contest,link,problems 2021 China Second Round,https://artofproblemsolving.com/community/c2480909_2021_china_second_round,"Let $k\ge 2$ be an integer and $a_1,a_2,\cdots,a_k$ be $k$ non-zero reals. Prove that there are finitely many pairs of pairwise distinct positive integers $(n_1,n_2,\cdots,n_k)$ such that $$a_1\cdot n_1!+a_2\cdot n_2!+\cdots+a_k\cdot n_k!=0.$$" 2021 China Second Round,https://artofproblemsolving.com/community/c2480909_2021_china_second_round,"In $\triangle ABC$, point $M$ is the middle point of $AC$. $MD//AB$ and meet the tangent of $A$ to $\odot(ABC)$ at point $D$. Point $E$ is in $AD$ and point $A$ is the middle point of $DE$. $\{P\}=\odot(ABE)\cap AC,\{Q\}=\odot(ADP)\cap DM$. Prove that $\angle QCB=\angle BAC$. " 2021 China Second Round,https://artofproblemsolving.com/community/c2480909_2021_china_second_round,"If $n\ge 4,\ n\in\mathbb{N^*},\ n\mid (2^n-2)$. Prove that $\frac{2^n-2}{n}$ is not a prime number." 2021 China Second Round,https://artofproblemsolving.com/community/c2480909_2021_china_second_round,"Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$" 2020 China Second Round Olympiad,https://artofproblemsolving.com/community/c1294613_2020_china_second_round_olympiad,"In triangle $ABC,$ $AB=BC,$ and let $I$ be the incentre of $\triangle ABC.$ $M$ is the midpoint of segment $BI.$ $P$ lies on segment $AC,$ such that $AP=3PC.$ $H$ lies on line $PI,$ such that $MH\perp PH.$ $Q$ is the midpoint of the arc $AB$ of the circumcircle of $\triangle ABC$. Prove that $BH\perp QH.$" 2020 China Second Round Olympiad,https://artofproblemsolving.com/community/c1294613_2020_china_second_round_olympiad,"Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$" 2020 China Second Round Olympiad,https://artofproblemsolving.com/community/c1294613_2020_china_second_round_olympiad,"Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$" 2020 China Second Round Olympiad,https://artofproblemsolving.com/community/c1294613_2020_china_second_round_olympiad,"Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(meaning any two edges in the subset doesn't cover the same vertex) calls a ""perfect matching"" of this T-diagram. Among all the T-diagrams, find the maximum number of ""perfect matching"" of a T-diagram." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"In acute triangle $\triangle ABC$, $M$ is the midpoint of segment $BC$. Point $P$ lies in the interior of $\triangle ABC$ such that $AP$ bisects $\angle BAC$. Line $MP$ intersects the circumcircles of $\triangle ABP,\triangle ACP$ at $D,E$ respectively. Prove that if $DE=MP$, then $BC=2BP$." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Let $a_1,a_2,\cdots,a_n$ be integers such that $1=a_1\le a_2\le \cdots\le a_{2019}=99$. Find the minimum $f_0$ of the expression $$f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}),$$and determine the number of sequences $(a_1,a_2,\cdots,a_n)$ such that $f=f_0$." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Let $m$ be an integer where $|m|\ge 2$. Let $a_1,a_2,\cdots$ be a sequence of integers such that $a_1,a_2$ are not both zero, and for any positive integer $n$, $a_{n+2}=a_{n+1}-ma_n$. Prove that if positive integers $r>s\ge 2$ satisfy $a_r=a_s=a_1$, then $r-s\ge |m|$." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that The two edges in each subset share a common vertice, Any of the two subsets do not intersect. " 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Suppose that $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ such that $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$ Let $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Prove that $$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$" 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Find all the positive integers $n$ such that: $(1)$ $n$ has at least $4$ positive divisors. $(2)$ if all positive divisors of $n$ are $d_1,d_2,\cdots ,d_k,$ then $d_2-d_1,d_3-d_2,\cdots ,d_k-d_{k-1}$ form a geometric sequence." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$ Prove that $A,K,L,E$ are concyclic." 2019 China Second Round Olympiad,https://artofproblemsolving.com/community/c947685_2019_china_second_round_olympiad,"Each side of a convex $2019$-gon polygon is dyed with red, yellow and blue, and there are exactly $673$ sides of each kind of color. Prove that there exists at least one way to draw $2016$ diagonals to divide the convex $2019$-gon polygon into $2017$ triangles, such that any two of the $2016$ diagonals don't have intersection inside the $2019$-gon polygon,and for any triangle in all the $2017$ triangles, the colors of the three sides of the triangle are all the same, either totally different." 2018 China Second Round Olympiad,https://artofproblemsolving.com/community/c868819_2018_china_second_round_olympiad,"Let $ a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B$ are positive reals such that $ a_i\leq b_i,a_i\leq A$ $(i=1,2,\cdots,n)$ and $\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.$ Prove that$$\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.$$" 2018 China Second Round Olympiad,https://artofproblemsolving.com/community/c868819_2018_china_second_round_olympiad,"In triangle $\triangle ABC$, $AB\textup{min} Y.$" 2018 China Second Round Olympiad,https://artofproblemsolving.com/community/c868819_2018_china_second_round_olympiad,"Prove that for any integer $a \ge 2$ and positive integer $n,$ there exist positive integer $k$ such that $a^k+1,a^k+2,\ldots,a^k+n$ are all composite numbers." 2017 China Second Round Olympiad,https://artofproblemsolving.com/community/c831699_2017_china__second_round_olympiad,"Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$." 2017 China Second Round Olympiad,https://artofproblemsolving.com/community/c831699_2017_china__second_round_olympiad,"Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$. Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$" 2017 China Second Round Olympiad,https://artofproblemsolving.com/community/c831699_2017_china__second_round_olympiad,"Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$." 2017 China Second Round Olympiad,https://artofproblemsolving.com/community/c831699_2017_china__second_round_olympiad,"Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{ \begin{array}{lcr} a_n+n,\quad a_n\le n, \\ a_n-n,\quad a_n>n, \end{array} \right. \quad n=1,2,\cdots.$ Find the number of positive integers $r$ satisfying $a_r3$ and $p+2$ are prime numbers,and define sequence $$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$ show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$" 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Let $a_1, a_2, \ldots, a_n$ be real numbers.Prove that you can select $\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}$ such that$$\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).$$" 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Let $S=\{A_1,A_2,\ldots ,A_n\}$, where $A_1,A_2,\ldots ,A_n$ are $n$ pairwise distinct finite sets $(n\ge 2)$, such that for any $A_i,A_j\in S$, $A_i\cup A_j\in S$. If $k= \min_{1\le i\le n}|A_i|\ge 2$, prove that there exist $x\in \bigcup_{i=1}^n A_i$, such that $x$ is in at least $\frac{n}{k}$ of the sets $A_1,A_2,\ldots ,A_n$ (Here $|X|$ denotes the number of elements in finite set $X$)." 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"$P$ is a point on arc $\overarc{BC}$ of the circumcircle of $\triangle ABC$ not containing $A$, $K$ lies on segment $AP$ such that $BK$ bisects $\angle ABC$. The circumcircle of $\triangle KPC$ meets $AC,BD$ at $D,E$ respectively. $PE$ meets $AB$ at $F$. Prove that $\angle ABC=2\angle FCB$." 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$." 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Let $a,b,c$ be nonnegative real numbers.Prove that$$\frac{(a-bc)^2+(b-ca)^2+(c-ab)^2}{(a-b)^2+(b-c)^2+(c-a)^2}\geq\frac{1}{2}.$$" 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$." 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Prove that there exist infinitely many positive integer triples $(a,b,c)(a,b,c>2015)$ such that $$ a|bc-1, b|ac+1, c|ab+1.$$" 2015 China Second Round Olympiad,https://artofproblemsolving.com/community/c266098_2015_china_second_round_olympiad,"Given positive integers $m,n(2\le m\le n)$, let $a_1,a_2,\ldots ,a_m$ be a permutation of any $m$ pairwise distinct numbers taken from $1,2,\ldots ,n$. If there exist $k\in\{1,2,\ldots ,m\}$ such that $a_k+k$ is odd, or there exist positive integers $k,l(1\le ka_l$, then call $a_1,a_2,\ldots ,a_m$ a good sequence. Find the number of good sequences." 2014 China Second Round Olympiad,https://artofproblemsolving.com/community/c266517_2014_china_second_round_olympiad,"Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that$$bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.$$" 2014 China Second Round Olympiad,https://artofproblemsolving.com/community/c266517_2014_china_second_round_olympiad,"Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$." 2014 China Second Round Olympiad,https://artofproblemsolving.com/community/c266517_2014_china_second_round_olympiad,"Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets." 2014 China Second Round Olympiad,https://artofproblemsolving.com/community/c266517_2014_china_second_round_olympiad,"Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among $$x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}$$ no two are congurent modulo $4028$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"$AB$ is a chord of circle $\omega$, $P$ is a point on minor arc $AB$, $E,F$ are on segment $AB$ such that $AE=EF=FB$. $PE,PF$ meets $\omega$ at $C,D$ respectively. Prove that $EF\cdot CD=AC\cdot BD$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$, $$\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}$$ Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"Let $n,k$ be integers greater than $1$, $n<2^k$. Prove that there exist $2k$ integers none of which are divisible by $n$, such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by $n$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"Let $n$ be a positive odd integer , $a_1,a_2,\cdots,a_n$ be any permutation of the positive integers $1,2,\cdots,n$ . Prove that :$(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n)$ is an even number." 2013 China Second Round Olympiad,https://artofproblemsolving.com/community/c272807_2013_china_second_round_olympiad,"The integers $n>1$ is given . The positive integer $a_1,a_2,\cdots,a_n$ satisfing condition : (1) $a_10)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"In $\triangle ABC$, the corresponding sides of angle $A,B,C$ are $a,b,c$ respectively. If $a\cos B-b\cos A=\frac{3}{5}c$, find the value of $\frac{\tan A}{\tan B}$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression $$\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.$$" 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Find the sum of all integers $n$ satisfying the following inequality: $$\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.$$" 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. (1) If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. (2) If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Given a sequence $\{a_n\}$ whose terms are non-zero real numbers. For any positive integer $n$, the equality $$(\sum_{i=1}^{n}a_i)^2=\sum_{i=1}^{n}a_i^3$$ holds. (1) If $n=3$, find all possible sequence $a_1,a_2,a_3$; (2) Does there exist such a sequence $\{a_n\}$ such that $a_{2011}=-2012$?" 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. (1) Prove that $|OA|\cdot |OB|$ is a constant. (2) Find the locus of $C$ if $A$ is a point on the semicircle $$(x-2)^2+y^2=4 \quad (2\le x\le 4).$$" 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"In an acute-angled triangle $ABC$, $AB>AC$. $M,N$ are distinct points on side $BC$ such that $\angle BAM=\angle CAN$. Let $O_1,O_2$ be the circumcentres of $\triangle ABC, \triangle AMN$, respectively. Prove that $O_1,O_2,A$ are collinear." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Prove that the set $\{2,2^2,\ldots,2^n,\ldots\}$ satisfies the following properties: (1) For every $a\in A, b\in\mathbb{N}$, if $b<2a-1$, then $b(b+1)$ isn't a multiple of $2a$; (2) For every positive integer $a\notin A,a\ne 1$, there exists a positive integer $b$, such that $b<2a-1$ and $b(b+1)$ is a multiple of $2a$." 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Let $P_0 ,P_1 ,P_2 , ... ,P_n$ be $n+1$ points in the plane. Let $d$($d>0$) denote the minimal value of all the distances between any two points. Prove that $$|P_0P_1|\cdot |P_0P_2|\cdot ... \cdot |P_0P_n|>(\frac{d}{3})^n\sqrt{(n+1)!}.$$" 2012 China Second Round Olympiad,https://artofproblemsolving.com/community/c4289_2012_china_second_round_olympiad,"Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that $$a...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i0$, compare $a_{n+1}$ with $a_n$." 2011 China Second Round Olympiad,https://artofproblemsolving.com/community/c4288_2011_china_second_round_olympiad,"A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$. (1) Prove that the locus of the incenter of triangle $PAB$ is a segment, (2) If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$." 2010 China Second Round Olympiad,https://artofproblemsolving.com/community/c4287_2010_china_second_round_olympiad,"Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic." 2010 China Second Round Olympiad,https://artofproblemsolving.com/community/c4287_2010_china_second_round_olympiad,"Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer." 2010 China Second Round Olympiad,https://artofproblemsolving.com/community/c4287_2010_china_second_round_olympiad,"let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$." 2010 China Second Round Olympiad,https://artofproblemsolving.com/community/c4287_2010_china_second_round_olympiad,"the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide. find the number of all possible codes(in terms of $n$)." 2009 China Second Round Olympiad,https://artofproblemsolving.com/community/c4286_2009_china_second_round_olympiad,"Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$. 1) Prove that $MP\cdot MT=NP\cdot NT$; 2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic." 2009 China Second Round Olympiad,https://artofproblemsolving.com/community/c4286_2009_china_second_round_olympiad,"Let $n$ be a positive integer. Prove that $$-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}$$" 2009 China Second Round Olympiad,https://artofproblemsolving.com/community/c4286_2009_china_second_round_olympiad,"Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$." 2009 China Second Round Olympiad,https://artofproblemsolving.com/community/c4286_2009_china_second_round_olympiad,"Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: Every row consists of distinct numbers; $\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; $x_{17}=x_{28}=x_{39}=0$; $x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$. Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists." 2008 China Second Round Olympiad,https://artofproblemsolving.com/community/c4285_2008_china_second_round_olympiad,"Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$. (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$." 2008 China Second Round Olympiad,https://artofproblemsolving.com/community/c4285_2008_china_second_round_olympiad,"Let $f(x)$ be a periodic function with periods $T$ and $1$($0a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$." 2008 China Second Round Olympiad,https://artofproblemsolving.com/community/c4285_2008_china_second_round_olympiad,"For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying (1)$0=f(0)log_x 2-1$. Then the range of $x$ is ${ \textbf{(A)}\ \frac{1}{2}\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $" 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is ${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $" 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$" 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Let $S$ be the set of all those 2007 place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is ${ \textbf{(A)}\ \frac{1}{2}(10^{2006}+8^{2006})\qquad\textbf{(B)}\ \frac{1}{2}(10^{2006}-8^{2006})\qquad\textbf{(C)}\ 10^{2006}+8^{2006}\qquad \textbf{(D)}}\ 10^{2006}-8^{2006}\qquad $" 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$. 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,Let complex number $z = (a+\cos\theta)+(2a-\sin \theta)i$. Find the range of real number $a$ if $|z|\ge 2$ for any $\theta\in \mathbb{R}$. 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value." 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls." 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,Find the number of real solutions to the equation $(x^{2006}+1)(1+x^2+x^4+\ldots +x^{2004})=2006x^{2005}$ 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,Suppose there are 8 white balls and 2 red balls in a packet. Each time one ball is drawn and replaced by a white one. Find the probability that the last red ball is drawn in the fourth draw. 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$." 2006 China Second Round Olympiad,https://artofproblemsolving.com/community/c4283_2006_china_second_round_olympiad,"Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S=\sum_{1\le iAC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively." 2005 China Second Round Olympiad,https://artofproblemsolving.com/community/c4282_2005_china_second_round_olympiad,"Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function $$ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. $$" 2005 China Second Round Olympiad,https://artofproblemsolving.com/community/c4282_2005_china_second_round_olympiad,"For each positive integer, define a function $$ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. $$ Find the value of $\sum_{k=1}^{200} f(k)$." 2004 China Second Round Olympiad,https://artofproblemsolving.com/community/c4281_2004_china_second_round_olympiad,"In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$." 2004 China Second Round Olympiad,https://artofproblemsolving.com/community/c4281_2004_china_second_round_olympiad,"In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$. The x-intercept of line $A_nB_n$ is $a_n$, and the x-coordinate of point $B_n$ is $b_n$, $n\in\mathbb{N}$. Prove that (1) $a_n>a_{n+1}>4$, $n\in\mathbb{N}$; (2) There is $n_0\in\mathbb{N}$, such that for any $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle." 2003 China Second Round Olympiad,https://artofproblemsolving.com/community/c4280_2003_china_second_round_olympiad,"Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$, $l\ge q^2(q+1)^2+1$, $q\ge2$, $q\in\mathbb{N}$. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$ in the figure." 2002 China Second Round Olympiad,https://artofproblemsolving.com/community/c4279_2002_china_second_round_olympiad,"In $\triangle ABC$, $\angle A = 60$, $AB>AC$, point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$. Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$. Determine the value of $\frac{MH+NH}{OH}$." 2002 China Second Round Olympiad,https://artofproblemsolving.com/community/c4279_2002_china_second_round_olympiad,"There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying $(1) x_2-x_1=\lambda$ $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$" 2002 China Second Round Olympiad,https://artofproblemsolving.com/community/c4279_2002_china_second_round_olympiad,"Before The World Cup tournament, the football coach of $F$ country will let seven players, $A_1, A_2, \ldots, A_7$, join three training matches (90 minutes each) in order to assess them. Suppose, at any moment during a match, one and only one of them enters the field, and the total time (which is measured in minutes) on the field for each one of $A_1, A_2, A_3$, and $A_4$ is divisible by $7$ and the total time for each of $A_5, A_6$, and $A_7$ is divisible by $13$. If there is no restriction about the number of substitutions of players during each match, then how many possible cases are there within the total time for every player on the field?" 2001 China Second Round Olympiad,https://artofproblemsolving.com/community/c920572_2001_china_second_round_olympiad,"Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. Line $AH$ and $BC$ intersect at $D,$ Line $BH$ and $AC$ intersect at $E,$ Line $CH$ and $AB$ intersect at $F,$ Line $AB$ and $ED$ intersect at $M,$ $AC$ and $FD$ intersect at $N.$ Prove that $(1)OB\perp DF,OC\perp DE;$ $(2)OH\perp MN.$" 2001 China Second Round Olympiad,https://artofproblemsolving.com/community/c920572_2001_china_second_round_olympiad,"If nonnegative reals $x_1, x_2, \ldots, x_n$ satisfy $$ \sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1 $$ what are the minimum and maximum values of $\sum_{i=1}^n x_i$?" 2001 China Second Round Olympiad,https://artofproblemsolving.com/community/c920572_2001_china_second_round_olympiad,"An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares." 2000 China Second Round Olympiad,https://artofproblemsolving.com/community/c922130_2000_china_second_round_olympiad,"In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$). Prove that $S_{AMDN}=S_{\triangle ABC}.$" 2000 China Second Round Olympiad,https://artofproblemsolving.com/community/c922130_2000_china_second_round_olympiad,"Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$ $$ \begin{cases} a_{n+1}=7a_n+6b_n-3, \\ b_{n+1}=8a_n+7b_n-4. \end{cases} $$Prove that for any non-negative integer $n,$ $a_n$ is a perfect square." 2000 China Second Round Olympiad,https://artofproblemsolving.com/community/c922130_2000_china_second_round_olympiad,"There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$" 1999 China Second Round Olympiad,https://artofproblemsolving.com/community/c925936_1999_china_second_round_olympiad,"In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$" 1999 China Second Round Olympiad,https://artofproblemsolving.com/community/c925936_1999_china_second_round_olympiad,"Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$." 1999 China Second Round Olympiad,https://artofproblemsolving.com/community/c925936_1999_china_second_round_olympiad,"$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$ $(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$ $(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined." 2012 Albania National Olympiad,https://artofproblemsolving.com/community/c3972_2012_albania_national_olympiad,Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes. 2012 Albania National Olympiad,https://artofproblemsolving.com/community/c3972_2012_albania_national_olympiad,The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$. 2012 Albania National Olympiad,https://artofproblemsolving.com/community/c3972_2012_albania_national_olympiad,"Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression $$\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)$$ does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$." 2012 Albania National Olympiad,https://artofproblemsolving.com/community/c3972_2012_albania_national_olympiad,"Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)$$ for all $x\in\mathbb{R}$." 2012 Albania National Olympiad,https://artofproblemsolving.com/community/c3972_2012_albania_national_olympiad,"Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$. a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$. b) In what ratio does the segment $EP$ divide the segment $OV$?" 2011 Albania National Olympiad,https://artofproblemsolving.com/community/c3971_2011_albania_national_olympiad,"(a) Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. (b) Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$." 2011 Albania National Olympiad,https://artofproblemsolving.com/community/c3971_2011_albania_national_olympiad,"Find all the values that can take the last digit of a ""perfect"" even number. (The natural number $n$ is called ""perfect"" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$)." 2011 Albania National Olympiad,https://artofproblemsolving.com/community/c3971_2011_albania_national_olympiad,"In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$)." 2011 Albania National Olympiad,https://artofproblemsolving.com/community/c3971_2011_albania_national_olympiad,"The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. (a) Prove that for every even $n$, $a_{n}$ is divisible by $n!$. (b) Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$." 2011 Albania National Olympiad,https://artofproblemsolving.com/community/c3971_2011_albania_national_olympiad,"The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that: (a) $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$ (b) $\angle ACB \geq 60^o$." 2010 Albania National Olympiad,https://artofproblemsolving.com/community/c3970_2010_albania_national_olympiad,"Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle). Albanian National Mathematical Olympiad 2010---12 GRADE Question 1." 2010 Albania National Olympiad,https://artofproblemsolving.com/community/c3970_2010_albania_national_olympiad,"We denote $N_{2010}=\{1,2,\cdots,2010\}$ (a)How many non empty subsets does this set have? (b)For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? (c)Same question as the (b) part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2." 2010 Albania National Olympiad,https://artofproblemsolving.com/community/c3970_2010_albania_national_olympiad,"(a)Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. (b)What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3." 2010 Albania National Olympiad,https://artofproblemsolving.com/community/c3970_2010_albania_national_olympiad,"The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. (a) Prove that $f_{2010} $ is divisible by $10$. (b) Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4." 2010 Albania National Olympiad,https://artofproblemsolving.com/community/c3970_2010_albania_national_olympiad,"All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. (a)Prove that at least $4S+10$ senators were left outside the commissions. (b)Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$, where $d$ is the length of the segment $I$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"The reals $b>0$ and $a$ are such that the quadratic $x^2+ax+b$ has two distinct real roots, exactly one of which lies in the interval $[-1;1]$. Prove that one of the roots lies in the interval $(-b;b)$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Given is a non-isosceles triangle $ABC$ with $20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {$1,2,3$}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$; else it is successful. Prove that the sages can create a strategy, by which they can win the game." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Find all sets of positive integers $\{x_1, x_2, \dots, x_{20}\}$ such that $$x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})$$for $i=1, 2, \dots, 20$ where $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?" 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle , such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean. 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Given is a cyclic pentagon $ABCDE$, inscribed in a circle $k$. The line $CD$ intersects $AB$ and $AE$ in $X$ and $Y$ respectively. Segments $EX$ and $BY$ intersect again at $P$, and they intersect $k$ in $Q$ and $R$, respectively. Point $A'$ is reflection of $A$ across $CD$. The circles $(PQR)$ and $(A'XY)$ intersect at $M$ and $N$. Prove that $CM$ and $DN$ intersect on $(PQR)$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"For some positive integer $n>m$, it turns out that it is representable as sum of $2021$ non-negative integer powers of $m$, and that it is representable as sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$, tangent to $G$ at $A_{2}$ passes through $A_{3}$, $\dots$, tangent to $G$ at $A_{n}$ passes through $A_{1}$?" 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Some language has only three letters - $A, B$ and $C$. A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all $A$. What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?" 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,In triangle $ABC$ angle bisectors $AA_{1}$ and $CC_{1}$ intersect at $I$. Line through $B$ parallel to $AC$ intersects rays $AA_{1}$ and $CC_{1}$ at points $A_{2}$ and $C_{2}$ respectively. Let $O_{a}$ and $O_{c}$ be the circumcenters of triangles $AC_{1}C_{2}$ and $CA_{1}A_{2}$ respectively. Prove that $\angle{O_{a}BO_{c}} = \angle{AIC} $ 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Find all permutations $(a_1, a_2,...,a_{2021})$ of $(1,2,...,2021)$, such that for every two positive integers $m$ and $n$ with difference bigger than $20^{21}$, the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$." 2021 All-Russian Olympiad,https://artofproblemsolving.com/community/c1988301_2021_allrussian_olympiad,"Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,There are 5 points on plane. Prove that you can chose some of them and shift them such that distances between shifted points won't change and as a result there will be symetric by some line set of 5 points. 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Find minimal natural $n$ for which there exist integers $a_1, a_2,\ldots, a_n$ such that quadratic trinom $$x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)$$has at least one integral root." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Circle $\Omega$ with center $O$ is the circumcircle of an acute triangle $\triangle ABC$ with $AB1$ identical cardboard rectangles and distributed them to $n$ children; every child got one rectangle. Every child cut his (her) rectangle into several identical squares (squares of different children could be different). Finally, the total number of squares was prime. Prove that initial rectangles was squares." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,There is point $D$ on edge $AC$ isosceles triangle $ABC$ with base $BC$. There is point $K$ on the smallest arc $CD$ of circumcircle of triangle $BCD$. Ray $CK$ intersects line parallel to line $BC$ through $A$ at point $T$. Let $M$ be midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$. 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Among 16 coins there are 8 heavy coins with weight of 11 g, and 8 light coins with weight of 10 g, but it's unknown what weight of any coin is. One of the coins is anniversary. How to know, is anniversary coin heavy or light, via three weighings on scales with two cups and without any weight?" 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"For $a,b,c$ be real numbers greater than $1$, prove that $$\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.$$" 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$" 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?" 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Let $ABC$ be an acute-angled triangle with $AC2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin? ThanksThanks to the user Vlados021 for translating the problem." