừa Thiên Huế Tiền Giang Tin Tức Toán Học Titu Andreescu Toán 12 Toán Cao Cấp Toán Chuyên Toán Rời Rạc Toán Tuổi Thơ Tôn Ngọc Minh Quân TOT TPHCM Trà Vinh Trắc Nghiệm Trắc Nghiệm Toán Trại Hè Trại Hè Hùng Vương Trại Hè Phương Nam Trần Đăng Phúc Trần Minh Hiền Trần Nam Dũng Trần Phương Trần Quang Hùng Trần Quốc Anh Trần Quốc Luật Trần Quốc Nghĩa Trần Tiến Tự Trịnh Đào Chiến Trường Đông Trường Hè Trường Thu Trường Xuân TST TST 2010-2011 TST 2011-2012 TST 2012-2013 TST 2013-2014 TST 2014-2015 TST 2015-2016 TST 2016-2017 TST 2017-2018 TST 2018-2019 TST 2019-2020 TST 2020-2021 TST 2021-2022 TST Bắc Giang TST Bình Phước TST Đồng Tháp TST Lạng Sơn TST Long An TST Quảng Nam TST Quảng Ninh TST Thái Nguyên TST Vĩnh Phúc Tuyên Quang Tuyển Sinh Tuyển Sinh 10 Tuyển Sinh 10 Bắc Giang Tuyển Sinh 10 Bình Phước Tuyển Sinh 10 Đồng Tháp Tuyển Sinh 10 Lạng Sơn Tuyển Sinh 10 Long An Tuyển Sinh 10 Quảng Nam Tuyển Sinh 10 Quảng Ninh Tuyển Sinh 10 Thái Nguyên Tuyển Sinh 10 Vĩnh Phúc Tuyển Sinh 2010-2011 Tuyển Sinh 2011-2012 Tuyển Sinh 2011-2022 Tuyển Sinh 2012-2013 Tuyển Sinh 2013-2014 Tuyển Sinh 2014-2015 Tuyển Sinh 2015-2016 Tuyển Sinh 2016-2017 Tuyển Sinh 2017-2018 Tuyển Sinh 2018-2019 Tuyển Sinh 2019-2020 Tuyển Sinh 2020-2021 Tuyển Sinh 2021-202 Tuyển Sinh 2021-2022 Tuyển Tập Tuymaada UK - Anh Undergraduate USA - Mỹ USA TSTST USAJMO USATST USEMO Uzbekistan Vasile Cîrtoaje Vật Lý Viện Toán Học Vietnam Viktor Prasolov VIMF Vinh Vĩnh Long Vĩnh Phúc Virginia Tech VLTT VMEO VMF VMO VNTST Võ Anh Khoa Võ Quốc Bá Cẩn Võ Thành Văn Vojtěch Jarník Vũ Hữu Bình Vương Trung Dũng WFNMC Journal Wiles Yên Bái Yên Định Yên Thành Zhautykov Zhou Yuan Zhe
MOlympiad.NET: [Shortlists & Solutions] Eric Larsen Math Olympiad 2012
[Shortlists & Solutions] Eric Larsen Math Olympiad 2012
Not found any posts Not found any related posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU Tag ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Contents See also related Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy THIS PREMIUM CONTENT IS LOCKED