[Shortlists] International Mathematical Olympiad 2014


  1. Let $a_0 < a_1 < a_2 \ldots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
  2. Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]
  3. For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$.
  4. Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
  5. Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$.
  6. Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that \[ n^2+4f(n)=f(f(n))^2 \] for all $n\in \mathbb{Z}$.


  1. Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.
  2. We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
  3. Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
  4. Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
  5. A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary.
    Note: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.
  6. We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions
    • The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
    • If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$.
    • If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$.
    How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other.
  7. Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves.
  8. A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy.
  9. There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or vice versa. Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.


  1. Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.
  2. Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$.
  3. Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.)
  4. Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
  5. Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
  6. Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $$\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}.$$
  7. Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W$, and $Z$ are also collinear.

Number Theory

  1. Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
  2. Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
  3. For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
  4. Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
  5. Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
  6. Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.
  7. Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ .
  8. For every real number $x$, let $||x||$ denote the distance between $x$ and the nearest integer. Prove that for every pair $(a, b)$ of positive integers there exist an odd prime $p$ and a positive integer $k$ satisfying \[\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.\]

Post a Comment





Kỷ Yếu$cl=violet$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0$hide=mobile



Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,3,Amsterdam,5,Ấn Độ,2,An Giang,23,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,53,Bắc Giang,50,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,48,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,38,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,14,Bình Định,46,Bình Dương,23,Bình Phước,38,Bình Thuận,34,Birch,1,Booklet,11,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,1,BxMO,13,Cà Mau,14,Cần Thơ,14,Canada,40,Cao Bằng,7,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,355,Chu Tuấn Anh,1,Chuyên Đề,124,Chuyên Sư Phạm,31,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,618,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,56,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,1773,Đề Thi JMO,1,Điện Biên,8,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,4,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,50,Đồng Tháp,52,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,17,ELMO,19,EMC,9,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,26,Gauss,1,GDTX,3,Geometry,12,Gia Lai,26,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,29,Hà Nội,232,Hà Tĩnh,73,Hà Trung Kiên,1,Hải Dương,50,Hải Phòng,42,Hàn Quốc,5,Hậu Giang,4,Hậu Lộc,1,Hilbert,1,Hình Học,33,HKUST,7,Hòa Bình,13,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,101,HSG 11,91,HSG 12,588,HSG 9,425,HSG Cấp Trường,78,HSG Quốc Gia,106,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,33,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,56,IMT,1,India,45,Inequality,13,InMC,1,International,315,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,17,KHTN,54,Kiên Giang,64,Kim Liên,1,Kon Tum,18,Korea,5,Kvant,2,Kỷ Yếu,42,Lai Châu,4,Lâm Đồng,33,Lạng Sơn,21,Langlands,1,Lào Cai,17,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,1,Leibniz,1,Long An,42,Lớp 10,10,Lớp 10 Chuyên,455,Lớp 10 Không Chuyên,229,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lý Thánh Tông,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,11,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,10,MYM,227,MYTS,4,Nam Định,33,Nam Phi,1,Nam Trung Bộ,1,National,249,Nesbitt,1,Newton,4,Nghệ An,52,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 Mình Hà,1,Nguyễn Minh Tuấn,8,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,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,43,Ninh Thuận,15,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,99,Olympic 10/3,5,Olympic 11,92,Olympic 12,30,Olympic 24/3,7,Olympic 27/4,20,Olympic 30/4,69,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,12,Olympic Toán,304,Olympic Toán Sơ Cấp,3,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 Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,30,Phú Yên,29,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,45,Putnam,25,Quảng Bình,44,Quảng Nam,32,Quảng Ngãi,34,Quảng Ninh,43,Quảng Trị,27,Quỹ Tích,1,Riemann,1,RMM,12,RMO,24,Romania,36,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,69,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia,7,Scholze,1,Serbia,17,Sharygin,24,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,27,Sóc Trăng,28,Sơn La,12,Spain,8,Star Education,5,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,14,Tập San,6,Tây Ban Nha,1,Tây Ninh,29,Thạch Hà,1,Thái Bình,39,Thái Nguyên,49,Thái Vân,2,Thanh Hóa,62,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,7,Thừa Thiên Huế,36,Tiền Giang,19,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,126,Trà Vinh,6,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,34,Trại Hè Hùng Vương,25,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,9,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,Trung Quốc,14,Trường Đông,19,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,56,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Sinh 10,680,Tuyển Tập,44,Tuymaada,4,Undergraduate,67,USA,44,USAJMO,10,USATST,7,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,2,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,27,Vĩnh Long,21,Vĩnh Phúc,64,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,47,VNTST,22,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,Yên Bái,20,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
MOlympiad: [Shortlists] International Mathematical Olympiad 2014
[Shortlists] International Mathematical Olympiad 2014
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home 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 THIS PREMIUM CONTENT IS LOCKED Please share to unlock 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