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"A triangular pyramid $ABCD$ is given. A sphere $\omega_A$ is tangent to the face $BCD$ and to the planes of other faces in points don't lying on faces. Similarly, sphere $\omega_B$ is tangent to the face $ACD$ and to the planes of other faces in points don't lying on faces. Let $K$ be the point where $\omega_A$ is tangent to $ACD$, and let $L$ be the point where $\omega_B$ is tangent to $BCD$. The points $X$ and $Y$ are chosen on the prolongations of $AK$ and $BL$ over $K$ and $L$ such that $\angle CKD = \angle CXD + \angle CBD$ and $\angle CLD = \angle CYD +\angle CAD$. Prove that the distances from the points $X$, $Y$ to the midpoint of $CD$ are the same. thanksThanks to the user Vlados021 for translating the problem." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$? thanksThanks to the user Vlados021 for translating the problem." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"In the segment $AC$ of an isosceles triangle $\triangle ABC$ with base $BC$ is chosen a point $D$. On the smaller arc $CD$ of the circumcircle of $\triangle BCD$ is chosen a point $K$. Line $CK$ intersects the line through $A$ parallel to $BC$ at $T$. $M$ is the midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$. (A.Kuznetsov)" 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"There are non-constant polynom $P(x)$ with integral coefficients and natural number $n$. Suppose that $a_0=n$, $a_k=P(a_{k-1})$ for any natural $k$. Finally, for every natural $b$ there is number in sequence $a_0, a_1, a_2, \ldots$ that is $b$-th power of some natural number that is more than 1. Prove that $P(x)$ is linear polynom." 2019 All-Russian Olympiad,https://artofproblemsolving.com/community/c868289_2019_allrussian_olympiad,"A positive integer $n$ is given. A cube $3\times3\times3$ is built from $26$ white and $1$ black cubes $1\times1\times1$ such that the black cube is in the center of $3\times3\times3$-cube. A cube $3n\times 3n\times 3n$ is formed by $n^3$ such $3\times3\times3$-cubes. What is the smallest number of white cubes which should be colored in red in such a way that every white cube will have at least one common vertex with a red one. thanksThanks to the user Vlados021 for translating the problem." 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers." 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"Circle $\omega$ is tangent to sides $AB, AC$ of triangle $ABC$. A circle $\Omega$ touches the side $AC$ and line $AB$ (produced beyond $B$), and touches $\omega$ at a point $L$ on side $BC$. Line $AL$ meets $\omega, \Omega$ again at $K, M$. It turned out that $KB \parallel CM$. Prove that $\triangle LCM$ is isosceles." 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that $$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$(As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)" 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"On the $n\times n$ checker board, several cells were marked in such a way that lower left ($L$) and upper right($R$) cells are not marked and that for any knight-tour from $L$ to $R$, there is at least one marked cell. For which $n>3$, is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?" 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?" 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,$a$ and $b$ are given positive integers. Prove that there are infinitely many positive integers $n$ such that $n^b+1$ doesn't divide $a^n+1$. 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card." 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,$ABCD$ is a convex quadrilateral. Angles $A$ and $C$ are equal. Points $M$ and $N$ are on the sides $AB$ and $BC$ such that $MN||AD$ and $MN=2AD$. Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of $\triangle ABC$. Prove that $HK$ is perpendicular to $CD$. 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,Determine the number of real roots of the equation $$|x|+|x+1|+\cdots+|x+2018|=x^2+2018x-2019$$ 2018 All-Russian Olympiad,https://artofproblemsolving.com/community/c654052_2018_allrussian_olympiad,"Let $\triangle ABC$ be an acute-angled triangle with $AB3$ different natural numbers, less than $(n-1)!$ For every pair of numbers Ivan divides bigest on lowest and write integer quotient (for example, $100$ divides $7$ $= 14$) and write result on the paper. Prove, that not all numbers on paper are different." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"$a,b,c$ - different natural numbers. Can we build quadratic polynomial $P(x)=kx^2+lx+m$, with $k,l,m$ are integer, $k>0$ that for some integer points it get values $a^3,b^3,c^3$ ?" 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"In the scalene triangle $ABC$,$\angle ACB=60$ and $\Omega$ is its cirumcirle.On the bisectors of the angles $BAC$ and $CBA$ points $A^\prime$,$B^\prime$ are chosen respectively such that $AB^\prime \parallel BC$ and $BA^\prime \parallel AC$.$A^\prime B^\prime$ intersects with $\Omega$ at $D,E$.Prove that triangle $CDE$ is isosceles.(A. Kuznetsov)" 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"Every cell of $100\times 100$ table is colored black or white. Every cell on table border is black. It is known, that in every $2\times 2$ square there are cells of two colors. Prove, that exist $2\times 2$ square that is colored in chess order." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"Let $ABC$ be an acute angled isosceles triangle with $AB=AC$ and circumcentre $O$. Lines $BO$ and $CO$ intersect $AC, AB$ respectively at $B', C'$. A straight line $l$ is drawn through $C'$ parallel to $AC$. Prove that the line $l$ is tangent to the circumcircle of $\triangle B'OC$." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"There are 3 heaps with $100,101,102$ stones. Ilya and Kostya play next game. Every step they take one stone from some heap, but not from same, that was on previous step. They make his steps in turn, Ilya make first step. Player loses if can not make step. Who has winning strategy?" 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"$n$ is composite. $11$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?" 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"In the $200\times 200$ table in some cells lays red or blue chip. Every chip ""see"" other chip, if they lay in same row or column. Every chip ""see"" exactly $5$ chips of other color. Find maximum number of chips in the table." 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board, erases it, then writes all divisors of $a$ except $a$( Can be same numbers on the board). After some time on the board there are $N^2$ numbers. For which $N$ is it possible?" 2017 All-Russian Olympiad,https://artofproblemsolving.com/community/c480484_2017_allrussian_olympiad,"Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)" 2016 All-Russian Olympiad,https://artofproblemsolving.com/community/c266043_2016_allrussian_olympiad,"A carpet dealer,who has a lot of carpets in the market,is available to exchange a carpet of dimensions $a\cdot b$ either with a carpet with dimensions $\frac{1}{a}\cdot \frac{1}{b}$ or with two carpets with dimensions $c\cdot b$ and $\frac{a}{c}\cdot b$ (the customer can select the number $c$).The dealer supports that,at the beginning he had a carpet with dimensions greater than $1$ and,after some exchanges like the ones we described above,he ended up with a set of carpets,each one having one dimension greater than $1$ and one smaller than $1$.Is this possible? Note:The customer can demand from the dealer to consider a carpet of dimensions $a\cdot b$ as one with dimensions $b\cdot a$." 2016 All-Russian Olympiad,https://artofproblemsolving.com/community/c266043_2016_allrussian_olympiad,"$\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$.An arbitrary line $ \ell $ intersects with $AB,AC$ at $K,L$,respectively and intersect with $\omega$ at $P,Q$.Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$.Prove that $P,Q,S,T$ are concyclic.(I.Bogdanov,P.Kozhevnikov)" 2016 All-Russian Olympiad,https://artofproblemsolving.com/community/c266043_2016_allrussian_olympiad,"Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_11$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges). Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$. Prove, that result of last major is multiple of $2016$ too." 2016 All-Russian Olympiad,https://artofproblemsolving.com/community/c266043_2016_allrussian_olympiad,"All russian olympiad 2016,Day 2 ,grade 9,P8 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2}$$All russian olympiad 2016,Day 2,grade 11,P7 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that $$\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3}$$Russia national 2016" 2016 All-Russian Olympiad,https://artofproblemsolving.com/community/c266043_2016_allrussian_olympiad,"Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov) P.Ssorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak:" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2+ax+b$ and $x^2+bx+a$ has two distinct real roots,and the product of these trinomials has exactly three distinct real roots.Determine all possible values of the sum of these three roots. (S.Berlov)" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$." 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,$100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such circle. (S.Berlov) 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. (S.Berlov,A.Magazinov)" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"An acute-angled $ABC \ (AB 1$ be a natural number. We write out the fractions $\frac{1}{n}$, $\frac{2}{n}$, $\dots$ , $\dfrac{n-1}{n}$ such that they are all in their simplest form. Let the sum of the numerators be $f(n)$. For what $n>1$ is one of $f(n)$ and $f(2015n)$ odd, but the other is even?" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$ teams. Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams." 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$" 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$." 2015 All-Russian Olympiad,https://artofproblemsolving.com/community/c255746_2015_allrussian_olympiad,"Given natural numbers $a$ and $b$, such that $aBC$. Let $\Omega $ be the circumcircle of $ ABC$. The tangents to $ \Omega $ at the points $A$ and $C$ meet at $P$, and $BP$ and $AC$ intersect at $S$. Let $AD$ be the altitude of the triangle $ABP$ and $\omega$ the circumcircle of the triangle $CSD$. Suppose $ \omega$ and $ \Omega $ intersect at $K\not= C$. Prove that $ \angle CKM=90^\circ $. V. Shmarov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. N. Agakhanov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic. I. Bogdanov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,Let $a$ be good if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers? 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Given a function $f\colon \mathbb{R}\rightarrow \mathbb{R} $ with $f(x)^2\le f(y)$ for all $x,y\in\mathbb{R} $, $x>y$, prove that $f(x)\in [0,1] $ for all $x\in \mathbb{R}$." 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"There are $n$ cells with indices from $1$ to $n$. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the $i$-th cell is now a card with the number $a_i$. Petya can swap any two cards with the numbers $x$ and $y$, but he must pay $2|x-y|$ coins. Show that Petya can return all the cards to their original position, not paying more than $|a_1-1|+|a_2-2|+\ldots +|a_n-n|$ coins." 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. M. Kungodjin" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Let $M$ be the midpoint of the side $AC$ of $ \triangle ABC$. Let $P\in AM$ and $Q\in CM$ be such that $PQ=\frac{AC}{2}$. Let $(ABQ)$ intersect with $BC$ at $X\not= B$ and $(BCP)$ intersect with $BA$ at $Y\not= B$. Prove that the quadrilateral $BXMY$ is cyclic. F. Ivlev, F. Nilov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons. 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Does there exist positive $a\in\mathbb{R}$, such that $$|\cos x|+|\cos ax| >\sin x +\sin ax $$ for all $x\in\mathbb{R}$? N. Agakhanov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Peter and Bob play a game on a $n\times n$ chessboard. At the beginning, all squares are white apart from one black corner square containing a rook. Players take turns to move the rook to a white square and recolour the square black. The player who can not move loses. Peter goes first. Who has a winning strategy?" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least $15$. A. Golovanov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. M. Kungodjin" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Call a natural number $n$ good if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers. S. Berlov" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere." 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?" 2014 All-Russian Olympiad,https://artofproblemsolving.com/community/c5174_2014_allrussian_olympiad,"Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card. E. Lakshtanov" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations $$(x-a)(x-b)=x-c$$ $$(x-c)(x-b)=x-a$$ $$(x-c)(x-a)=x-b$$ have real solutions." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$. 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Squares $CAKL$ and $CBMN$ are constructed on the sides of acute-angled triangle $ABC$, outside of the triangle. Line $CN$ intersects line segment $AK$ at $X$, while line $CL$ intersects line segment $BM$ at $Y$. Point $P$, lying inside triangle $ABC$, is an intersection of the circumcircles of triangles $KXN$ and $LYM$. Point $S$ is the midpoint of $AB$. Prove that angle $\angle ACS=\angle BCP$." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--(0,1)); draw ((0,0)--(1,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1,.5)); draw ((1,.5)--(1,0)); draw ((1,.5)--(1,0)); [/asy]" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ new points , dividing the circle into $ n $ new arcs. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) I. Mitrophanov" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. V. Senderov" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Inside the inscribed quadrilateral $ABCD$ are marked points $P$ and $Q$, such that $\angle PDC + \angle PCB,$ $\angle PAB + \angle PBC,$ $\angle QCD + \angle QDA$ and $\angle QBA + \angle QAD$ are all equal to $90^\circ$. Prove that the line $PQ$ has equal angles with lines $AD$ and $BC$. A. Pastor" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Does exist natural $n$, such that for any non-zero digits $a$ and $b$ $$\overline {ab}\ |\ \overline {anb}\ ?$$ (Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.) V. Senderov" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out? A. Golovanov" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. L. Emelyanov, A. Polyansky" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be? I. Bogdanov, D. Fon-Der-Flaass" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle. 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one)." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$." 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that $$ a^2+b^2+c^2+d^2 \ge abcd .$$" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,"The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)" 2013 All-Russian Olympiad,https://artofproblemsolving.com/community/c5173_2013_allrussian_olympiad,Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$. 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Let $a_1,\ldots a_{11}$ be distinct positive integers, all at least $2$ and with sum $407$. Does there exist an integer $n$ such that the sum of the $22$ remainders after the division of $n$ by $a_1,a_2,\ldots ,a_{11},4a_1,4a_2,\ldots ,4a_{11}$ is $2012$?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from given $2012$ and construct a convex $k$-gon without parallel sides. 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"The positive real numbers $a_1,\ldots ,a_n$ and $k$ are such that $a_1+\cdots +a_n=3k$, $a_1^2+\cdots +a_n^2=3k^2$ and $a_1^3+\cdots +a_n^3>3k^3+k$. Prove that the difference between some two of $a_1,\ldots,a_n$ is greater than $1$." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Initially, ten consecutive natural numbers are written on the board. In one turn, you may pick any two numbers from the board (call them $a$ and $b$) and replace them with the numbers $a^2-2011b^2$ and $ab$. After several turns, there were no initial numbers left on the board. Could there, at this point, be again, ten consecutive natural numbers?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$. 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Any two of the real numbers $a_1,a_2,a_3,a_4,a_5$ differ by no less than $1$. There exists some real number $k$ satisfying $$a_1+a_2+a_3+a_4+a_5=2k$$$$a_1^2+a_2^2+a_3^2+a_4^2+a_5^2=2k^2$$ Prove that $k^2\ge 25/3$." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"On a Cartesian plane, $n$ parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than $2(n-1)$ corners (i.e. the intersections of a pair of parabolas)." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Initially, there are $111$ pieces of clay on the table of equal mass. In one turn, you can choose several groups of an equal number of pieces and push the pieces into one big piece for each group. What is the least number of turns after which you can end up with $11$ pieces no two of which have the same mass?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let $\ell$ be a line not parallel to the sides of any squares. For every line segment $I$ that is parallel to $\ell$, we can count the difference between the length of its red and its blue areas. Prove that for every such line $\ell$ there exists a number $C$ that exceeds all those differences that we can calculate." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have $$P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)$$ Prove that the polynomial $P(x)$ has at least one real root." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too." 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,"On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends. Who has a winning strategy: the starting player or his opponent?" 2012 All-Russian Olympiad,https://artofproblemsolving.com/community/c5172_2012_allrussian_olympiad,For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$. 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers? 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in his notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"There are some counters in some cells of $100\times 100$ board. Call a cell nice if there are an even number of counters in adjacent cells. Can exactly one cell be nice? K. Knop" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"In every cell of a table with $n$ rows and ten columns, a digit is written. It is known that for every row $A$ and any two columns, you can always find a row that has different digits from $A$ only when it intersects with two columns. Prove that $n\geq512$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"The graph $G$ is not $3$-coloured. Prove that $G$ can be divided into two graphs $M$ and $N$ such that $M$ is not $2$-coloured and $N$ is not $1$-coloured. V. Dolnikov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Perimeter of triangle ABC is 4. Point X is marked at ray AB and point Y is marked at ray AC such that AX=AY=1. BC intersects XY at point M. Prove that perimeter of one of triangles ABM or ACM is 2. (V. Shmarov)." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"For positive integers $a>b>1$, define $$x_n = \frac {a^n-1}{b^n-1}$$ Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. V. Senderov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells. I. Bogdanov & O. Podlipsky" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme. A. Magazinov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order. S. Berlov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$." 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"There are more than $n^2$ stones on the table. Peter and Vasya play a game, Peter starts. Each turn, a player can take any prime number less than $n$ stones, or any multiple of $n$ stones, or $1$ stone. Prove that Peter always can take the last stone (regardless of Vasya's strategy). S Berlov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. A. Golovanov" 2011 All-Russian Olympiad,https://artofproblemsolving.com/community/c5171_2011_allrussian_olympiad,"Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle. M. Kungojin" 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams. 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Let us call a natural number $unlucky$ if it cannot be expressed as $\frac{x^2-1}{y^2-1} $ with natural numbers $x,y >1$. Is the number of $unlucky$ numbers finite or infinite?" 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"In a acute triangle $ABC$, the median, $AM$, is longer than side $AB$. Prove that you can cut triangle $ABC$ into $3$ parts out of which you can construct a rhombus." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$. 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more. Find maximum $d$ for which we can guaranteed paint at least $d$ points." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$. Prove, that $P$, $Q$, $M$ lies at one line." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other. Proved, that we can divided all country on $4$ regions, such that any flight connected towns from other regions." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which $$(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? $$" 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"On an $n\times n$ chart, where $n \geq 4$, stand ""$+$"" signs in the cells of the main diagonal and ""$-$"" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true? For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$. (The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)" 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes." 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Could the four centers of the circles inscribed into the faces of a tetrahedron be coplanar? (vertexes of tetrahedron not coplanar)" 2010 All-Russian Olympiad,https://artofproblemsolving.com/community/c5170_2010_allrussian_olympiad,"Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_11$ and $ a$ so that $ a>n^2$, and among the integers $ a+1, a+2, \ldots, a+n$ one can find a multiple of each of the numbers $ n^2+1, n^2+2, \ldots, n^2+n$. Prove that $ a>n^4-n^3$." 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?" 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"Let $ a$, $ b$, $ c$ be three real numbers satisfying that $$ \left\{\begin{array}{c c c} \left(a+b\right)\left(b+c\right)\left(c+a\right)&=&abc\\ \left(a^3+b^3\right)\left(b^3+c^3\right)\left(c^3+a^3\right)&=&a^3b^3c^3\end{array}\right.$$ Prove that $ abc=0$." 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one? 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence? 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"Triangles $ ABC$ and $ A_1B_1C_1$ have the same area. Using compass and ruler, can we always construct triangle $ A_2B_2C_2$ equal to triangle $ A_1B_1C_1$ so that the lines $ AA_2$, $ BB_2$, and $ CC_2$ are parallel?" 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax + b)^{100} - (cx + d)^{100}$ has exactly $ n$ nonzero coefficients." 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"How many times changes the sign of the function $$ f(x)=\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}$$ at the interval $ \left[0, \frac{2009\pi}{2}\right]$?" 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?" 2009 All-Russian Olympiad,https://artofproblemsolving.com/community/c5169_2009_allrussian_olympiad,"Given strictly increasing sequence $ a_14$). What is the largest possible number of scientist that could work in Niichavo that day,in terms of $ x$?" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells? 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$? 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x + 1$ and $ \frac {x}{x + 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k+1$ weighings?" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"The columns of an $ n\times n$ board are labeled $ 1$ to $ n$. The numbers $ 1,2,...,n$ are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell ""good"" if the number in it is greater than the label of its column. For which $ n$ is there an arrangement in which each row contains equally many good cells?" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM = ON$. 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"The sequences $ (a_n),(b_n)$ are defined by $ a_1=1,b_1=2$ and $$a_{n + 1} = \frac {1 + a_n + a_nb_n}{b_n}, \quad b_{n + 1} = \frac {1 + b_n + a_nb_n}{a_n}.$$ Show that $ a_{2008} < 5$." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"Determine all triplets of real numbers $ x,y,z$ satisfying $$1+x^4\leq 2(y-z)^2,\quad 1+y^4\leq 2(x-z)^2,\quad 1+z^4\leq 2(x-y)^2.$$" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1+k)(b_2+k)\cdots(b_n+k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines. 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p + b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$. 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 - ax + b = 0$ and $ x^2 - bx + a = 0$ has two integral roots?" 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle." 2008 All-Russian Olympiad,https://artofproblemsolving.com/community/c5168_2008_allrussian_olympiad,"In a chess tournament $ 2n+3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match." 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$ has a real root. O. Podlipsky" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves. N. Agakhanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$. S. Berlov, F. Petrov, A. Akopyan" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark $2007$ points on this circle, after that Amayak removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick." 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"The distance between Maykop and Belorechensk is $24$ km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most $6$ kmph) or on a bike (with velocity at most $18$ kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after $2$ hours $40$ minutes. (Only one guy may seat on the bike simultaneously). Folclore" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$. S. Berlov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"For an integer $n>3$ denote by $n?$ the product of all primes less than $n$. Solve the equation $n?=2n+16$. V. Senderov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him. A. Badzyan" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) = 0$ and $ g(f(x)) = 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) = 0$ and $ g(g(x)) = 0$ does not have real roots too. S. Berlov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. N. Agakhanov, I. Bogdanov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Two players by turns draw diagonals in a regular $(2n+1)$-gon ($n>1$). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy? K. Sukhov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. V. Astakhov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. F. Petrov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Let $ABC$ be an acute triangle. The points $M$ and $N$ are midpoints of $AB$ and $BC$ respectively, and $BH$ is an altitude of $ABC$. The circumcircles of $AHN$ and $CHM$ meet in $P$ where $P\ne H$. Prove that $PH$ passes through the midpoint of $MN$. V. Filimonov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} = 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? A. Golovanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd. A. Poliansky" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$. A. Khrabrov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal $N$ they may show such a trick? K. Knop, O. Leontieva" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given a set of $n>2$ planar vectors. A vector from this set is called long, if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. N. Agakhanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Two circles $ \omega_{1}$ and $ \omega_{2}$ intersect in points $ A$ and $ B$. Let $ PQ$ and $ RS$ be segments of common tangents to these circles (points $ P$ and $ R$ lie on $ \omega_{1}$, points $ Q$ and $ S$ lie on $ \omega_{2}$). It appears that $ RB\parallel PQ$. Ray $ RB$ intersects $ \omega_{2}$ in a point $ W\ne B$. Find $ RB/BW$. S. Berlov " 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given a convex polyhedron $F$. Its vertex $A$ has degree $5$, other vertices have degree $3$. A colouring of edges of $F$ is called nice, if for any vertex except $A$ all three edges from it have different colours. It appears that the number of nice colourings is not divisible by $5$. Prove that there is a nice colouring, in which some three consecutive edges from $A$ are coloured the same way. D. Karpov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Prove that for $k>10$ Nazar may replace in the following product some one $\cos$ by $\sin$ so that the new function $f_{1}(x)$ would satisfy inequality $|f_{1}(x)|\le 3\cdot 2^{-1-k}$ for all real $x$. $$f(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x $$ N. Agakhanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"The incircle of triangle $ABC$ touches its sides $BC$, $AC$, $AB$ at the points $A_{1}$, $B_{1}$, $C_{1}$ respectively. A segment $AA_{1}$ intersects the incircle at the point $Q\ne A_{1}$. A line $\ell$ through $A$ is parallel to $BC$. Lines $A_{1}C_{1}$ and $A_{1}B_{1}$ intersect $\ell$ at the points $P$ and $R$ respectively. Prove that $\angle PQR=\angle B_{1}QC_{1}$. A. Polyansky" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer. A. Golovanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots? N. Agakhanov, I. Bogdanov" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? A. Zaslavsky" 2007 All-Russian Olympiad,https://artofproblemsolving.com/community/c5167_2007_allrussian_olympiad,"Given an undirected graph with $N$ vertices. For any set of $k$ vertices, where $1\le k\le N$, there are at most $2k-2$ edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges). I. Bogdanov, G. Chelnokov" 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Show that there exist four integers $a$, $b$, $c$, $d$ whose absolute values are all $>1000000$ and which satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?" 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Given a triangle $ABC$. Let a circle $\omega$ touch the circumcircle of triangle $ABC$ at the point $A$, intersect the side $AB$ at a point $K$, and intersect the side $BC$. Let $CL$ be a tangent to the circle $\omega$, where the point $L$ lies on $\omega$ and the segment $KL$ intersects the side $BC$ at a point $T$. Show that the segment $BT$ has the same length as the tangent from the point $B$ to the circle $\omega$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1b_2>...>b_{10}$. Show that $a_{10}>500$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Let $P$, $Q$, $R$ be points on the sides $AB$, $BC$, $CA$ of a triangle $ABC$ such that $AP=CQ$ and the quadrilateral $RPBQ$ is cyclic. The tangents to the circumcircle of triangle $ABC$ at the points $C$ and $A$ intersect the lines $RQ$ and $RP$ at the points $X$ and $Y$, respectively. Prove that $RX=RY$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"A $100\times 100$ chessboard is cut into dominoes ($1\times 2$ rectangles). Two persons play the following game: At each turn, a player glues together two adjacent cells (which were formerly separated by a cut-edge). A player loses if, after his turn, the $100\times 100$ chessboard becomes connected, i. e. between any two cells there exists a way which doesn't intersect any cut-edge. Which player has a winning strategy - the starting player or his opponent?" 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be neighbours if a cell of one domino has a common edge with a cell of the other one.)" 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0 y_k$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"Assume that the polynomial $\left(x+1\right)^n-1$ is divisible by some polynomial $P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$, whose degree $k$ is even and whose coefficients $c_{k-1}$, $c_{k-2}$, ..., $c_1$, $c_0$ all are odd integers. Show that $k+1\mid n$." 2006 All-Russian Olympiad,https://artofproblemsolving.com/community/c5166_2006_allrussian_olympiad,"At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Given a parallelogram $ABCD$ with $AB1,\,S=a_1+a_2+a_3$. Provided ${a_i^2\over a_i-1}>S$ for every $i=1,\,2,\,3$ prove that $$\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.$$" 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?" 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty)." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"100 people from 50 countries, two from each countries, stay on a circle. Prove that one may partition them onto 2 groups in such way that neither no two countrymen, nor three consecutive people on a circle, are in the same group." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$. 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$. 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"A white plane is partitioned onto cells (in a usual way). A finite number of cells are coloured black. Each black cell has an even (0, 2 or 4) adjacent (by the side) white cells. Prove that one may colour each white cell in green or red such that every black cell will have equal number of red and green adjacent cells." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$? 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?" 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$." 2005 All-Russian Olympiad,https://artofproblemsolving.com/community/c5165_2005_allrussian_olympiad,"100 people from 25 countries, four from each countries, stay on a circle. Prove that one may partition them onto 4 groups in such way that neither no two countrymans, nor two neighbours will be in the same group." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes for sure, in which white balls lie?" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality $$ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. $$" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m + n = p + q$ and $ \sqrt{m} + \sqrt[3]{n} = \sqrt{p} + \sqrt[3]{q} > 2004$ ?" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"In the cabinet 2004 telephones are located; each two of these telephones are connected by a cable, which is colored in one of four colors. From each color there is one cable at least. Can one always select several telephones in such a way that among their pairwise cable connections exactly 3 different colors occur?" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called ""good"", if you cancel such a pair, the above property remains still valid. What is the smallest possible number of good pairs?" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $O$ be the circumcenter of an acute-angled triangle $ABC$, let $T$ be the circumcenter of the triangle $AOC$, and let $M$ be the midpoint of the segment $AC$. We take a point $D$ on the side $AB$ and a point $E$ on the side $BC$ that satisfy $\angle BDM = \angle BEM = \angle ABC$. Show that the straight lines $BT$ and $DE$ are perpendicular." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $ ABCD$ be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a tangent quadrilateral, we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral $ ABCD$ touch its sides $ AB$, $ BC$, $ CD$, and $ DA$ in the points $ K$, $ L$, $ M$, and $ N$, respectively. The exterior angle bisectors of the angles $ DAB$ and $ ABC$ intersect each other at a point $ K^{\prime}$. The exterior angle bisectors of the angles $ ABC$ and $ BCD$ intersect each other at a point $ L^{\prime}$. The exterior angle bisectors of the angles $ BCD$ and $ CDA$ intersect each other at a point $ M^{\prime}$. The exterior angle bisectors of the angles $ CDA$ and $ DAB$ intersect each other at a point $ N^{\prime}$. Prove that the straight lines $ KK^{\prime}$, $ LL^{\prime}$, $ MM^{\prime}$, and $ NN^{\prime}$ are concurrent." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"A sequence of non-negative rational numbers $ a(1), a(2), a(3), \ldots$ satisfies $ a(m) + a(n) = a(mn)$ for arbitrary natural $ m$ and $ n$. Show that not all elements of the sequence can be distinct." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"A country has 1001 cities, and each two cities are connected by a one-way street. From each city exactly 500 roads begin, and in each city 500 roads end. Now an independent republic splits itself off the country, which contains 668 of the 1001 cities. Prove that one can reach every other city of the republic from each city of this republic without being forced to leave the republic." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change. 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) - P(y) = R(x, y) (Q(x) - Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) = S(Q(x)) \quad \forall x.$" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"A rectangular array has 9 rows and 2004 columns. In the 9 * 2004 cells of the table we place the numbers from 1 to 2004, each 9 times. And we do this in such a way that two numbers, which stand in exactly the same column in and differ around at most by 3. Find the smallest possible sum of all numbers in the first row." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Let $ M = \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$" 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"Prove that there is no finite set which contains more than $ 2N,$ with $ N > 3,$ pairwise non-collinear vectors of the plane, and to which the following two characteristics apply: 1) for $ N$ arbitrary vectors from this set there are always further $ N-1$ vectors from this set so that the sum of these is $ 2N-1$ vectors is equal to the zero-vector; 2) for $ N$ arbitrary vectors from this set there are always further $ N$ vectors from this set so that the sum of these is $ 2N$ vectors is equal to the zero-vector." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,"In a country there are several cities; some of these cities are connected by airlines, so that an airline connects exactly two cities in each case and both flight directions are possible. Each airline belongs to one of $k$ flight companies; two airlines of the same flight company have always a common final point. Show that one can partition all cities in $k+2$ groups in such a way that two cities from exactly the same group are never connected by an airline with each other." 2004 All-Russian Olympiad,https://artofproblemsolving.com/community/c5164_2004_allrussian_olympiad,A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$ 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The tangents at $A$ to $S_1$ and $S_2$ meet segments $BO_2$ and $BO_1$ at $K$ and $L$ respectively. Show that $KL \parallel O_1O_2.$ 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"On a line are given $2k -1$ white segments and $2k -1$ black ones. Assume that each white segment intersects at least $k$ black segments, and each black segment intersects at least $k$ white ones. Prove that there are a black segment intersecting all the white ones, and a white segment intersecting all the black ones." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality $$\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.$$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Let $B$ and $C$ be arbitrary points on sides $AP$ and $PD$ respectively of an acute triangle $APD$. The diagonals of the quadrilateral $ABCD$ meet at $Q$, and $H_1,H_2$ are the orthocenters of triangles $APD$ and $BPC$, respectively. Prove that if the line $H_1H_2$ passes through the intersection point $X \ (X \neq Q)$ of the circumcircles of triangles $ABQ$ and $CDQ$, then it also passes through the intersection point $Y \ (Y \neq Q)$ of the circumcircles of triangles $BCQ$ and $ADQ.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b, c \in M$, the number $a^2 + bc$ is rational. Prove that there is a positive integer $n$ such that $a\sqrt n$ is rational for all $a \in M.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Consider a tree (i.e. a connected graph with no cycles) with $n$ vertices. Its vertices are assigned numbers $x_1,x_2,\cdots x_n$ and each edge is assigned the product of the numbers at its endpoints. Let $S$ denote the sum of the numbers at the deges. Prove that $$ 2S \leq \sqrt{n-1}\left(x_1^2+x_2^2+\cdots+x_n^2\right). $$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,A finite set of points $X$ and an equilateral triangle $T$ are given on a plane. Suppose that every subset $X'$ of $X$ with no more than $9$ elements can be covered by two images of $T$ under translations. Prove that the whole set $X$ can be covered by two images of $T$ under translations. 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Let $ a_0$ be a natural number. The sequence $ (a_n)$ is defined by $ a_{n+1}=\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$ and $ a_{n+1}=[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"In a triangle $ABC, O$ is the circumcenter and $I$ the incenter. The excircle $\omega_a$ touches rays $AB,AC$ and side $BC$ at $K,M,N$, respectively. Prove that if the midpoint $P$ of $KM$ lies on the circumcircle of $\triangle ABC$, then points $O,N, I$ lie on a line." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients. 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"Is it possible to write a positive integer in every cell of an infinite chessboard, in such a manner that, for all positive integers $m, n$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n$ ?" 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them." 2003 All-Russian Olympiad,https://artofproblemsolving.com/community/c5163_2003_allrussian_olympiad,"The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?" 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"Point $A$ lies on one ray and points $B,C$ lie on the other ray of an angle with the vertex at $O$ such that $B$ lies between $O$ and $C$. Let $O_1$ be the incenter of $\triangle OAB$ and $O_2$ be the center of the excircle of $\triangle OAC$ touching side $AC$. Prove that if $O_1A = O_2A$, then the triangle $ABC$ is isosceles." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"On a plane are given $6$ red, $6$ blue, and $6$ green points, such that no three of the given points lie on a line. Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which are no joined. Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"There are eight rooks on a chessboard, no two attacking each other. Prove that some two of the pairwise distances between the rooks are equal. (The distance between two rooks is the distance between the centers of their cell.)" 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?" 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0m$, where $n$,$m$ are natural numbers, $$ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.$$" 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"There are some markets in a city. Some of them are joined by one-way streets, such that for any market there are exactly two streets to leave it. Prove that the city may be partitioned into $1014$ districts such that streets join only markets from different districts, and by the same one-way for any two districts (either only from first to second, or vice-versa)." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,"The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle]." 2002 All-Russian Olympiad,https://artofproblemsolving.com/community/c5162_2002_allrussian_olympiad,Prove that there exist infinitely many natural numbers $ n$ such that the numerator of $ 1 + \frac {1}{2} + \frac {1}{3} + \frac {1}{4} + ... + \frac {1}{n}$ in the lowest terms is not a power of a prime number. 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?" 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$. Let $N$ be the midpoint of $BK$. Prove that the angles $NAK$ and $NCK$ are equal." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Consider a convex $2000$-gon, no three of whose diagonals have a common point. Each of its diagonals is colored in one of $999$ colors. Prove that there exists a triangle all of whose sides lie on diagonals of the same color. (Vertices of the triangle need not be vertices of the original polygon.)" 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?" 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"In a party, there are $2n + 1$ people. It's well known that for every group of $n$ people, there exist a person(out of the group) who knows all them(the $n$ people of the group). Show that there exist a person who knows all the people in the party." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,Let $N$ be a point on the longest side $AC$ of a triangle $ABC$. The perpendicular bisectors of $AN$ and $NC$ intersect $AB$ and $BC$ respectively in $K$ and $M$. Prove that the circumcenter $O$ of $\triangle ABC$ lies on the circumcircle of triangle $KBM$. 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a+b-1$ divides $ n$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Let $A_1, A_2, ... , A_{100}$ be subsets of a line, each a union of $100$ pairwise disjoint closed segments. Prove that the intersection of all hundred sets is a union of at most $9901$ disjoint closed segments." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"The polynomial $ P(x)=x^3+ax^2+bx+d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)=x^2+x+2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)" 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,The total mass of $100$ given weights with positive masses equals $2S$. A natural number $k$ is called middle if some $k$ of the given weights have the total mass $S$. Find the maximum possible number of middle numbers. 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons. 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Participants to an olympiad worked on $n$ problems. Each problem was worth a positive integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$. 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2+ab+b^2)|ab(a+b)$. Prove that: $ |a-b|>\sqrt [3] {ab}$." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns dominant if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road." 2001 All-Russian Olympiad,https://artofproblemsolving.com/community/c5161_2001_allrussian_olympiad,"A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Let $O$ be the center of the circumcircle $\omega$ of an acute-angle triangle $ABC$. A circle $\omega_1$ with center $K$ passes through $A$, $O$, $C$ and intersects $AB$ at $M$ and $BC$ at $N$. Point $L$ is symmetric to $K$ with respect to line $NM$. Prove that $BL \perp AC$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Some pairs of cities in a certain country are connected by roads, at least three roads going out of each city. Prove that there exists a round path consisting of roads whose number is not divisible by $3$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"The sequence $a_1 = 1$, $a_2, a_3, \cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$; otherwise, $a_{n+1} = a_n + 3$. Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"On some cells of a $2n \times 2n$ board are placed white and black markers (at most one marker on every cell). We first remove all black markers which are in the same column with a white marker, then remove all white markers which are in the same row with a black one. Prove that either the number of remaining white markers or that of remaining black markers does not exceed $n^2$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,Let $E$ be a point on the median $CD$ of a triangle $ABC$. The circle $\mathcal S_1$ passing through $E$ and touching $AB$ at $A$ meets the side $AC$ again at $M$. The circle $S_2$ passing through $E$ and touching $AB$ at $B$ meets the side $BC$ at $N$. Prove that the circumcircle of $\triangle CMN$ is tangent to both $\mathcal S_1$ and $\mathcal S_2$. 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle. An allowed operation is to add to a number the greatest common divisor of its two neighhbors. Prove that we can make all the numbers pairwise copirme in a finite number of moves. 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,Evaluate the sum $$ \left\lfloor \frac{2^0}{3} \right\rfloor + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor. $$ 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then $$ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. $$" 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is yes or no.) Can we always arrange the masses of the weights in the increasing order with at most nine measurings?" 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are midpoint of arcs $AB$ and $BC$ respectively. Circumcircles of triangles $BQK$ and $BPM$ are intersect at $L$. Show that $BPLQ$ is a parallelogram. 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Some paper squares of $k$ distinct colors are placed on a rectangular table, with sides parallel to the sides of the table. Suppose that for any $k$ squares of distinct colors, some two of them can be nailed on the table with only one nail. Prove that there is a color such that all squares of that color can be nailed with $2k-2$ nails." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that $$f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)$$ for all $x, y, z \in \mathbb R$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote $$ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. $$ Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$" 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,Prove the inequality $$ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. $$ 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line." 2000 All-Russian Olympiad,https://artofproblemsolving.com/community/c5160_2000_allrussian_olympiad,"All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"There are several cities in a country. Some pairs of the cities are connected by a two-way airline of one of the $N$ companies, so that each company serves exactly one airline from each city, and one can travel between any two cities, possibly with transfers. During a financial crisis, $N-1$ airlines have been canceled, all from different companies. Prove that it is still possible to travel between any two cities." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves? Edited by pbornsztein" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Prove that for all natural numbers $n$, $$ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. $$ Here, $\{x\}$ denotes the fractional part of $x$." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"A circle through vertices $A$ and $B$ of triangle $ABC$ meets side $BC$ again at $D$. A circle through $B$ and $C$ meets side $AB$ at $E$ and the first circle again at $F$. Prove that if points $A$, $E$, $D$, $C$ lie on a circle with center $O$ then $\angle BFO$ is right." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, $$ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. $$" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$. The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn. Show that these three lines are concurrent." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side). Prove that at least $ \left[\frac{n^2}{3}\right]$ moves are needed to reach a configuration where no more moves are possible. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"In triangle $ABC$, a circle passes through $A$ and $B$ and is tangent to $BC$. Also, a circle that passes through $B$ and $C$ is tangent to $AB$. These two circles intersect at a point $K$ other than $B$. If $O$ is the circumcenter of $ABC$, prove that $\angle{BKO}=90^\circ$." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Positive numbers $x,y$ satisfy $x^2+y^2 \ge x^3+y^4$. Prove that $x^3+y^3 \le 2$." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"In a group of 12 persons, among any 9 there are 5 which know each other. Prove that there are 6 persons in this group which know each other" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,Each rational point on a real line is assigned an integer. Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus." 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers. Prove that at least three of these numbers are equal. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line. 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"Through vertex $A$ of a tetrahedron $ABCD$ passes a plane tangent to the circumscribed sphere of the tetrahedron. Show that the lines of intersection of the plane with the planes $ABC$, $ABD$, $ACD$, form six equal angles if and only if: $$AB\cdot CD=AC\cdot BD=AD\cdot BC$$" 1999 All-Russian Olympiad,https://artofproblemsolving.com/community/c5159_1999_allrussian_olympiad,"There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc. 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,A convex polygon is partitioned into parallelograms. A vertex of the polygon is called good if it belongs to exactly one parallelogram. Prove that there are more than two good vertices. 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Let $S(x)$ denote the sum of the decimal digits of $x$. Do there exist natural numbers $a,b,c$ such that $$ S(a+b)<5, \quad S(b+c)<5, \quad S(c+a)<5, \quad S(a+b+c)> 50? $$" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A maze is an $8 \times 8$ board with some adjacent squares separated by walls, so that any two squares can be connected by a path not meeting any wall. Given a command LEFT, RIGHT, UP, DOWN, a pawn makes a step in the corresponding direction unless it encounters a wall or an edge of the chessboard. God writes a program consisting of a finite sequence of commands and gives it to the Devil, who then constructs a maze and places the pawn on one of the squares. Can God write a program which guarantees the pawn will visit every square despite the Devil's efforts?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"We are given five watches which can be winded forward. What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"In triangle $ABC$ with $AB>BC$, $BM$ is a median and $BL$ is an angle bisector. The line through $M$ and parallel to $AB$ intersects $BL$ at point $D$, and the line through $L$ and parallel to $BC$ intersects $BM$ at point $E$. Prove that $ED$ is perpendicular to $BL$." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A jeweller makes a chain consisting of $N>3$ numbered links. A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized. What is the maximum number?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points. By the way, can two sides of a polygon intersect?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$. Points $K_b$ and $K_c$ are analogously defined. Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element. 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Initially the numbers $19$ and $98$ are written on a board. Every minute, each of the two numbers is either squared or increased by $1$. Is it possible to obtain two equal numbers at some time?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Let n be an integer at least 4. In a convex n-gon, there is NO four vertices lie on a same circle. A circle is called circumscribed if it passes through 3 vertices of the n-gon and contains all other vertices. A circumscribed circle is called boundary if it passes through 3 consecutive vertices, a circumscribed circle is called inner if it passes through 3 pairwise non-consecutive points. Prove the number of boundary circles is 2 more than the number of inner circles." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called successful if each number is the product of its neighbors. Find the number of successful arrangements. 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A set $\mathcal S$ of translates of an equilateral triangle is given in the plane, and any two have nonempty intersection. Prove that there exist three points such that every triangle in $\mathcal S$ contains one of these points." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"yeah you're right,the official problem is the following one: there are 1998 cities in Russia, each being connected (in both directions) by flights to three other cities. any city can be reached by any other city by a sequence of flights. the KGB plans to close off 200 cities, no two joined by a single flight. show that this can be done so that any open city can be reached from any other open city by a sequence of flights only passing through open cities. Solution.we begin with some terminology. define a trigraph to be a connected undirected graph in which every vertex has degree at most $ 3$. a trivalent vertex of such a graph is a vertex of degree $ 3$. in this wording, the problem becomes: we have a trigraph $ G$ with $ 1998$ vertices, all of which are trivalent. we want to remove $ 200$ vertices, no two of which are adjacent, such that the remaining vertices stay connected. we remove the vertices one at a time. suppose we have deleted $ k$ of the $ 1998$ vertices, no two of which are adjacent, such the trigraph $ G'$ induced by the remaining vertices is connected. we will show that if $ K < 200$, we can always delete a trivalent vertex of $ G'$ such that the graph remains connected. this vertex cannot be adjacent in $ G$ to any of the other $ k$ deleted vertices, because then its degree in $ G'$ would be less than $ 3$. hence repeating this $ 200$ times gives us the desired set of vertices. Lemma. let $ G$ be a trigraph such that the removal of any trivalent vertex disconnects $ G$. then $ G$ is planer. moreover $ G$ can be drawn in such a way that every vertex lies on the ""outside"" face; in other words, for any point $ P$ outside some bounded set, each vertex $ v$ of $ G$ can be joined to $ P$ by a curve which does not intersect any edges of $ G$ (except at $ v$). Proof. we induct on the number of trivalent vertices of $ G$. if $ G$ does not contain any trivalent vertices, then $ G$ must be a path or a cycle and the claim is obvious. so suppose $ G$ contains $ n\geq 1$ trivalent vertices and that every trigraph with fewer trivalent vertices can be drawn as described. if $ G$ is a tree the claim is obvious, so suppose $ G$ contains a cycle; let $ v_1,\ldots ,v_k$ $ (k\geq 3)$ be a minimal cycle. let $ S = \{v_1,\ldots ,v_k\}$, and let $ T = \{i\mid v_i \textrm{ is trivalent }\}$. ($ T$ cannot be empty, because then no $ v_i$ would be connected to a vertex of degree $ 3$.) for each $ i\in T$, let $ w_i$ denote the third vertex which is adjacent to $ v_i$ (other than $ v_{i - 1}$ and $ v_{i + 1}$), and let $ S_i$ be the set of vertices in $ G$ which can be reached from $ w_i$ without passing though $ S$. (for $ i\not\in T$, let $ S_i = \emptyset$.) we claim that the sets $ S,S_1,\ldots ,S_k$ partition the vertices of $ G$. first, note that if $ v$ is a vertex of $ G$ not in $ S$ then there is a shortest path joining $ v$ to a vertex $ v_i$ of $ S$; the penultimate vertex on this path must be $ w_i$, so $ v\in S_i$. now suppose that $ v\in S_i\cap S_j$ for some $ i\neq j$; then $ v_i$ and $ v_j$ are trivalent and there exist paths $ w_i\to v$, $ w_j\to v$ which do not pass though $ S$. we will show that there exists a path from every vertex of $ G - \{v_i\}$ to $ v$ which does not pass though $ v_i$. for $ k\neq i$ there is a path $ v_k\to v_j\to w_j\to v$; if $ w\in S_k$ for $ k\neq i$, then there is a path $ w\to w_k\to v_k\to v$; if $ w\in S_i$, there is a path $ w\to w_i\to v$. since $ S\cup S_1\cup\ldots\cup S_k = G$, we have shown that the graph obtained from $ G$ by deleting $ v_i$ is connected,a contradiction, as $ v_i$ is trivalent. therefore $ S_i\cap S_j = \emptyset$ for $ i\neq j$. obviously $ S_i\cap S$ is empty for all $ i$; hence $ S,S_1,\ldots ,S_k$ partition the vertices of $ G$. let $ G',G_1,\ldots ,G_k$ be the induced subgraphs of $ S,S_1,\ldots ,S_k$ in $ G$, respectively. by construction, the only edges in $ G$ which are not in one of the graphs $ G',G_1,\ldots ,G_k$ are the edges $ v_iw_i$ for $ i\in T$. now $ G_i$ is a trigraph with fewer than $ n$ trivalent vertices, since at least one of the $ n$ trivalent vertices in $ G$ is in $ S$. hence by the inductive hypothesis, we can draw each $ G_i$ in the plane in such a way that every vertex lies on the outside face. since $ v_1,\ldots ,v_k$ was a minimal cycle, there are no ""extra"" edges between these vertices, so the graph $ G'$ is a $ k$-cycle. now place the vertices of $ S$ at the vertices of a small regular $ k$-gon far from all the graphs $ G_i$; then we can draw a curve joining each pair $ v_i,w_i$. it is east to check that this gives us a drawing of $ G$ with the desired properties. $ \square$ now suppose we have removed $ k$ vertices from $ G$, no two of which are adjacent, such that the trigraph $ G'$ induced by the remaining vertices is connected, and suppose that removing any trivalent vertex of $ G'$ disconnects the graph; we must show $ k\geq 200$. by the lemma, $ G'$ is planner. we will call a dace other than the outside one a ""proper face"". let $ F$ be the number of proper faces of $ G'$; since $ G'$ has $ 1998 - k$ vertices and $ 2997 - 3k$ edges, $ F\geq 1 - (1998 - k) + (2997 - 3k) = 1000 - 2k$. we now show that no two proper faces can share a vertex. observe that each vertex belongs to at most as many faces as its degree; thus vertices of degree $ 1$ lie only on the outside face. no two proper faces can intersect in a vertex of degree $ 2$, or that vertex would not lie on the outside face, contracting the lemma. if two proper daces intersected in a trivalent vertex $ v$, each face would give a path between two of $ v$'s neighbors, so removing $ v$ would not disconnect the graph, by an argument similar to that of the lemma. since each proper face contains at least $ 3$ vertices and no two share a vertex, we have $ 3F\leq 1998 - k$. combining this with the previous inequality gives $ 3000 - 6k\leq 3F\leq 1998 - k$ so $ 1002\leq 5k$ and $ k\geq 200$, as desired." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"Are there $1998$ different positive integers, the product of any two being divisible by the square of their difference?" 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$." 1998 All-Russian Olympiad,https://artofproblemsolving.com/community/c5158_1998_allrussian_olympiad,"A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures." 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. E. Malinnikova" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M. A. Khrabrov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. D. Karpov, C. Gukshin, D. Fon-der-Flaas" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case? See also " 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Do there exist real numbers $b$ and $c$ such that each of the equations $x^2+bx+c = 0$ and $2x^2+(b+1)x+c+1 = 0$ have two integer roots? N. Agakhanov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name. A. Shapovalov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"The incircle of triangle $ABC$ touches sides $AB$;$BC$;$CA$ at $M$;$N$;$K$, respectively. The line through $A$ parallel to $NK$ meets $MN$ at $D$. The line through $A$ parallel to $MN$ meets $NK$ at $E$. Show that the line $DE$ bisects sides $AB$ and $AC$ of triangle $ABC$. M. Sonkin" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible. D. Hramtsov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. M. Sonkin" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. E. Poroshenko" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) D. Tereshin" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles. A. Shapovalov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? N. Agakhanov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"A circle centered at $O$ and inscribed in triangle $ABC$ meets sides $AC$;$AB$;$BC$ at $K$;$M$;$N$, respectively. The median $BB_1$ of the triangle meets $MN$ at $D$. Show that $O$;$D$;$K$ are collinear. M. Sonkin" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Find all triples $m$; $n$; $l$ of natural numbers such that $m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$: S. Tokarev" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. D. Fon-der-Flaas" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat, black hat or a red hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case? K. Knop P.S. Of course, the sages hear the previous guesses. See also " 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color. M. Smurov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"Of the quadratic trinomials $x^2 + px + q$ where $p$; $q$ are integers and $1\leqslant p, q \leqslant 1997$, which are there more of: those having integer roots or those not having real roots? M. Evdokimov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"We are given a polygon, a line $l$ and a point $P$ on $l$ in general position: all lines containing a side of the polygon meet $l$ at distinct points diering from $P$. We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the line $l$ on opposite sides of $P$. Show that $P$ lies inside the polygon if and only if on each side of $l$ there are an odd number of marked vertices. O. Musin" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. N. Agakhanov" 1997 All-Russian Olympiad,https://artofproblemsolving.com/community/c5157_1997_allrussian_olympiad,"In an $m\times n$ rectangular grid, where m and n are odd integers, $1\times 2$ dominoes are initially placed so as to exactly cover all but one of the $1\times 1$ squares at one corner of the grid. It is permitted to slide a domino towards the empty square, thus exposing another square. Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle. A. Shapovalov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be? A. Golovanov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides. D. Tereshin" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$. Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$. A. Kovaldji, V. Senderov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. A. Skopenkov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10. L. Kuptsov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"In the isosceles triangle $ABC$ ($AC = BC$) point $O$ is the circumcenter, $I$ the incenter, and $D$ lies on $BC$ so that lines $OD$ and $BI$ are perpendicular. Prove that $ID$ and $AC$ are parallel. M. Sonkin" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number $k$, not exceeding the number of coins in either pile, the sum of the weights of the $k$ heaviest coins in the first pile is not more than that of the second pile. Show that for any natural number $x$, if each coin (in either pile) of weight not less than $x$ is replaced by a coin of weight $x$, the first pile will not be lighter than the second. D. Fon-der-Flaas" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Can a $5\times 7$ checkerboard be covered by L's (figures formed from a $2\times2$ square by removing one of its four $1\times1$ corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's? M. Evdokimov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Points $E$ and $F$ are given on side $BC$ of convex quadrilateral $ABCD$ (with $E$ closer than $F$ to $B$). It is known that $\angle BAE = \angle CDF$ and $\angle EAF = \angle FDE$. Prove that $\angle FAC = \angle EDB$. M. Smurov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. Р. Sadykov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$. A. Kovaldji, V. Senderov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. O. Musin" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges? A. Shapovalov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders: (a) Each day, at least one task must be issued to a soldier. (b) No soldier may have more than two task or receive more than one tasks in a single day. (c) The lists of soldiers receiving tasks for two different days must not be the same. (d) The first sergeant violating any of these orders will be jailed. Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed? M. Kulikov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals $180^\circ$. M. Smurov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values? I. Rubanov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? N. Agakhanov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period. A. Shapovalov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. N. Agakhanov, N. Tereshin" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others? A. Golovanov" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$. M. Sonkin" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? E. Malinnikova" 1996 All-Russian Olympiad,https://artofproblemsolving.com/community/c5156_1996_allrussian_olympiad,"The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers? S. Tokarev" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. S. Tokarev" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ V. Gordon" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? S. Tokarev" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Can the numbers from 1 to 81 be written in a 9×9 board, so that the sum of numbers in each 3×3 square is the same? S. Tokarev" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"We call natural numbers similar if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third. S. Dvoryaninov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"In an acute-angled triangle ABC, points $A_2$, $B_2$, $C_2$ are the midpoints of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Compute the sum of angles $B_2A_1C_2$, $C_2B_1A_2$ and $A_2C_1B_2$. D. Tereshin" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later). After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? I. Izmest’ev" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. D. Karpov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$? A. Shapovalov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides. A. Berzin’sh, O. Musin" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$." 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right. L. Kuptsov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$. A. Galochkin, O. Ljashko" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions? A. Golovanov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry. D. Tereshin" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Two points on the distance 1 are given in a plane. It is allowed to draw a line through two marked points, as well as a circle centered in a marked point with radius equal to the distance between some two marked points. By marked points we mean the two initial points and intersection points of two lines, two circles, or a line and a circle constructed so far. Let $C(n)$ be the minimum number of circles needed to construct two points on the distance $n$ if only a compass is used, and let $LC(n)$ be the minimum total number of circles and lines needed to do so if a ruler and a compass are used, where $n$ is a natural number. Prove that the sequence $C(n)/LC(n)$ is not bounded. A. Belov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. A. Golovanov" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths? V. New" 1995 All-Russian Olympiad,https://artofproblemsolving.com/community/c5155_1995_allrussian_olympiad,"The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere. D. Tereshin" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear. (A. Kalinin)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"There are three piles of matches on the table: one with $100$ matches, one with $200$, and one with $300$. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy? (K. Kokhas’)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"On a line are given $n$ blue and $n$ red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors. (O. Musin)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Prove the equality $\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}$ =$\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $ (R. Zhenodarov)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than half a mllion." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"A trapezoid $ABCD$ ($AB ///CD$) has the property that there are points $P$ and $Q$ on sides $AD$ and $BC$ respectively such that $\angle APB = \angle CPD$ and $\angle AQB = \angle CQD$. Show that the points $P$ and $Q$ are equidistant from the intersection point of the diagonals of the trapezoid. (M. Smurov)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square). (A. Perlin)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Let be given three quadratic polynomials: $P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$. Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle. Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"In a regular $ 6n+1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"I'll post some nice combinatorics problems here, taken from the wonderful training book ""Les olympiades de mathmatiques"" (in French) written by Tarik Belhaj Soulami. Here goes the first one: Let $\mathbb{I}$ be a non-empty subset of $\mathbb{Z}$ and let $f$ and $g$ be two functions defined on $\mathbb{I}$. Let $m$ be the number of pairs $(x,\;y)$ for which $f(x) = g(y)$, let $n$ be the number of pairs $(x,\;y)$ for which $f(x) = f(y)$ and let $k$ be the number of pairs $(x,\;y)$ for which $g(x) = g(y)$. Show that $$2m \leq n + k.$$" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$. Prove that lines $ AA_1,BB_1,CC_1$ are concurrent." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends. PS. Here majority means larger than half." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"All Russian MO 1994 Grade 11 ""Natural numbers $a$ and $b$ are such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer. If $d$ is the greatest common divisor of $a$ and $b$, prove that $d^2 \le a + b$." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Inside a convex $100$-gon are selected $k$ points, $2 \leq k \leq 50$. Show that one can choose $2k$ vertices of the $100$-gon so that the convex $2k$-gon determined by these vertices contains all the selected points." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at A and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that the common chord of the circumcircles of triangles $ABC$ and $BDE$ passes through point $F$. (A. Kalinin)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive." 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two. (N. Agakhanov)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)" 1994 All-Russian Olympiad,https://artofproblemsolving.com/community/c691133_1994_allrussian_olympiad,"Players $ A,B$ alternately move a knight on a $ 1994\times 1994$ chessboard. Player $ A$ makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while $ B$ makes only vertical moves. Initally player $ A$ places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player $ A$ has a winning strategy." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?" 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$. 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?" 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures are different." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers? 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?" 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters. 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: ""Is your neighbor to the right smart or dumb?"" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?" 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?" 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares. 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' = \frac {a_k + a_{k + 1}}2$ the average sequence of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - average sequence of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called Good. Prove that if $ \{x_k\}$ is a good sequence, then $ \{x_k^2\}$ is also good." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Prove that the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle is half the measure of the angle formed by the hypotenuses. 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"Prove that there exists a positive integer $n$, such that if an equilateral triangle with side lengths $n$ is split into $n^2$ triangles with side lengths 1 with lines parallel to its sides, then among the vertices of the small triangles it is possible to choose $1993n$ points so that no three of them are vertices of an equilateral triangle." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple. 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,"In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$." 1993 All-Russian Olympiad,https://artofproblemsolving.com/community/c5154_1993_allrussian_olympiad,Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area. 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Show that $x^4 + y^4 + z^2\ge xyz \sqrt8$ for all positive reals $x, y, z$." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"A country contains $n$ cities and some towns. There is at most one road between each pair of towns and at most one road between each town and each city, but all the towns and cities are connected, directly or indirectly. We call a route between a city and a town a gold route if there is no other route between them which passes through fewer towns. Show that we can divide the towns and cities between $n$ republics, so that each belongs to just one republic, each republic has just one city, and each republic contains all the towns on at least one of the gold routes between each of its towns and its city." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?" 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits? 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Find all real $x, y$ such that $\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\ (1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}$ ." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?" 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Show that for any real numbers $x, y > 1$, we have $\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$" 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat? 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $Q$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$. 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Define the sequence $a_1 = 1, a_2, a_3, ...$ by $a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$. Show that $1$ is the only square in the sequence." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Let $f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$. 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?" 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits." 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.] 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,$1992$ vectors are given in the plane. Two players pick unpicked vectors alternately. The winner is the one whose vectors sum to a vector with larger magnitude (or they draw if the magnitudes are the same). Can the first player always avoid losing? 1992 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909856_1992_all_soviet_union_mathematical_olympiad,"If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $(a - d)^2 \ge 4d + 8$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Find all integers $a, b, c, d$ such that $ab - 2cd = 3, ac + bd = 1$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Find unequal integers $m, n$ such that $mn + n$ and $mn + m$ are both squares. Can you find such integers between $988$ and $1991$?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie. 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"A minus sign is placed on one square of a $5 \times 5$ board and plus signs are placed on the remaining squares. A move is to select a $2 \times 2, 3 \times 3, 4 \times 4$ or $5 \times 5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Does there exist a triangle in which two sides are integer multiples of the median to that side? Does there exist a triangle in which every side is an integer multiple of the median to that side?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same. 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table. 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$. Show that the product of the numbers in each row is $-1$. b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular." 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?" 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$. 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members? 1991 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909855_1991_all_soviet_union_mathematical_olympiad,"The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$. What is the maximum possible value of $|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$, where $s_n = \frac{x_1 + x_2 + ... + x_n}{n}$?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Show that $x^4 > x - \frac12$ for all real $x$. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c""$ (in that order), and $BC$ into three parts length $a, a', a""$ (in that order), and $CA$ into three parts length $b, b', b""$ (in that order). Show that $abc = a'b'c' = a""b""c""$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color? 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"$ABCD$ is a convex quadrilateral. $X$ is a point on the side $AB. AC$ and $DX$ intersect at $Y$. Show that the circumcircles of $ABC, CDY$ and $BDX$ have a common point." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001$. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that angle $XDY = \frac{\pi}{2n}$. Show that DY bisects angle $\angle XYC$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)? 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"Given a point $X$ and $n$ vectors $\overrightarrow{x_i}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $X_1$ such that $XX_1 = \overrightarrow{x_1}, X_2$ such that $X_1X_2 = \overrightarrow{x_2}$ and so on. Show that for some permutation we can find two points $Y, Z$ with angle $\angle YXZ = 60^o $, so that all the points lie inside or on the triangle $XYZ$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$. 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"A quadratic polynomial $p(x)$ has positive real coefficients with sum $1$. Show that given any positive real numbers with product $1$, the product of their values under $p$ is at least $1$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,A cube side $100$ is divided into a million unit cubes with faces parallel to the large cube. The edges form a lattice. A prong is any three unit edges with a common vertex. Can we decompose the lattice into prongs with no common edges? 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite? 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$." 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,"A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?" 1990 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c909854_1990_all_soviet_union_mathematical_olympiad,Given $2n$ genuine coins and $2n$ fake coins. The fake coins look the same as genuine coins but weigh less (but all fake coins have the same weight). Show how to identify each coin as genuine or fake using a balance at most $3n$ times. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block? 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"One bird lives in each of $n$ bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds $A, B, C, D$ (not necessarily distinct), if the distance $AB < CD$ before the change, then $AB > CD$ after the change. Find all possible values of $n$." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"We are given $1998$ normal coins, $1$ heavy coin and $1$ light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance $4$ times or less." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?" 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?" 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,An insect is on a square ceiling side $1$. The insect can jump to the midpoint of the segment joining it to any of the four corners of the ceiling. Show that in $8$ jumps it can get to within $1/100$ of any chosen point on the ceiling 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"$ABCD$ has $AB = CD$, but $AB$ not parallel to $CD$, and $AD$ parallel to $BC$. The triangle is $ABC$ is rotated about $C$ to $A'B'C$. Show that the midpoints of $BC, B'C$ and $A'D$ are collinear." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$." 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,"Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?" 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$. 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges). 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$? 1989 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907033_1989_all_soviet_union_mathematical_olympiad,A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$. 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,A book contains $30$ stories. Each story has a different number of pages under $31$. The first story starts on page $1$ and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers? 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area. 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Show that there are infinitely many triples of distinct positive integers $a, b, c$ such that each divides the product of the other two and $a + b = c + 1$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"The numbers $1$ and $2$ are written on an empty blackboard. Whenever the numbers $m$ and $n$ appear on the blackboard the number $m + n + mn$ may be written. Can we obtain : (1) $13121$, (2) $12131$?" 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns? 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$. 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$" 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?" 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"In the triangle $ABC$, the angle $C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ ." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$ 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge a + d$? 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column h becomes $a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"In the acute-angled triangle $ABC$, the altitudes $BD$ and $CE$ are drawn. Let $F$ and $G$ be the points of the line $ED$ such that $BF$ and $CG$ are perpendicular to $ED$. Prove that $EF = DG$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have? 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis. 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"What is the smallest $n$ for which there is a solution to $\sin x_1 + \sin x_2 + ... + \sin x_n = 0, \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100$ ?" 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$." 1988 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907028_1988_all_soviet_union_mathematical_olympiad,"Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$" 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,Given right heptagon ($7$-angle) $A_1...A_7$. Prove that $\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$. 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The ""Sea battle"" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$. 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"An L is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many Ls can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with Ls (with no interior points overlapping)." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Three lines are drawn parallel to the sides of the triangles in the opposite to the vertex, not belonging to the side, part of the plane. The distance from each side to the corresponding line equals the length of the side. Prove that six intersection points of those lines with the continuations of the sides are situated on one circumference." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,Given a convex pentagon. The angles $ABC$ and $ADE$ are equal. The angles $AEC$ and $ADB$ are equal too. Prove that the angles $BAC$ and $DAE$ are equal also. 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Prove such $a$, that all the numbers $\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)$ are negative." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The positive numbers $a,b,c,A,B,C$ satisfy a condition $a + A = b + B = c + C = k$. Prove that $aB + bC + cA \le k^2$." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The $B$ vertex of the $ABC$ angle lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. The $K$ point belongs to the intersection of the $[BA)$ beam and the circumference. The $KP$ chord is orthogonal to the angle $ABC$ bisector. The $(KP)$ line intersects the $BC$ beam in the M point. Prove that the $[PM]$ segment is twice as long as the distance from the circle centre to the angle $ABC$ bisector." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Two players are writting in turn natural numbers not exceeding $p$. The rules forbid to write the divisors of the numbers already having been written. Those who cannot make his move looses. a) Who, and how, can win if $p=10$? b) Who wins if $p=1000$?" 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 ""knights"" in the ""night guard"". When it can happen, for the first time, that every knight has been on duty the same number of times?" 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The convex $n$-angle ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"The plot of the $y=f(x)$ function, being rotated by the (right) angle around the $(0,0)$ point is not changed. a) Prove that the equation $f(x)=x$ has the unique solution. b) Give an example of such a function." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,"All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only." 1987 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907027_1987_all_soviet_union_mathematical_olympiad,Prove that for every natural $n$ the following inequality is held: $(2n + 1)^n \ge (2n)^n + (2n - 1)^n$. 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,The point $M$ belongs to the $[AC]$ side of the acute-angle triangle $ABC$. Two circles are outscribed around $ABM$ and $BCM$ triangles. What $M$ position corresponds to the minimal area of those circles intersection? 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for a) $n=6$ b) $n=1986$ ?" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Two circumferences, with the distance $d$ between centres, intersect in $P$ and $Q$ points. Two lines are drawn through the $A$ point on the first circumference ($Q\ne A\ne P$) and $P$ and $Q$ points. They intersect the second circumference in the $B$ and $C$ points. a) Prove that the radius of the circle, circumscribed around the $ABC$ triangle, equals $d$. b) Describe the set of the new circle's centres, if the $A$ point moves along all the first circumference." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Given right hexagon. Each side is divided onto $1000$ equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one. Prove that the unpainted node is not a hexagon vertex." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,Find all the natural numbers equal to the square of its divisors number. 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"A line is drawn through the $A$ vertex of $ABC$ triangle with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and the angles $ABM$ and $ACM$ are equal. What lines do not contain such a point $M$ at all?" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Given two points inside a convex dodecagon (twelve sides) situated 1$0$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Given right $n$-angle $A_1A_2...A_n$. Prove that if a) $n$ is even number, than for the arbitrary point $M$ in the plane, it is possible to choose signs in an expression $\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}$to make it equal to the zero vector . b) $n$ is odd, than the abovementioned expression equals to the zero vector for the finite set of $M$ points only." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules: At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column. a) What is the minimal b) What is the maximal possible number of $+1$ in the obtained table?" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,Prove that for every natural $n$ the following inequality is valid $|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$ 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"A triangle and a square are outscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?" 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Let us call a polynomial admissible if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the admissible polynomials $P$ such, that $P(2) = n$." 1986 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907026_1986_all_soviet_union_mathematical_olympiad,"Consider all the tetrahedrons $AXBY$, outscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on $X$ and $Y$ points." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,Two perpendiculars are drawn from the middles of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area. 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"What maximal number of the dames can be put on the chess-board $8\times 8$ so, that every dame can be taken by at least one other dame?" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"You should paint all the sides and diagonals of the right $n$-angle so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The senior coefficient a in the square polynomial $P(x) = ax^2 + bx + c$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$? j h i e f g a b c d" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$ b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The convex pentagon $ABCDE$ was drawn in the plane. $A_1$ was symmetric to $A$ with respect to $B$. $B_1$ was symmetric to $B$ with respect to $C$. $C_1$ was symmetric to $C$ with respect to $D$. $D_1$ was symmetric to $D$ with respect to $E$. $E_1$ was symmetric to $E$ with respect to $A$. How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The sequence $a_1, a_2, ... , a_k, ...$ is built according to the rules: $a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$. Prove that it is non-periodical sequence." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Given a cube, a cubic box, that exactly suits for the cube, and six colours. First man paints each side of the cube with its (side's) unique colour. Another man does the same with the box. Prove that the third man can put the cube in the box in such a way, that every cube side will touch the box side of different colour." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points. Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle outscribed around $AMD$ triangle has radius $R$. 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and middles of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes. Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Solve the equation (""$2$"" encounters $1985$ times): $$\frac{x}{2+ \frac{x}{2+\frac{x}{2+... \frac{x}{2+\sqrt {1+x}}}}}=1$$" 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"All the points situated more close than $1$ cm to ALL the vertices of the right pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it." 1985 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907021_1985_all_soviet_union_mathematical_olympiad,"The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through $A,C$ and $B_1$ points." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,Prove that every positive $a$ and $b$ satisfy inequality $\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$ 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Given four colours and enough square plates $1\times 1$. We have to paint four edges of every plate with four different colours and combine plates, putting them with the edges of the same colour together. Describe all the pairs $m,n$, such that we can combine those plates in a $n\times m$ rectangle, that has every edge of one colour, and its four edges have different colours." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Prove that every positive $x,y$ and real $a$ satisfy inequality $x^{\sin ^2a} y^{\cos^2a} < x + y$." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win? 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"$n$ natural numbers ($n>3$) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number. Prove that the sum of those relations is a) not less than $2n$ b) less than $3n$" 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The circle with the centre $O$ is inscribed in the $ABC$ triangle. The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ lines intersect in one point." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$. 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"$n$ real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order. Prove that the second line coincides with the first one." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Given $ABC$ triangle. From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the outscribed circumference in $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\ y^2/ 3 + z^2 = 9 \\ z^2 + zx + x^2 = 16 \end{cases}$ Find the value of expression $xy + 2yz + 3zx$." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The teacher wrote on a blackboard: "" $x^2 + 10x + 20$ "" Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: "" $x^2 + 20x + 10$ "". Is it true that some time during the process there was written the square polynomial with the integer roots?" 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is outscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring)." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"There are scales and $(n+1)$ weights with the total weight $2n$. Each weight is an integer. We put all the weights in turn on the lighter side of the scales, starting from the heaviest one, and if the scales is in equilibrium -- on the left side. Prove that when all the weights will be put on the scales, they will be in equilibrium." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Let us call ""absolutely prime"" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $xx...x6yy...y4$ ($n$ times $x$ and $n$ times $y$) is an exact square. Find all possible $x$ and $y$." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Given a sequence $\{x_n\},x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$. Prove that the sequence has limit and find it." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The white fields of $1983\times 1984 $1983x1984 are filled with either $+1$ or $-1$. For every black field, the product of neighbouring numbers is $+1$. Prove that all the numbers are $+1$." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed). 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is outscribed around some circle and find its radius." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"Prove that every cube's cross-section, containing its centre, has the area not less then its face's area." 1984 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907020_1984_all_soviet_union_mathematical_olympiad,"The white fields of $3x3$ chess-board are filled with either $+1$ or $-1$. For every field, let us calculate the product of neighbouring numbers. Then let us change all the numbers by the respective products. Prove that we shall obtain only $+1$'s, having repeated this operation finite number of times." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$. Is it possible to admit that the initial numbers were a) $\{2, 2, 2\}$? b) $\{3, 3, 3\}$?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved. Prove that the $XYZ$ triangle is completely covered by the expanded disks." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Find all the solutions of the system $y^2 = x^3 - 3x^2 + 2x$ $x^2 = y^3 - 3y^2 + 2y$" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,The point $D$ is the middle of the $[AB]$ side of the $ABC$ triangle. The points $E$ and F belong to $[AC]$ and $[BC]$ respectively. Prove that the $DEF$ triangle area does not exceed the sum of the $ADE$ and $BDF$ triangles areas. 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical? 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Two acute angles $a$ and $b$ satisfy condition $\sin^2a+\sin^2b = \sin(a+b)$ Prove that $a + b = \pi /2$." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ lines." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The Abba tribe language alphabet contains two letters only. Not a word of this language is a beginning of another word. Can this tribe vocabulary contain $3$ four-letter, $10$ five-letter, $30$ six-letter and $5$ seven-letter words?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight. 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Given a point $O$ inside $ABC$ triangle. Prove that $S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$, where $S_A, S_B, S_C$ denote $BOC, COA, AOB$ triangles areas respectively." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"Given $(2m+1)$ different integers, each absolute value is not greater than $(2m-1)$. Prove that it is possible to choose three numbers among them, with their sum equal to zero." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the $ABC$ triangle respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the $DEF, DEA, DBF, CEF$, triangles respectively. Prove that $d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$. When the equality takes place?" 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The $M$ set consists of $k$ non-intersecting segments on the line. It is possible to put an arbitrary segment shorter than $1$ cm on the line in such a way, that his ends will belong to $M$. Prove that the total sum of the segment lengths is not less than $1/k$ cm." 1983 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907019_1983_all_soviet_union_mathematical_olympiad,"The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $MO_1K$ angle. Find the $MO_1KO_2$ quadrangle area." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $2^{k_1}+2^{k_2}+...+2^{k_n}$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can You find a number, divisible by $111...1$ ($m$ times ""$1$""), that has the sum of its digits less than $m$?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"A nonnegative real number is written at every cube's vertex. The sum of those numbers equals to $1$. Two players choose in turn faces of the cube, but they cannot choose the face parallel to already chosen one (the first moves twice, the second -- once). Prove that the first player can provide the number, at the common for three chosen faces vertex, to be not greater than $1/6$." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Once upon a time, three boys visited a library for the first time. The first decided to visit the library every second day. The second decided to visit the library every third day. The third decided to visit the library every fourth day. The librarian noticed, that the library doesn't work on Wednesdays. The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases. They strictly obeyed these rules. Some Monday later I met them all in that library. What day of week was when they visited a library for the first time?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,The parallelogram $ABCD$ isn't a diamond. The relation of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.) Find the $|AM|/|BM|$ relation. 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$. Sort those numbers in increasing order." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence. Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the middles of the first line links: $B_1$ is the middle of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$. Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than a) not less than $[n/2]$ times ($[ ...]$ mean the whole part), b) not less than $n$ times." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Prove that the following inequality is valid for the positive $x$: $$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property: none of the remained numbers equals to the product of two other remained numbers?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Every square on the infinite sheet of cross-lined paper contains some real number. Prove that some square contains a number that does not exceed at least four of eight neighbouring numbers." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Prove that the following inequality holds for all real $a$ and natural $n$: $|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$ . $F(a)$ is the distance from $a$ to the closest integer." 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,"Can You find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition: a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$, b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$, for all $x,y,z$?" 1982 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907018_1982_all_soviet_union_mathematical_olympiad,The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that $KLMN$ perimeter is less than $4/3$ $ABCD$ perimeter. 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to $(NA)$ and $(NB)$ lines respectively. Prove that $(AA_1)$ and $(BB_1)$ lines are parallel." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$. a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously. b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The rectangular table has four rows. The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number $n$ and write the number $k$ if $n$ was met $k$ times. Prove that the second row coincides with the fourth one." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $y \le -x^2 $ and $y \ge x^2 - 2x + a$" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"There are $1000$ inhabitants in a settlement. Every evening every inhabitant tells all his friends all the news he had heard the previous day. Every news becomes finally known to every inhabitant. Prove that it is possible to choose $90$ of inhabitants so, that if you tell them a news simultaneously, it will be known to everybody in $10$ days." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"It is known about real $a$ and $b$ that the inequality $a \cos x + b \cos (3x) > 1$ has no real solutions. Prove that $|b|\le 1$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The quadrangles $AMBE, AHBT, BKXM$, and $CKXP$ are parallelograms. Prove that the quadrangle $ABTE$ is also parallelogram. (the vertices are mentioned counterclockwise)" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,Find the natural solutions of the equation $x^3 - y^3 = xy + 61$. 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Eighteen soccer teams have played $8$ tours of a one-round tournament. Prove that there is a triple of teams, having not met each other yet." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the $ABC$ triangle. $\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$. Prove that the perimeter $P$ of the $ABC$ triangle and the perimeter $p$ of the $A_1B_1C_1$ triangle, satisfy inequality $\frac{P}{2} < p < \frac{3P}{4}$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Positive numbers $x,y$ satisfy equality $x^3+y^3=x-y$. Prove that $x^2+y^2<1$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures: 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?" 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"Six points are marked inside the $3x4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"a) Find the minimal value of the polynomial $P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$." 1981 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907017_1981_all_soviet_union_mathematical_olympiad,"The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The vertical side of a square is divided onto $n$ segments. The sum of the segments with even numbers lengths equals to the sum of the segments with odd numbers lengths. $n-1$ lines parallel to the horizontal sides are drawn from the segments ends, and, thus, $n$ strips are obtained. The diagonal is drawn from the lower left corner to the upper right one. This diagonal divides every strip onto left and right parts. Prove that the sum of the left parts of odd strips areas equals to the sum of the right parts of even strips areas." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The load for the space station ""Salute"" is packed in containers. There are more than $35$ containers, and the total weight is $18$ metric tons. There are $7$ one-way transport spaceships ""Progress"", each able to bring $3$ metric tons to the station. It is known that they are able to take an arbitrary subset of $35$ containers. Prove that they are able to take all the load." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The points $M$ and $P$ are the middles of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that the $ABCD$ area is less than $\frac{a^2}{2}$." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected. Prove that You can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The six-digit decimal number contains six different non-zero digits and is divisible by $37$. Prove that having transposed its digits you can obtain at least $23$ more numbers divisible by $37$" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Find real solutions of the system : $\sin x + 2 \sin (x+y+z) = 0$ $\sin y + 3 \sin (x+y+z) = 0$ $\sin z + 4 \sin (x+y+z) = 0$" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Let us denote with $S(n)$ the sum of all the digits of $n$. a) Is there such an $n$ that $n+S(n)=1980$? b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares? 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time. b) If nobody had the immunity on the first day, the epidemic influenza will stop some day." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $n_{k+1} = n_k + P(n_k)$. Can it be unbounded for some $n_1$?" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ - the middle of the $[AP]$ segment. Find the angles of $DEC$ triangle." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$. What is the least possible number of elements in $A$?" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)" 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between $(AC)$ and $(BD)$ lines is less than $|CD|/|AB|$." 1980 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c907012_1980_all_soviet_union_mathematical_olympiad,"The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"What is the least possible relation of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?" 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Every member of a certain parliament has not more than $3$ enemies. Prove that it is possible to divide it onto two subparliaments so, that everyone will have not more than one enemy in his subparliament. ($A$ is the enemy of $B$ if and only if $B$ is the enemy of $A$.)" 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain : a) $1/5$, b) an arbitrary rational number between $0$ and $1$." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$. Prove that for every $n$, it also satisfies $x_1+x_2/2+x_3/3+...+x_n/n\le 3$." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"What is the least possible number of the checkers being required a) for the $8\times 8$ chess-board, b) for the $n\times n$ chess-board, to provide the property: Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker ?" 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Find $x$ and $y$ ($a$ and $b$ parameters): $$\frac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a$$$$\frac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b$$" 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,Given some square carpets with the total area $4$. Prove that they can fully cover the unit square. 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Prove that for the arbitrary numbers $x_1, x_2, ... , x_n$ from the $[0,1]$ segment $(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)$." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments. Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it. Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$. Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2, A_2A_4$ and $l_3, ... , A_{i-1}A_{i+1}$ and $l_i, ... , A_{1977}A_{1979}$ and $l_{1978}, A_{1978}A_1$ and $l_{1979}, A_{1979}A_2$ and $l_1$" 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"The finite sequence $a_1, a_2, ... , a_n$ of ones and zeroes should satisfy a condition: for every $k$ from $0$ to $(n-1)$ the sum a_1a_{k+1} + a_2a_{k+2} + ... + a_{n-k}a_n should be odd. a) Build such a sequence for $n=25$. b) Prove that there exists such a sequence for some $n > 1000$." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"The convex quadrangle is divided by its diagonals onto four triangles. The circles inscribed in those triangles are equal. Prove that the given quadrangle is a diamond." 1979 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901988_1979_all_soviet_union_mathematical_olympiad,"Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2, A_2A_3, ... A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible a) for $k=3$, b) for every $k$ less than $(n-1)$." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Let $a_n$ be the closest to $\sqrt n$ integer. Find the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. the angle $CBM$ equals to $CDM$. Prove that the angle $ACD$ equals to $BCM$. 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$). 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Given a finite set $K_0$ of points (in the plane or space). The sequence of sets $K_1, K_2, ... , K_n, ...$ is build according to the rule: we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$. a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$? b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices. c) How many faces contain the minimal convex polyhedron containing $K_1$? d) What is the volume of the above mentioned polyhedron? e) What is the volume of the minimal convex polyhedron containing $K_n$?" 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Given two heaps of checkers. the bigger contains $m$ checkers, the smaller -- $n$ ($m>n$). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins. a) Prove that if $m > 2n$, than the first can always win. b) Find all $x$ such that if $m > xn$, than the first can always win." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $|x_m-x_k|>1/|m-k|$." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Let $f(x) = x^2 + x + 1$. Prove that for every natural $m>1$ the numbers $m, f(m), f(f(m)), ...$ are relatively prime." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.) 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, having got the pair of cards with $(a,b)$ and $(b,c)$ produces new card with $(a,c)$. All the automates return the initial cards also. Suppose there was $(5,19)$ card initially. Is it possible to obtain a) $(1,50)$? b) $(1,100)$? c) Suppose there was $(a,b)$ card initially $(a3$. Consider the set $M$ of the pairs $(x,y)$ with the integer coordinates in the plane such that $0 \le x < p, 0 \le y < p$. Prove that it is possible to mark $p$ points of $M$ such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,Prove that for every tetrahedron there exist two planes such that the projection areas on those planes relation is not less than $\sqrt 2$. 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Given $a_1, a_2, ... , a_n$. Define $b_k = \frac{a_1 + a_2 + ... + a_k}{k}$ for $1 \le k\le n$. Let $C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2$, $D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$. Prove that $C \le D \le 2C$." 1978 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901987_1978_all_soviet_union_mathematical_olympiad,"Consider a sequence $x_n=(1+\sqrt2+\sqrt3)^n$. Each member can be represented as $x_n=qn+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$, where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call ""special"" a couple of line's segments if the one's continuation intersects another. Prove that there is even number of special pairs." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the middles of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the middles of the arcs $AB$ and $AC$ of the circle outscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?" 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given infinite sequence $a_n$. It is known that the limit of $b_n=a_{n+1}-a_n/2$ equals zero. Prove that the limit of $a_n$ equals zero." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"There are direct routes from every city of a certain country to every other city. The prices are known in advance. Two tourists (they do not necessary start from one city) have decided to visit all the cities, using only direct travel lines. The first always chooses the cheapest ticket to the city, he has never been before (if there are several -- he chooses arbitrary destination among the cheapests). The second -- the most expensive (they do not return to the first city). Prove that the first will spend not more money for the tickets, than the second." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Every vertex of a convex polyhedron belongs to three edges. It is possible to outscribe a circle around all its faces. Prove that the polyhedron can be inscribed in a sphere." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"The polynomial $x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots. Who wins?" 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk. How much milk did every cup contain, if there was three litres of milk total?" 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Let us call ""fine"" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also. a) Find all two- and four-digit fine numbers. b) Is there any six-digit fine number? c) Prove that there exists $20$-digit fine number. d) Prove that there exist at least ten $100$-digit fine numbers. e) Prove that there exists $30$-digit fine number." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"There are $1000$ tickets with the numbers $000, 001, ... , 999$, and $100$ boxes with the numbers $00, 01, ... , 99$. You may put a ticket in a box, if you can obtain the box number from the ticket number by deleting one digit. Prove that: a) You can put all the tickets in $50$ boxes; b) $40$ boxes is not enough for that; c) it is impossible to use less than $50$ boxes. d) Consider $10000$ $4$-digit tickets, and you are allowed to delete two digits. Prove that $34$ boxes is enough for storing all the tickets. e) What is the minimal used boxes set in the case of $k$-digit tickets?" 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given a square $100\times 100$ on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior. Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line." 1977 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901986_1977_all_soviet_union_mathematical_olympiad,"Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m) 2$, d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$" 1976 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901985_1976_all_soviet_union_mathematical_olympiad,"Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called ""equator"". We shall use the words ""pole"", ""parallel"",""meridian"" as self-explanatory. a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $g(x_1)^2 + g(x_2)2 + g(x_3)^2 = 1$. (*) Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property. b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$. c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$. d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$. e) Prove that for all $x , f(x) = g(x)$." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"a) The triangle $ABC$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$. b) The quadrangle $ABCD$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?" 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"What is the smallest perimeter of the convex $32$-angle, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?" 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution? b) The same problem for $13\times 13$ square." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Denote the middles of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2, A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only ""$1$"" and ""$2$"" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Prove that for all the positive numbers $a,b,c$ the following inequality is valid: $a^3+b^3+c^3+3abc>ab(a+b)+bc(b+c)+ac(a+c)$" 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Three flies are crawling along the perimeter of the $ABC$ triangle in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the $ABC$ triangle. (The centre of masses for the triangle is the point of medians intersection." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Given a horizontal strip on the plane (its sides are parallel lines) and $n$ lines intersecting the strip. Every two of them intersect inside the strip, and not a triple has a common point. Consider all the paths along the segments of those lines, starting on the lower side of the strip and ending on the upper side with the properties: moving along such a path we are constantly rising up, and, having reached the intersection, we are obliged to turn to another line. Prove that: a) there are not less than $n/2$ such a paths without common points; b) there is a path consisting of not less than of $n$ segments; c) there is a path that goes along not more than along $n/2+1$ lines; d) there is a path that goes along all the $n$ lines." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"For what $k$ is it possible to construct a cube $kxkxk$ of the black and white cubes $1x1x1$ in such a way that every small cube has the same colour, that have exactly two his neighbours. (Two cubes are neighbours, if they have the common face.)" 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"Given a polynomial $P(x)$ with a) natural coefficients; b) integer coefficients; Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times." 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if: a) it is hockey (draws allowed)? b) it is volleyball (no draws)?" 1975 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901984_1975_all_soviet_union_mathematical_olympiad,"a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$ $(a_1 + b_1)/(p_1 + q_1)$ ,$ (a_1 + b_2)/(p_1 + q_2) $ $(a_2 + b_1)/(p_2 + q_1)$ ,$ (a_2 + b_2)/(p_2 + q_2)$ there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We build the table $n\times n$, with the numbers ($0 < i,j \le n$) $(a_i + b_j)/(p_i + q_j)$ in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point)." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given some cards with either ""$-1$"" or ""$+1$"" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards. What is the minimal number of questions allowing to determine all the numbers on the cards ... a) for $30$ cards, b) for $31$ cards, c) for $32$ cards. (You should prove, that you cannot manage with less questions.) d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the $50$ numbers. What is the minimal number of questions allowing to determine it?" 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest. Prove that it is really the smallest." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"a) Each of the side of the convex hexagon ($6$-angle) is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?" 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: for every $k, k\le n$, the length of the sum of the first $k$ vectors is less than $2$." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Find all the real $a,b,c$ such that the equality $|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$ is valid for all the real $x,y,z$." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,Given a square $ABCD$. Points $P$ and $Q$ are in the sides $[AB]$ and $[BC]$ respectively. $|BP|=|BQ|$. Let $H$ be the base of the perpendicular from the point $B$ to the segment $[PC]$. Prove that the angle $DHQ$ is a right one. 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given some red and blue points. Some of them are connected by the segments. Let us call ""exclusive"" the point, if its colour differs from the colour of more than half of the connected points. Every move one arbitrary ""exclusive"" point is repainted to the other colour. Prove that after the finite number of moves there will remain no ""exclusive"" points." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits. 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$" 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Two are playing the game ""cats and rats"" on the chess-board $8\times 8$. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions. a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat? b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"a) Prove that you can rearrange the numbers $1, 2, ... , 32$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean. b) Can you rearrange the numbers $1, 2, ... , 100$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?" 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits. 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$." 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $f(x_1+x_2)\ge f(x_1)+f(x_2)$. a) Prove that for all $x$ holds $f(x) \le 2x$. b) Is the inequality $f(x) \le 1.9x$ valid?" 1974 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901979_1974_all_soviet_union_mathematical_olympiad,"Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the middles of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Fourteen coins are submitted to the judge. An expert knows, that the coins from number one to seven are false, and from $8$ to $14$ -- normal. The judge is sure only that all the true coins have the same weight and all the false coins weights equal each other, but are less then the weight of the true coins. The expert has the scales without weights. a) The expert wants to prove, that the coins $1--7$ are false. How can he do it in three weighings? b) How can he prove, that the coins $1--7$ are false and the coins $8--14$ are true in three weighings?" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,Given an angle with the vertex $O$ and a circle touching its sides in the points $A$ and $B$. A ray is drawn from the point $A$ parallel to $[OB)$. It intersects with the circumference in the point $C$. The segment $[OC]$ intersects the circumference in the point $E$. The straight lines $(AE)$ and $(OB)$ intersect in the point $K$. Prove that $|OK| = |KB|$. 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"The real numbers $a,b,c$ satisfy the condition: for all $x$, such that for $ -1 \le x \le 1$, the inequality$| ax^2 + bx + c | \le 1$ is held. Prove that for the same $x$ , $| cx^2 + bx + a | \le 2$." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less. What is the greatest possible number of the winner?" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"The square polynomial $f(x)=ax^2+bx+c$ is of such a sort, that the equation $f(x)=x$ does not have real roots. Prove that the equation $f(f(x))=0$ does not have real roots also." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"$n$ squares of the infinite cross-lined sheet of paper are painted with black colour (others are white). Every move all the squares of the sheet change their colour simultaneously. The square gets the colour, that had the majority of three ones: the square itself, its neighbour from the right side and its neighbour from the upper side. a) Prove that after the finite number of the moves all the black squares will disappear. b) Prove that it will happen not later than on the $n$-th move" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are built on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.) a) Prove that the circumferences outscribed around the outer triangles intersect in one point. b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"The king have revised the chess-board $8\times 8$ having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared. a) Give an example that the king could make $28$ steps parallel the sides of the board only. b) Prove that he could not make less than $28$ such a steps. c) What is the maximal and minimal length of the broken line if the side of a field is $1$?" 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,Given a convex $n$-angle with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-angle. 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Prove that for every positive $x_1, x_2, x_3, x_4, x_5$ holds inequality: $(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)$." 1973 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901978_1973_all_soviet_union_mathematical_olympiad,"Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Given a rectangle $ABCD$, points $M$ -- the middle of $[AD]$ side, $N$ -- the middle of $[BC]$ side. Let us take a point $P$ on the continuation of the $[DC]$ segment over the point $D$. Let us denote the point of intersection of lines $(PM)$ and $(AC)$ as $Q$. Prove that the angles $QNM$ and $MNP$ are equal." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Given $50$ segments on the line. Prove that one of the following statements is valid: 1. Some $8$ segments have the common point. 2. Some $8$ segments do not intersect each other." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square. 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"The triangle table is built according to the rule: You put the natural number $a>1$ in the upper row, and then you write under the number $k$ from the left side $k^2$, and from the right side -- $(k+1)$. For example, if $a = 2$, you get the table on the picture. Prove that all the numbers on each particular line are different. 2 / \ / \ 4 3 / \ / \ 16 5 9 4 / \ / \ /\ / \" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Let $O$ be the intersection point of the of the convex quadrangle $ABCD$ diagonals. Prove that the line drawn through the points of intersection of the medians of $AOB$ and $COD$ triangles is orthogonal to the line drawn through the points of intersection of the heights of $BOC$ and $AOD$ triangles." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Each of the $9$ straight lines divides the given square onto two quadrangles with the areas related as $2:3$. Prove that there exist three of them intersecting in one point" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"The $7$-angle $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle. Prove that if the centre of the circle is inside the $7$-angle, than the sum of $A_1,A_2$ and $A_3$ angles is less than $450$ degrees." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"A game for two. One gives a digit and the second substitutes it instead of a star in the following difference: $**** - **** = $ Then the first gives the next digit, and so on $8$ times. The first wants to obtain the greatest possible difference, the second -- the least. Prove that: 1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour. 2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Let $x,y$ be positive numbers, $s$ -- the least of $\{ x, (y+ 1/x), 1/y\}$. What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the outscribed circle." 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Is it possible to put the numbers $0,1$ or $2$ in the unit squares of the cross-lined paper $100x100$ in such a way, that every rectangle $3x4$ (and $4x3$) would contain three zeros, four ones and five twos?" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $\big\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\big\}$. What is the minimal possible $s$? What $x_i $correspond it?" 1972 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901977_1972_all_soviet_union_mathematical_olympiad,"One-round hockey tournament is finished (each plays with each one time, the winner gets $2$ points, looser -- $0$, and $1$ point for draw). For arbitrary subgroup of teams there exists a team (may be from that subgroup) that has got an odd number of points in the games with the teams of the subgroup. Prove that there was even number of the participants." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"Prove that for every natural $n$ there exists a number, containing only digits ""$1$"" and ""$2$"" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two )." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"a) Given a triangle $A_1A_2A_3$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A3], B_3$ and $D_1$ on the side $[A_3A_1]$. If you build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2$ and $A_3B_3C_3D_3$, the lines $(A_1C_1), (A_2C_2)$ and $(A_3C_3)$, will cross in one point $O$. Prove that if $|A_1B_1| = |A_2D_2|$ and $|A_2B_2| = |A_3D_3|$, than $|A_3B_3| = |A_1D_1|$. b) Given a convex polygon $A_1A_2 ... A_n$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A_3], ... B_n$ and $D_1$ on the side $[A_nA_1]$. Ifyou build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2 ... , A_nB_nC_nD_n$, the lines $(A_1C_1), (A_2C_2), ..., (A_nC_n)$, will cross in one point $O$. Prove that $|A_1B_1| \cdot |A_2B_2|\cdot ... \cdot |A_nB_n| = |A_1D_1|\cdot |A_2D_2|\cdot ...\cdot |A_nD_n|$." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$. The second player has to determine all the numbers. He is allowed to ask the questions like ""What number is written in the $x$ place in the $y$ row?"" What is the minimal number of the questions asked by the second player before he founds out all the numbers? b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?" 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"Given an unite square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$, then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,The projections of the body on two planes are circles. Prove that they have the same radius. 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary polygon outscribed around the circle. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"Given $25$ different positive numbers. Prove that you can choose two of them such, that none of the other numbers equals neither to the sum nor to the difference between the chosen numbers." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"a) The vertex $A_1$ of the right $12$-angle (dodecagon) $A_1A_2...A_{12}$ is marked with ""$-$"" and all the rest $--$ with ""$+$"". You are allowed to change the sign simultaneously in the $6$ vertices in succession. Prove that is impossible to obtain dodecagon with $A_2$ marked with ""$-$"" and the rest of the vertices $--$ with ""$+$"". b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession. c) Prove the same statement if it is allowed to change the signs in three vertices in succession." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"A cube with the edge of length $n$ is divided onto $n^3$ unit ones. Let us choose some of them and draw three lines parallel to the edges through their centres. What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes? a) Find the answer for the small $n$ ($n = 2,3,4$). b) Try to find the answer for $n = 10$. c) If You can not solve the general problem, try to estimate that value from the upper and lower side. d) Note, that You can reformulate the problem in such a way: Consider all the triples $(x_1,x_2,x_3)$, where $x_i$ can be one of the integers $1,2,...,n$. What is the minimal number of the triples necessary to provide the property: for each of the triples there exist the chosen one, that differs only in one coordinate. Try to find the answer for the situation with more than three coordinates, for example, with four." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"a) Consider the function $f(x,y) = x^2 + xy + y^2$. Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $f((x-m),(y-n)) \le 1/2$. b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le 1/2$. Prove that in fact, $g(x,y) \le 1/3$. Find all the points $(x,y)$, where $g(x,y)=1/3$. c) Consider function $f_a(x,y) = x^2 + axy + y^2$ ($0 \le a \le 2$). Find any $c$ such that $g_a(x,y) \le c$. Try to obtain the closest estimation." 1971 All Soviet Union Mathematical Olympiad,https://artofproblemsolving.com/community/c901976_1971_all_soviet_union_mathematical_olympiad,"A switch has two inputs $1, 2$ and two outputs $1, 2$. It either connects $1$ to $1$ and $2$ to $2$, or $1$ to $2$ and $2$ to 1. If you have three inputs $1, 2, 3$ and three outputs $1, 2, 3$, then you can use three switches, the first across $1$ and $2$, then the second across $2$ and $3$, and finally the third across $1$ and $2$. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for $4$ inputs, so that by suitably setting the switches the output can be any permutation of the inputs?" "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,"$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line." "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,"Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called good if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number." "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,"Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$." "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,"Consider the following $50$-term sums: $S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$, $T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$. Express $\frac{S}{T}$ as an irreducible fraction." "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,An integer $n \geq 3$ is called special if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$. "2014 Argentine National Olympiad, Level 3",https://artofproblemsolving.com/community/c175919_2014_argentine_national_olympiad_level_3,"Determine whether there exists positive integers $a_{1}1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$" "2005 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3777_2005_federal_competition_for_advanced_students_part_1,"We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent." "2006 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3778_2006_federal_competition_for_advanced_students_part_1,"Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k=k(n)\geq2$ known. In how many different ways (as a function of $ k=k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?" "2006 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3778_2006_federal_competition_for_advanced_students_part_1,"Show that the sequence $ a_n = \frac {(n + 1)^nn^{2 - n}}{7n^2 + 1}$ is strictly monotonically increasing, where $ n = 0,1,2, \dots$." "2006 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3778_2006_federal_competition_for_advanced_students_part_1,"In the triangle $ ABC$ let $ D$ and $ E$ be the boundary points of the incircle with the sides $ BC$ and $ AC$. Show that if $ AD=BE$ holds, then the triangle is isoceles." "2006 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3778_2006_federal_competition_for_advanced_students_part_1,"Given is the function $ f= \lfloor x^2 \rfloor + \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} = x - \lfloor x \rfloor$.) Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$." "2007 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3779_2007_federal_competition_for_advanced_students_part_1,"In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A+B\neq0$" "2007 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3779_2007_federal_competition_for_advanced_students_part_1,"For every positive integer $ n$ determine the highest value $ C(n)$, such that for every $ n$-tuple $ (a_1,a_2,\ldots,a_n)$ of pairwise distinct integers $ (n + 1)\sum_{j = 1}^n a_j^2 - \left(\sum_{j = 1}^n a_j\right)^2\geq C(n)$" "2007 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3779_2007_federal_competition_for_advanced_students_part_1,"Let $ M(n )=\{-1,-2,\ldots,-n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?" "2007 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3779_2007_federal_competition_for_advanced_students_part_1,"Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n - 1}A_n$ $ (A_n = A_0)$ where the side $ A_{i - 1}A_i = i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i + 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $ \Phi = \sum_{i = 0}^{n - 1} \phi_i$ of these $ n$ angles." "2008 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938800_2008_federal_competition_for_advanced_students_p1,What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$? "2008 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938800_2008_federal_competition_for_advanced_students_p1,"Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ ." "2008 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938800_2008_federal_competition_for_advanced_students_p1,"Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$. Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$." "2008 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938800_2008_federal_competition_for_advanced_students_p1,In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles. "2009 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938844_2009_federal_competition_for_advanced_students_p1,"Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ ." "2009 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938844_2009_federal_competition_for_advanced_students_p1,"For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$ $(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$ Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square." "2009 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938844_2009_federal_competition_for_advanced_students_p1,"There are $n$ bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a segment the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly $n+1$ segments." "2009 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c938844_2009_federal_competition_for_advanced_students_p1,"Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent." "2010 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3780_2010_federal_competition_for_advanced_students_part_1,"Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. (41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)" "2010 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3780_2010_federal_competition_for_advanced_students_part_1,"For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$. (41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)" "2010 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3780_2010_federal_competition_for_advanced_students_part_1,"Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called outstanding if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$. Determine the number $a(n)$ of outstanding subsets of $M_n$. (41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)" "2010 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3780_2010_federal_competition_for_advanced_students_part_1,"The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? (41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)" "2011 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3781_2011_federal_competition_for_advanced_students_part_1,"Determine all integer triplets $(x,y,z)$ such that $$x^4+x^2=7^zy^2\mbox{.}$$" "2011 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3781_2011_federal_competition_for_advanced_students_part_1,"For a positive integer $k$ and real numbers $x$ and $y$, let $$f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}$$ If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$." "2011 Federal Competition For Advanced Students, Part 1",https://artofproblemsolving.com/community/c3781_2011_federal_competition_for_advanced_students_part_1,"A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two. How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011 AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)" "2016 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c854175_2016_federal_competition_for_advanced_students_p1,"Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point? (Gerd Baron)" "2016 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c854175_2016_federal_competition_for_advanced_students_p1,"Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)" 2017 Federal Competition For Advanced Students,https://artofproblemsolving.com/community/c831741_2017_federal_competition_for_advanced_students,"Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions : (a) $P(2017) = 2016$ and (b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$. proposed by Walther Janous" 2017 Federal Competition For Advanced Students,https://artofproblemsolving.com/community/c831741_2017_federal_competition_for_advanced_students,"Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ (different from $M$) is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$. Prove that $AR$ and $QR$ have same length. proposed by Stephan Wagner" 2017 Federal Competition For Advanced Students,https://artofproblemsolving.com/community/c831741_2017_federal_competition_for_advanced_students,"Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table. Determine the smallest number $N\ge100000$ which Berta has wining strategy. proposed by Gerhard Woeginger" 2017 Federal Competition For Advanced Students,https://artofproblemsolving.com/community/c831741_2017_federal_competition_for_advanced_students,"Find all pairs $(a,b)$ of non-negative integers such that: $$2017^a=b^6-32b+1$$ proposed by Walther Janous" "2018 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c881354_2018_federal_competition_for_advanced_students_p1,"Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? (Proposed by Walther Janous)" "2018 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c881354_2018_federal_competition_for_advanced_students_p1,"Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear. (Proposed by Karl Czakler)" "2018 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c881354_2018_federal_competition_for_advanced_students_p1,"Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? (Proposed by Richard Henner)" "2018 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c881354_2018_federal_competition_for_advanced_students_p1,"Let $M$ be a set containing positive integers with the following three properties: (1) $2018 \in M$. (2) If $m \in M$, then all positive divisors of m are also elements of $M$. (3) For all elements $k, m \in M$ with $1 < k < m$, the number $km + 1$ is also an element of $M$. Prove that $M = Z_{\ge 1}$. (Proposed by Walther Janous)" "2019 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c1087271_2019_federal_competition_for_advanced_students_p1,"We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as $a_n = 14a_{n-1} + a_{n-2}$ , $b_n = 6b_{n-1}-b_{n-2}$. Determine whether there are infinite numbers that occur in both sequences" "2019 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c1087271_2019_federal_competition_for_advanced_students_p1,"Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$. The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$. Show that the segments $AY$ and $BX$ have equal length." "2019 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c1087271_2019_federal_competition_for_advanced_students_p1,"Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$. At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$. The two players take turns, with Ariane starting. Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently. Depending on $n$, determine which of the two has a winning strategy." "2019 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c1087271_2019_federal_competition_for_advanced_students_p1,"Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers$ n$. (For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)" "2021 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c2016100_2021_federal_competition_for_advanced_students_p1,"Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$(Marian Dinca)" "2021 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c2016100_2021_federal_competition_for_advanced_students_p1,"Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$, such that $AX = AB$. Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$. Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$. (Theresia Eisenkölbl)" "2021 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c2016100_2021_federal_competition_for_advanced_students_p1,"Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$. (Walther Janous)" "2021 Federal Competition For Advanced Students, P1",https://artofproblemsolving.com/community/c2016100_2021_federal_competition_for_advanced_students_p1,"On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)" 2019 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1085914_2019_regional_competition_for_advanced_students,"Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$" 2019 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1085914_2019_regional_competition_for_advanced_students,The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$. 2019 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1085914_2019_regional_competition_for_advanced_students,"Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$. (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$." 2019 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1085914_2019_regional_competition_for_advanced_students,Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors. 2018 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881329_2018_regional_competition_for_advanced_students,"If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$and determine all $a, b$ yielding equality. Proposed by Gottfried Perz" 2018 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881329_2018_regional_competition_for_advanced_students,"Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. Proposed by Stefan Leopoldseder" 2018 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881329_2018_regional_competition_for_advanced_students,"Let $n \ge 3$ be a natural number. Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two. Proposed by Walther Janous" 2018 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881329_2018_regional_competition_for_advanced_students,"Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. Proposed by Richard Henner" 2017 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c668635_2017_regional_competition_for_advanced_students,"Let $x_1, x_2, \dots, x_n$ be non-negative real numbers such that $$x_1^2+x_2^2 + \dots x_9^2 \ge 25.$$Prove that one can choose three of these numbers such that their sum is at least $5$. Proposed by Karl Czakler" 2017 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c668635_2017_regional_competition_for_advanced_students,"Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$. Proposed by Theresia Eisenkölbl" 2017 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c668635_2017_regional_competition_for_advanced_students,"The nonnegative integers $2000$, $17$ and $n$ are written on the blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed, where all three numbers remain unchanged. The player who is in turn and cannot make an allowed move loses the game. Prove that the game will end for every number $n$. Who wins the game in the case $n = 2017$? Proposed by Richard Henner" 2017 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c668635_2017_regional_competition_for_advanced_students,"Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. Proposed by Walther Janous" 2016 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854170_2016_regional_competition_for_advanced_students,"Determine all positive integers $k$ and $n$ satisfying the equation $$k^2 - 2016 = 3^n$$ (Stephan Wagner)" 2016 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854170_2016_regional_competition_for_advanced_students,"Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality $$(a+2)(b+2) \ge cd$$holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds. (Walther Janous)" 2016 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854170_2016_regional_competition_for_advanced_students,"On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move. Prove that Bob wins, no matter how they play. (Richard Henner)" 2016 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854170_2016_regional_competition_for_advanced_students,"Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$. (a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle. (b) Prove that the line $ST$ is parallel to the side $BC$. (Karl Czakler)" 2015 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854086_2015_regional_competition_for_advanced_students,"Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$$$\gcd(b,15) = c$$$$\gcd(a,c) = 5$$ (Richard Henner)" 2015 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854086_2015_regional_competition_for_advanced_students,"Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$. Prove that at least one of the three numbers $$x(x+y-z)$$$$y(y+z-x)$$$$z(z+x-y)$$is less or equal $1$. (Karl Czakler)" 2015 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854086_2015_regional_competition_for_advanced_students,"Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board. Determine the least integer that can be reached at the end by an appropriate sequence of moves. (Theresia Eisenkölbl)" 2015 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c854086_2015_regional_competition_for_advanced_students,"Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$. (a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular. (Walther Janous)" 2014 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c939049_2014_regional_competition_for_advanced_students,"Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ ." 2014 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c939049_2014_regional_competition_for_advanced_students,"You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $" 2014 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c939049_2014_regional_competition_for_advanced_students,"The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number." 2014 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c939049_2014_regional_competition_for_advanced_students,"For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area." 2013 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1077524_2013_regional_competition_for_advanced_students,For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest? 2013 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1077524_2013_regional_competition_for_advanced_students,"Determine all integers $x$ satisfying $$ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. $$($[y]$ is the largest integer which is not larger than $y.$)" 2013 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1077524_2013_regional_competition_for_advanced_students,"For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$ (a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$). (b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$" 2013 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1077524_2013_regional_competition_for_advanced_students,We call a pentagon distinguished if either all side lengths or all angles are equal. We call it very distinguished if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very distinguished pentagon has an axis of symmetry. 2012 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3773_2012_regional_competition_for_advanced_students,Prove that the inequality $$ a + a^3 - a^4 - a^6 < 1$$ holds for all real numbers $a$. 2012 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3773_2012_regional_competition_for_advanced_students,"Determine all integer solutions $(x, y)$ of the equation $$(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}$$" 2012 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3773_2012_regional_competition_for_advanced_students,"In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?" 2012 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3773_2012_regional_competition_for_advanced_students,"In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal." 2011 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3772_2011_regional_competition_for_advanced_students,"Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum $$\sum_{j=1}^{42}\frac{1}{p_j^2+1}$$ is not a unit fraction $\frac{1}{n^2}$ of some integer square number." 2011 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3772_2011_regional_competition_for_advanced_students,"Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\ \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}" 2011 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3772_2011_regional_competition_for_advanced_students,"Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$. Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$. Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$. Prove that $Q$, $T$, $U$ and $V$ are vertices of a trapezoid." 2011 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3772_2011_regional_competition_for_advanced_students,"Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that $$\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\mbox{.}$$ Determine the set of elements of $(a_n)$." 2010 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881334_2010_regional_competition_for_advanced_students,"Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality: $$\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.$$ (41th Austrian Mathematical Olympiad, regional competition, problem 1)" 2010 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881334_2010_regional_competition_for_advanced_students,"Solve the following in equation in $\mathbb{R}^3$: $$x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.$$ (41st Austrian Mathematical Olympiad, regional competition, problem 2)" 2010 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881334_2010_regional_competition_for_advanced_students,"Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar. (41th Austrian Mathematical Olympiad, regional competition, problem 3)" 2010 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c881334_2010_regional_competition_for_advanced_students,"Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. (41th Austrian Mathematical Olympiad, regional competition, problem 4)" 2009 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c938810_2009_regional_competition_for_advanced_students,"Find the largest interval $ M \subseteq \mathbb{R^ + }$, such that for all $ a$, $ b$, $ c$, $ d \in M$ the inequality $$ \sqrt {ab} + \sqrt {cd} \ge \sqrt {a + b} + \sqrt {c + d}$$ holds. Does the inequality $$ \sqrt {ab} + \sqrt {cd} \ge \sqrt {a + c} + \sqrt {b + d}$$ hold too for all $ a$, $ b$, $ c$, $ d \in M$? ($ \mathbb{R^ + }$ denotes the set of positive reals.)" 2009 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c938810_2009_regional_competition_for_advanced_students,"How many integer solutions $ (x_0$, $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6)$ does the equation $$ 2x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=9$$ have?" 2009 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c938810_2009_regional_competition_for_advanced_students,"Let $ D$, $ E$, $ F$ be the feet of the altitudes wrt sides $ BC$, $ CA$, $ AB$ of acute-angled triangle $ \triangle ABC$, respectively. In triangle $ \triangle CFB$, let $ P$ be the foot of the altitude wrt side $ BC$. Define $ Q$ and $ R$ wrt triangles $ \triangle ADC$ and $ \triangle BEA$ analogously. Prove that lines $ AP$, $ BQ$, $ CR$ don't intersect in one common point." 2009 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c938810_2009_regional_competition_for_advanced_students,"Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences. How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2=40 \cdot 2009$ are there?" 2008 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3771_2008_regional_competition_for_advanced_students,"Show: For all real numbers $ a,b,c$ with $ 0x_2>x_3>x_4>x_5>0$ and $ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$" 2007 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3770_2007_regional_competition_for_advanced_students,"Let $ a$ be a positive real number and $ n$ a non-negative integer. Determine $ S-T$, where $ S= \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$ and $ T= \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$" 2007 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3770_2007_regional_competition_for_advanced_students,"Let $ M$ be the intersection of the diagonals of a convex quadrilateral $ ABCD$. Determine all such quadrilaterals for which there exists a line $ g$ that passes through $ M$ and intersects the side $ AB$ in $ P$ and the side $ CD$ in $ Q$, such that the four triangles $ APM$, $ BPM$, $ CQM$, $ DQM$ are similar." 2006 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3769_2006_regional_competition_for_advanced_students,"Let $ 0 < x 1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1+ax_2=0$ $ x_2+a^2x_3=0$ … $ x_k+a^kx_{k+1}=0$ … $ x_n+a^nx_1=0$" 2006 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3769_2006_regional_competition_for_advanced_students,"In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$. Given are the points $ A$ and $ B$. Determine the set of points $ P=P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle." 2006 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3769_2006_regional_competition_for_advanced_students,"Let $ $ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $ h_n$ is the harmonic mean of its two neighbours $ h_{n-1}$ and $ h_{n+1}$ : $ h_n=\frac{2h_{n-1}h_{n+1}}{h_{n-1}+h_{n+1}}$) Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers." 2005 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3768_2005_regional_competition_for_advanced_students,"Show for all integers $ n \ge 2005$ the following chaine of inequalities: $ (n+830)^{2005}1$, then the sequence contains an infinite geometric sequence ($ b_n=b_0q^n$) of real numbers." 2004 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3767_2004_regional_competition_for_advanced_students,"Determine all integers $ a$ and $ b$, so that $ (a^3+b)(a+b^3)=(a+b)^4$" 2004 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3767_2004_regional_competition_for_advanced_students,"Solve the following equation for real numbers: $ \sqrt{4-x\sqrt{4-(x-2)\sqrt{1+(x-5)(x-7)}}}=\frac{5x-6-x^2}{2}$ (all square roots are non negative)" 2004 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3767_2004_regional_competition_for_advanced_students,"Given is a convex quadrilateral $ ABCD$ with $ \angle ADC=\angle BCD>90^{\circ}$. Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$" 2004 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3767_2004_regional_competition_for_advanced_students,"The sequence $ < x_n >$ is defined through: $ x_{n + 1} = \left(\frac {n}{2004} + \frac {1}{n}\right)x_n^2 - \frac {n^3}{2004} + 1$ for $ n > 0$ Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence." 2003 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044444_2003_regional_competition_for_advanced_students,"Find the minimum value of the expression $ \frac{a+1}{a(a+2)}+\frac{b+1}{b(b+2)}+\frac{c+1}{c(c+2)}$, where $ a,b,c$ are positive real numbers with $ a+b+c \le 3$." 2003 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044444_2003_regional_competition_for_advanced_students,Find all prime numbers $ p$ with $ 5^p+4p^4$ is the square of an integer. 2003 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044444_2003_regional_competition_for_advanced_students,"Given are two parallel lines $ g$ and $ h$ and a point $ P$, that lies outside of the corridor bounded by $ g$ and $ h$. Construct three lines $ g_1$, $ g_2$ and $ g_3$ through the point $ P$. These lines intersect $ g$ in $ A_1,A_2, A_3$ and $ h$ in $ B_1, B_2, B_3$ respectively. Let $ C_1$ be the intersection of the lines $ A_1B_2$ and $ A_2B_1$, $ C_2$ be the intersection of the lines $ A_1B_3$ and $ A_3B_1$ and let $ C_3$ be the intersection of the lines $ A_2B_3$ and $ A_3B_2$. Show that there exists exactly one line $ n$, that contains the points $ C_1,C_2,C_3$ and that $ n$ is parallel to $ g$ and $ h$." 2003 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044444_2003_regional_competition_for_advanced_students,For every real number $ b$ determine all real numbers $ x$ satisfying $ x-b= \sum_{k=0}^{\infty}x^k$. 2002 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044486_2002_regional_competition_for_advanced_students,"Find the smallest natural number $x> 0$ so that all following fractions are simplified $\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime." 2002 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044486_2002_regional_competition_for_advanced_students,"Solve the following system of equations over the real numbers: $2x_1 = x_5 ^2 - 23$ $4x_2 = x_1 ^2 + 7$ $6x_3 = x_2 ^2 + 14$ $8x_4 = x_3 ^2 + 23$ $10x_5 = x_4 ^2 + 34$" 2002 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044486_2002_regional_competition_for_advanced_students,"In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons." 2002 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044486_2002_regional_competition_for_advanced_students,"Let $a_0, a_1, ..., a_{2002}$ be real numbers. a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies: it is smaller or equal to $1/4$. b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ? c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ : the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$." 2001 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044495_2001_regional_competition_for_advanced_students,"Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?" 2001 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044495_2001_regional_competition_for_advanced_students,"Find all real solutions to the equation $(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)(x-2)^{2000}+(x-2)^{2001}=0$" 2001 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044495_2001_regional_competition_for_advanced_students,"In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?" 2001 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c1044495_2001_regional_competition_for_advanced_students,"Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$." 1970 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3766_1970_regional_competition_for_advanced_students,"Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$ When does equality hold?" 1970 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3766_1970_regional_competition_for_advanced_students,"In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections." 1970 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3766_1970_regional_competition_for_advanced_students,"$E_1$ and $E_2$ are parallel planes and their distance is $p$. (a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$? (b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$ Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way" 1970 Regional Competition For Advanced Students,https://artofproblemsolving.com/community/c3766_1970_regional_competition_for_advanced_students,"Find all real solutions of the following set of equations: $$72x^3+4xy^2=11y^3$$ $$27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5$$" 2015 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c166174_2015_azerbaijan_national_olympiad,"Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$.Then prove that $$a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}$$" 2015 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c166174_2015_azerbaijan_national_olympiad,"Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle." 2015 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c166174_2015_azerbaijan_national_olympiad,Find all polynomials $P(x)$ with real coefficents such that $$P(P(x))=(x^2+x+1)\cdot P(x)$$where $x \in \mathbb{R}$ 2015 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c166174_2015_azerbaijan_national_olympiad,"Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$." 2015 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c166174_2015_azerbaijan_national_olympiad,"In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$" 2020 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c1257110_2020_azerbaijan_national_olympiad,"$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.Example:$\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?" 2020 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c1257110_2020_azerbaijan_national_olympiad,"$a,b,c$ are positive integer. Solve the equation: $ 2^{a!}+2^{b!}=c^3 $" 2020 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c1257110_2020_azerbaijan_national_olympiad,"$a,b,c$ are positive numbers.$a+b+c=3$ Prove that: $\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $" 2020 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c1257110_2020_azerbaijan_national_olympiad,There is a non-equilateral triangle $ABC$.Let $ABC$'s Incentri $I$.Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$.Prove that $EM\parallel FN $. 2020 Azerbaijan National Olympiad,https://artofproblemsolving.com/community/c1257110_2020_azerbaijan_national_olympiad,"$a,b,c$ are non-negative integers. Solve: $a!+5^b=7^c$ Proposed by Serbia" 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers. 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"Prove that,if $a,b,c$ are positive real numbers, $$ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}$$" 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$.$\omega$ intersects $\alpha$ at $A$ and $B$.Let $P$ any point on the circumference $\omega$.The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$. 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"$A$ is a positive real number.$n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$." 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"Prove that for all positive integers $n$ we can find a permutation of {$1,2,...,n$} such that the average of two numbers doesn't appear in-between them.For example {$1,3,2,4$}works,but {$1,4,2,3$} doesn't because $2$ is between $1$ and $3$." 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of $$\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}$$" 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$." 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$." 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$. 2019 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854532_2019_bangladesh_mathematical_olympiad,"Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction." 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"Solve: $x^2(2-x)^2=1+2(1-x)^2$ Where $x$ is real number." 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"BdMO National 2018 Higher Secondary P2 $AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$" 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?" 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"Yukihira is counting the minimum number of lines $m$, that can be drawn on the plane so that they intersect in exactly $200$ distinct points.What is $m$?" 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof." 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"Find all the pairs of integers $(m,n)$ satisfying the equality $3(m^2+n^2)-7(m+n)=-4$" 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"Evaluate $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$" 2018 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854514_2018_bangladesh_mathematical_olympiad,"a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s. prove that 1. $n$ is greater than or equal to $2^{k+1}-1$ 2.Find all $n$ such that $2$ is a n-good" 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"BdMO National $2016$ Higher Secondary Problem 1: (a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$." 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"Bangladesh National Mathematical Olympiad 2016 Higher Secondary Problem 2: (a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?" 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$." 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"BdMO National 2016 Higher Secondary Problem 4: Consider the set of integers $ \left \{ 1, 2, ......... , 100 \right \} $. Let $ \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum, $S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $." 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"Bangladesh National Mathematical Olympiad 2016 Higher Secondary Problem 5: Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling. (a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line? (b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?" 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"BdMO National 2016 Higher Secondary Problem 6 $\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$. (a) Do the lines $AC$ and $DI$ intersect? Give a proof. (b) What is the angle of intersection between the lines $OD$ and $IB$?" 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"Juli is a mathematician and devised an algorithm to find a husband. The strategy is: • Start interviewing a maximum of $1000$ prospective husbands. Assign a ranking $r$ to each person that is a positive integer. No two prospects will have same the rank $r$. • Reject the first $k$ men and let $H$ be highest rank of these $k$ men. • After rejecting the first $k$ men, select the next prospect with a rank greater than $H$ and then stop the search immediately. If no candidate is selected after $999$ interviews, the $1000th$ person is selected. Juli wants to find the value of $k$ for which she has the highest probability of choosing the highest ranking prospect among all $1000$ candidates without having to interview all $1000$ prospects. (a) (6 points:) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)th$ prospect? (b) (6 points:) Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected? (c) (6 points:) What is the probability that the prospect with the highest rank is the $(m+1)th$ person and that Juli will choose the $(m+1)th$ man using this algorithm? (d) (16 points:) The total probability that Juli will choose the highest ranking prospect among the $1000$ prospects is the sum of the probability for each possible value of $m+1$ with $m+1$ ranging between $k+1$ and $1000$. Find the sum. To simplify your answer use the formula $In N \approx \frac{1}{N-1}+\frac{1}{N-2}+...+\frac{1}{2}+1$ (e) (6 points:) Find that value of $k$ that maximizes the probability of choosing the highest ranking prospect without interviewing all $1000$ candidates. You may need to know that the maximum of the function $x ln \frac{A}{x-1}$ is approximately $\frac{A + 1}{e}$, where $A$ is a constant and $e$ is Euler’s number, $e = 2.718....$" 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$." 2016 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854508_2016_bangladesh_mathematical_olympiad,"The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$ (a)(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$ Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$ (b)(10 POINTS):Show that, $\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$ Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$ (c)(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the previous part of the problem in term of this answer." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P1 A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P2 Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P3 Let $ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of polygon $X$." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P5 Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n.$ Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice. 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an ""awesome prime"". Find the largest ""awesome prime"" and prove that it is indeed the largest such prime." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof." 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$" 2013 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c854534_2013_bangladesh_mathematical_olympiad,"Higher Secondary P10 $X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$. Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions: (A) each element of $P_m(X)$ occurs exactly once in the sequence, (B) for any $i$ such that $0 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ friendship/enmity. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$. Problem 8: $ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$ Problem 9: The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square. Problem 10: Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$. The problems of the Junior categories are available in BdMO Online forum: " 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"BdMO National 2015 Secondary Problem 1. A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness One said exactly one of the four witnesses is a liar. Witness Two said exactly two of the four witnesses is a liar. Witness Three said exactly three of the four witnesses is a liar. Witness Four said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?" 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"BdMO National Higher Secondary Problem 3 Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?" 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$? Post no:$100$" 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)" 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height." 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?" 2015 Bangladesh Mathematical Olympiad,https://artofproblemsolving.com/community/c566957_2015_bangladesh_mathematical_olympiad,"In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$." Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"A pair of positive integers $(m,n)$ is called 'steakmaker' if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$" Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$. Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,Prottasha has a 10 sided dice. She throws the dice two times and sum the numbers she gets. Which number has the most probability to come out? Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"Once in a restaurant Dr. Strange found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?" Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$." Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$ABP = $\angle$ADP. What is the radius of the circumcircle of $ABCD$?" Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"Tiham is trying to find 6 digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the 3 digit number$ PQR$, and the 3 digit number $STU$ is divisible by 37. How many such numbers Tiham can find?" Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,Let $ABC$ be a triangle where$\angle$B=55 and $\angle$ C = 65. D is the mid-point of BC. Circumcircle of ACD and ABD cuts AB and AC at point F and E respectively. Center of circumcircle of AEF is O. $\angle$FDO = ? Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer k, in which piles contain at least k coins, take away exact k coins from these piles. Find the minimum number of turns you need to take way all of these coins?" Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number." Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"A prime number$ q $is called 'Kowai' number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers." Bangladesh Mathematical Olympiad 2020 Final,https://artofproblemsolving.com/community/c1278575_bangladesh_mathematical_olympiad_2020_final,"$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?" 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle. 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors. 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,"Some students of a group were friends of some others. One day all students of the group take part in a picnic. During the picnic some friends had a quarrel with each other, but some other students became friends. After the picnic, the number of friends for each student changed by $1$. Prove that the number of students in the group was even." 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,"Given a triangle $ABC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$." 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,"Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$." 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root. 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,"The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win. Original WordingThe expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win." 1995 Belarus National Olympiad,https://artofproblemsolving.com/community/c1980320_1995_belarus_national_olympiad,"Five numbers 1,2,3,4,5 are written on a blackboard. A student may erase any two of the numbers a and b on the board and write the numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?" 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem1:$$A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?" 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 2:$$Points $D$ and $E$ are taken on side $CB$ of triangle $ABC$, with $D$ between $C$ and $E$, such that $\angle BAE =\angle CAD$. If $AC < AB$, prove that $AC.AE < AB.AD$." 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 3:$$Is it possible to mark 10 red, 10 blue and 10 green points on a plane such that: For each red point A, the point (among the marked ones) closest to A is blue; for each blue point B, the point closest to B is green; and for each green point C, the point closest to C is red?" 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them." 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 1$$;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$." 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$, prove that $BD$ bisects $\angle ABC$." 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$ prove that $x$ is a root of a quadratic equation with integer coefficients." 1997 Belarusian National Olympiad,https://artofproblemsolving.com/community/c447860_1997_belarusian_national_olympiad,"$$Problem 4 $$Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero). Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?" 1999 Belarusian National Olympiad,https://artofproblemsolving.com/community/c571629_1999_belarusian_national_olympiad,Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$ 1999 Belarusian National Olympiad,https://artofproblemsolving.com/community/c571629_1999_belarusian_national_olympiad,"Let $m, n$ be positive integers. Starting with all positive integers written in a line, we can form a list of numbers in two ways: $(1)$ Erasing every $m$-th and then, in the obtained list, erasing every $n$-th number; $(2)$ Erasing every $n$-th number and then, in the obtained list, erasing every $m$-th number. A pair $(m,n)$ is called good if, whenever some positive integer $k$ occurs in both these lists, then it occurs in both lists on the same position. (a) Show that the pair $(2, n)$ is good for any $n\in \mathbb{N}$. (b) Is there a good pair $(m, n)$ with $2a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle." 1999 Belarusian National Olympiad,https://artofproblemsolving.com/community/c571629_1999_belarusian_national_olympiad,Solve in integer:$x^6+x^2y=y^3+2y^2$ 1999 Belarusian National Olympiad,https://artofproblemsolving.com/community/c571629_1999_belarusian_national_olympiad,Let O be the center of circle W. Two equal chords AB and CD of W intersect at L such that AL>LB and DL>LC. Let M and N be points on AL and DL respectively such that (ALC)=2*(MON). Prove that the chord of W passing through M and N is equal to AB and CD. 1999 Belarusian National Olympiad,https://artofproblemsolving.com/community/c571629_1999_belarusian_national_olympiad,"Let $n$ be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be unattainable. a.) Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. b.) Prove that if $n=3$, all positive integers except 7 are attainable." 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"Pit and Bill play the following game. First Pit writes down a number $a$, then Bill writes a number $b$, then Pit writes a number $c$. Can Pit always play so that the three equations $$x^3+ax^2+bx+c, x^3+bx^2+cx+a, x^3+cx^2+ax+b$$have (a) a common real root; (b) a common negative root?" 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$" 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$." 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?" 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"Nine points are given on a plane, no three of which lie on a line. Any two of these points are joined by a segment. Is it possible to color these segments by several colors in such a way that, for each color, there are exactly three segments of that color and these three segments form a triangle?" 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have? 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"(a) Find all positive integers $n$ for which the equation $(a^a)^n = b^b$ has a solution in positive integers $a,b$ greater than $1$. (b) Find all positive integers $a, b$ satisfying $(a^a)^5=b^b$" 2000 Belarusian National Olympiad,https://artofproblemsolving.com/community/c568904_2000_belarusian_national_olympiad,"To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"Prove for postitive $a$ and natural $n$ $$a^n+\frac{1}{a^n}-2 \geq n^2(a+\frac{1}{a}-2)$$" 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common. (a) If $n = 14$, determine the largest possible number of variants the problem committee can prepare. (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"In the increasing sequence of positive integers $a_1$, $a_2$,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence. (a) Prove that all but finitely terms of the sequence are funny. (b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?" 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"Let $n$ be a positive integer. Each square of a $(2n-1) \times (2n - 1)$ square board contains an arrow, either pointing up, down,to the left, or to the right. A beetle sits in one of the cells. Each year it creeps from one square in the direction of the arrow in that square, either reaching another square or leaving the board. Each time the beetle moves, the arrow in the square it leaves turns $\frac{\pi}{2}$ clockwise. Prove that the beetle leaves the board in at most $2^{3n-1}(n-1)!-4$ years after it first moves." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$. Prove that $O$, $R$,$T$, and $B$ are concyclic." 2001 Belarusian National Olympiad,https://artofproblemsolving.com/community/c1198798_2001_belarusian_national_olympiad,"There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all citizens (including himself) make and wear a necklace with zero or more stones so that: (i) given a pair of friends, there exists a color such that each has a stone of that color; (ii) given a pair of enemies,there does not exist a color such that each a stone of that color. (a) Prove that the aborigines can carry out the chieftain’s order. (b) What is the minimum number of colors of stones required for the aborigines to carry out the chieftain’s order?" 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"A connected graph with at least one vertex of an odd degree is given. Show that one can color the edges of the graph red and blue in such a way that, for each vertex, the absolute difference between the numbers of red and blue edges at that vertex does not exceed 1." 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$" 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,The cells of an $n\times n$ table ($n\ge 3$) are painted black and white in the chess-like manner. Per move one can choose any $2\times 2$ square and reverse the color of the cells inside it. Find all $n$ for which one can obtain a table with all cells of the same color after finitely many such moves. 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$." 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"Suppose that $A$ and $B$ are sets of real numbers such that $$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$(where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$ (a) Does it follow that $A=B$ (b) The same question, with the assumption that $B$ is bounded" 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have." 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$." 2004 Belarusian National Olympiad,https://artofproblemsolving.com/community/c567891_2004_belarusian_national_olympiad,"Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise aunt Polly is right." 2005 Belarusian National Olympiad,https://artofproblemsolving.com/community/c470033_2005_belarusian_national_olympiad,"Prove for positive numbers: $$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$" 2005 Belarusian National Olympiad,https://artofproblemsolving.com/community/c470033_2005_belarusian_national_olympiad,"A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$. Determine $\angle MTB$." 2005 Belarusian National Olympiad,https://artofproblemsolving.com/community/c470033_2005_belarusian_national_olympiad,"Solve in positive integers $a>b$: $$(a-b)^{ab}=a^bb^a$$" 2005 Belarusian National Olympiad,https://artofproblemsolving.com/community/c470033_2005_belarusian_national_olympiad,"An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color. a)Show, that exists good $9 \times 9$ good table. b)Prove, that fif $n \times n$ table is good, then $n<11$" 2005 Belarusian National Olympiad,https://artofproblemsolving.com/community/c470033_2005_belarusian_national_olympiad,"For $0 \frac{6}{11}$." 2007 Belarusian National Olympiad,https://artofproblemsolving.com/community/c467451_2007_belarusian_national_olympiad,Find solution in positive integers : $$n^5+n^4=7^m-1$$ 2007 Belarusian National Olympiad,https://artofproblemsolving.com/community/c467451_2007_belarusian_national_olympiad,"Let $(m,n)$ be a pair of positive integers. (a) Prove that the set of all positive integers can be partitioned into four pairwise disjoint nonempty subsets such that none of them has two numbers with absolute value of their difference equal to either $m$, $n$, or $m+n$. (b) Find all pairs $(m,n)$ such that the set of all positive integers can not be partitioned into three pairwise disjoint nonempty subsets satisfying the above condition." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair one number is divisible by another. 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$where $k!=1\cdot2\cdot\ldots\cdot k$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$. Prove that $AF$ is an altitude of triangle $ABC$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$.) Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD. Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"For all positive integers $m$ and $n$ prove the inequality $$ |n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1. $$" 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$1 and $R$2 there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$1 with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$2 with the columns from $S$ there are odd number of $1's$. Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$. 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?" 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$. 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$has a solution in integers $a$ and $b$. 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$. 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$for all real $x$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$. Prove that $\angle BXC=90^{\circ}$." 2018 Belarusian National Olympiad,https://artofproblemsolving.com/community/c854334_2018_belarusian_national_olympiad,"For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots1$. (V. Karamzin)" 2019 Belarusian National Olympiad,https://artofproblemsolving.com/community/c939475_2019_belarusian_national_olympiad,"The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$. Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$. (I. Voronovich)" 2019 Belarusian National Olympiad,https://artofproblemsolving.com/community/c939475_2019_belarusian_national_olympiad,"For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1b \right) \Rightarrow a+b \in S$ or $a-b \in S$ Prove that if we sort these numbers in ascending order, then they form an arithmetic progression" 2017 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c732292_2017_bosnia_and_herzegovina__regional_olympiad,"It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$" 2017 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c732292_2017_bosnia_and_herzegovina__regional_olympiad,"In triangle $ABC$ on side $AC$ are points $K$, $L$ and $M$ such that $BK$ is an angle bisector of $\angle ABL$, $BL$ is an angle bisector of $\angle KBM$ and $BM$ is an angle bisector of $\angle LBC$, respectively. Prove that $4 \cdot LM 10^4$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that: $a)$ $\mid c \mid \leq 1$ $b)$ $\mid a+c \mid \leq 1$ $c)$ $a^2+b^2+c^2 \leq 5$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$ 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers." 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"$h_a$, $h_b$ and $h_c$ are altitudes, $t_a$, $t_b$ and $t_c$ are medians of acute triangle, $r$ radius of incircle, and $R$ radius of circumcircle of acute triangle $ABC$. Prove that $$\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}$$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,It is given circle with center in center of coordinate center with radius of $2016$. On circle and inside it are $540$ points with integer coordinates such that no three of them are collinear. Prove that there exist two triangles with vertices in given points such that they have same area 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that $a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$ $b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$" 2016 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734667_2016_bosnia_and_herzegovina__regional_olympiad,"Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$ 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"In parallelogram $ABCD$ holds $AB=BD$. Let $K$ be a point on $AB$, different from $A$, such that $KD=AD$. Let $M$ be a point symmetric to $C$ with respect to $K$, and $N$ be a point symmetric to point $B$ with respect to $A$. Prove that $DM=DN$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,Solve the inequation: $$5\mid x\mid \leq x(3x+2-2\sqrt{8-2x-x^2})$$ 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points. $a)$ Prove that there exist two students with exactly the same answers to problems $b)$ Prove that there exist at least $4$ students with same number of points" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$$$y^3+z^3=3z+3x+4$$$$x^3+z^3=3x+3y+4$$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$is prime" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: i.) Each pair of students are in exactly one club. ii.) For each student and each society, the student is in exactly one club of the society. iii.) Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. Proposed by Guihua Gong, Puerto Rico" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality: $$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$" 2015 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c734978_2015_bosnia_and_herzegovina__regional_olympiad,"It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$ 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Determine the set $S$ with minimal number of points defining $7$ distinct lines 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$where $x$, $y$ and $z$ are integers" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. Alternative formulation. Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$." 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$." 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$ 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"Solve the equation $$x^2+y^2+z^2=686$$where $x$, $y$ and $z$ are positive integers" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$ 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,"At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances Find number of students in this class" 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$$$y=\frac{2x^2}{1+x^2}$$$$z=\frac{2y^2}{1+y^2}$$ 2014 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c735466_2014_bosnia_and_herzegovina__regional_olympiad,Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$ $b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$ " 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$if and only if $a=b$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus. 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"If $a$, $b$ and $c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$, prove that $$\frac{1}{2} \leq \frac{a}{1+a^4}+\frac{b}{1+b^4}+\frac{c}{1+c^4} \leq \frac{9\sqrt{3}}{10}$$" 2013 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736203_2013_bosnia_and_herzegovina__regional_olympiad,"If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$ 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,Solve equation $$x^2-\sqrt{a-x}=a$$where $x$ is real number and $a$ is real parameter 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have?" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$, line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$. Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$, line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$. If $PQ=QR$, prove that $ST=TU$" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$ 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"On football toornament there were $4$ teams participating. Every team played exactly one match with every other team. For the win, winner gets $3$ points, while if draw both teams get $1$ point. If at the end of tournament every team had different number of points and first place team had $6$ points, find the points of other teams" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$ 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$" 2012 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c736518_2012_bosnia_and_herzegovina__regional_olympiad,"Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,Factorise $$(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3$$ 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$is not perfect square" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$Prove that $p$ divides $m$ 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$ $a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$ $b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$has root in interval $\left(0,\frac{\pi}{2}\right)$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$ 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational" 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$ 2011 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738345_2011_bosnia_and_herzegovina__regional_olympiad,"If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$$$a^3+b^3+c^3+d^3=0$$Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$ 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$$$y-\frac{x+3y}{x^2+y^2}=0$$" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that $i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements $ii)$ Upper bound can be reached for $n=3$" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$where $x$, $y$ and $z$ are real numbers" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$ 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,"In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points" 2010 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c738806_2010_bosnia_and_herzegovina__regional_olympiad,It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$$$ 2 1$ such that $ \frac {b^{n} - 1}{b - 1}$ is perfect square. Prove that $ b$ is divisible by 8." 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"Given are two disjoint sets $ A$ and $ B$ such that their union is $ \mathbb N$. Prove that for all positive integers $ n$ there exist different numbers $ a$ and $ b$, both greater than $ n$, such that either $ \{ a,b,a + b \}$ is contained in $ A$ or $ \{ a,b,a + b \}$ is contained in $ B$." 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD=\frac{1}{2}(b+c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D=OM+OL$" 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}+b^{2}+c^{2}=1$ prove the inequality: $$ \frac{a^{5}+b^{5}}{ab(a+b)}+ \frac {b^{5}+c^{5}}{bc(b+c)}+\frac {c^{5}+a^{5}}{ca(a+b)}\geq 3(ab+bc+ca)-2.$$" 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,Prove that equation $ p^{4}+q^{4}=r^{4}$ does not have solution in set of prime numbers. 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$" 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$ at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$. Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$." 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"If $ a$, $ b$ and $ c$ are positive reals prove inequality: $$ \left(1+\frac{4a}{b+c}\right)\left(1+\frac{4b}{a+c}\right)\left(1+\frac{4c}{a+b}\right) > 25.$$" 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}+a^{3}+1}{a^{2}b^{2}+ab^{2}+1}$ is an integer. 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?" 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$. Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$ and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$. If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear." 2008 Bosnia And Herzegovina - Regional Olympiad,https://artofproblemsolving.com/community/c3642_2008_bosnia_and_herzegovina__regional_olympiad,"Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$" 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates. PS: The rays can be lines and the problem is still true." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always speak the truth or always lie. Arnaldo can ask questions for any of these people provided these questions are of the type: “In set $A$, what is the parity of people who speak to true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only be “$even$” or “$odd$”. a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and which people always lie? b) What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite?" 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer . (a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$. (b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"The following sentece is written on a board: The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions. Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?" 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$: If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$; If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$. Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence." 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"A positive integer is brazilian if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is superbrazilian if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, but $561=484+77$ is not superbrazilian, because $561$ is not brazilian. How many $4$-digit numbers are superbrazilians?" 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$. 2020 Brazil National Olympiad,https://artofproblemsolving.com/community/c1968245_2020_brazil_national_olympiad,"Let $k$ be a positive integer. Arnaldo and Bernaldo play a game in a table $2020\times 2020$, initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are $k$ consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't $k$ consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of $k$, Arnaldo has the winning strategy." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Let $\omega_1$ and $\omega_2$ be two circles with centers $C_1$ and $C_2$, respectively, which intersect at two points $P$ and $Q$. Suppose that the circumcircle of triangle $PC_1C_2$ intersects $\omega_1$ at $A \neq P$ and $\omega_2$ at $B \neq P$. Suppose further that $Q$ is inside the triangle $PAB$. Show that $Q$ is the incenter of triangle $PAB$." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Given are the real line and the two unique marked points $0$ and $1$. We can perform as many times as we want the following operation: we take two already marked points $a$ and $b$ and mark the reflection of $a$ over $b$. Let $f(n)$ be the minimum number of operations needed to mark on the real line the number $n$ (which is the number at a distance $\left| n\right|$ from $0$ and it is on the right of $0$ if $n>0$ and on the left of $0$ if $n<0$). For example, $f(0)=f(1)=0$ and $f(-1)=f(2)=1$. Find $f(n)$." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,Let $\mathbb{R}_{>0}$ be the set of the positive real numbers. Find all functions $f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that $$f(xy+f(x))=f(f(x)f(y))+x$$for all positive real numbers $x$ and $y$. 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Prove that for every positive integer $m$ there exists a positive integer $n_m$ such that for every positive integer $n \ge n_m$, there exist positive integers $a_1, a_2, \ldots, a_n$ such that $$\frac{1}{a_1^m}+\frac{1}{a_2^m}+\ldots+\frac{1}{a_n^m}=1.$$" 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"(a) Prove that given constants $a,b$ with $1b$. (b) Find all pairs of real numbers $(a,b)$ with $1b$." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Let $A_1A_2A_3A_4A_5$ be a convex, cyclic pentagon with $\angle A_i + \angle A_{i+1} >180^{\circ}$ for all $i \in \{1,2,3,4,5\}$ (all indices modulo $5$ in the problem). Define $B_i$ as the intersection of lines $A_{i-1}A_i$ and $A_{i+1}A_{i+2}$, forming a star. The circumcircles of triangles $A_{i-1}B_{i-1}A_i$ and $A_iB_iA_{i+1}$ meet again at $C_i \neq A_i$, and the circumcircles of triangles $B_{i-1}A_iB_i$ and $B_iA_{i+1}B_{i+1}$ meet again at $D_i \neq B_i$. Prove that the ten lines $A_iC_i, B_iD_i$, $i \in \{1,2,3,4,5\}$, have a common point." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"An eight-digit number is said to be 'robust' if it meets both of the following conditions: (i) None of its digits is $0$. (ii) The difference between two consecutive digits is $4$ or $5$. Answer the following questions: (a) How many are robust numbers? (b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all the super robust numbers." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Let $a, b$ and $k$ be positive integers with $k> 1$ such that $lcm (a, b) + gcd (a, b) = k (a + b)$. Prove that $a + b \geq 4k$" 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre" 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference." 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"In the picture below, a white square is surrounded by four black squares and three white squares. They are surrounded by seven black squares. What is the maximum number of white squares that can be surrounded by $ n $ black squares?" 2019 Brazil National Olympiad,https://artofproblemsolving.com/community/c1007703_2019_brazil_national_olympiad,"In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon they are said to be mutually visible if the line segment connecting them has no other blue dots. Prove that There is a set of $ 2019$ blue dots that are mutually visible two by two." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"We say that a polygon $P$ is inscribed in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. a) Show that if the initial number written is $0$, then Azambuja cannot reach his goal. b) Find all initial numbers for which Azambuja can achieve his goal." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo: Rule 1: Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise. Rule 2: Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed. Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?" 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"We say that a quadruple $(A,B,C,D)$ is dobarulho when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that: $1.$ $A \leq 8$ $2.$ $D>1$ $3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$. Find all such quadruples." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"a) In a $ XYZ$ triangle, the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side. The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $ b) Show that the triangles $ IMK $ and $ KNJ $ are congruent c) Show that the $ IDJK $ quad is inscritibed" 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum value of $x$." 2018 Brazil National Olympiad,https://artofproblemsolving.com/community/c774916_2018_brazil_national_olympiad,"Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression $S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers." 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,"1. For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$. a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$. b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$." 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,"2. Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements." 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,3. A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$. 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,"4. We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules: i. The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3). ii. Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet. iii. If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used. iv. Every password has at least four digits. Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password. Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order." 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,"5. In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$." 2017 Brazil National Olympiad,https://artofproblemsolving.com/community/c575420_2017_brazil_national_olympiad,"6. Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer." 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"Let $ABC$ be a triangle. $r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively. The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$. The lines $BD$ and $CE$ cut each other at $F$. $I$ is the incenter of $ABC$. Show that if $A,F,I$ are collinear, then $AB=AC$." 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\)." 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a winner number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it loser. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)" 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?" 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials \(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\). Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\). Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\). " 2016 Brazil National Olympiad,https://artofproblemsolving.com/community/c387202_2016_brazil_national_olympiad,"Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides. The lines \(AB\) and \(CD\) meet in \(E\). Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\). The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\). The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\). Show that \(I,J,M\) are colinear." 2015 Brazil National Olympiad,https://artofproblemsolving.com/community/c168606_2015_brazil_national_olympiad,"Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$. Prove that $A$, $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$." 2015 Brazil National Olympiad,https://artofproblemsolving.com/community/c168606_2015_brazil_national_olympiad,"Consider $S=\{1, 2, 3, \cdots, 6n\}$, $n>1$. Find the largest $k$ such that the following statement is true: every subset $A$ of $S$ with $4n$ elements has at least $k$ pairs $(a,b)$, $a1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its false derived is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$." 2015 Brazil National Olympiad,https://artofproblemsolving.com/community/c168606_2015_brazil_national_olympiad,"Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$iff $n$ is prime or $n=4$. b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$" 2015 Brazil National Olympiad,https://artofproblemsolving.com/community/c168606_2015_brazil_national_olympiad,"Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$, a polynomial $g(x)$, with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$?" 2015 Brazil National Olympiad,https://artofproblemsolving.com/community/c168606_2015_brazil_national_olympiad,"Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"Let $N$ be an integer, $N>2$. Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$. Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins. For each value of $N$, find which player has a winning strategy and describe it." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as $$P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.$$ Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal." 2014 Brazil National Olympiad,https://artofproblemsolving.com/community/c5127_2014_brazil_national_olympiad,"Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Let $r_A$ be the line $A_1A_2$. Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$, $r_B$ and $r_C$ determine a triangle $XYZ$. Prove that the incenter of $XYZ$, the circumcenter of $XYZ$ and $I$ are collinear." 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"Let $\Gamma$ be a circle and $A$ a point outside $\Gamma$. The tangent lines to $\Gamma$ through $A$ touch $\Gamma$ at $B$ and $C$. Let $M$ be the midpoint of $AB$. The segment $MC$ meets $\Gamma$ again at $D$ and the line $AD$ meets $\Gamma$ again at $E$. Given that $AB=a$, $BC=b$, compute $CE$ in terms of $a$ and $b$." 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo." 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that $$f(x+y) \left(f(x) + f(y)\right) = f(xy)$$ for all non-zero reals $x, y$ such that $x+y \neq 0$." 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$. P.S.: This is basically the same problem as " 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$, let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$. Prove that $p(k) \ge k+1$ for every positive integer $k$." 2013 Brazil National Olympiad,https://artofproblemsolving.com/community/c5126_2013_brazil_national_olympiad,"The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$." 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,"In a culturing of bacteria, there are two species of them: red and blue bacteria. When two red bacteria meet, they transform into one blue bacterium. When two blue bacteria meet, they transform into four red bacteria. When a red and a blue bacteria meet, they transform into three red bacteria. Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria." 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,"$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously). Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$." 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s. 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,"There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that $$ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} $$ where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?" 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors? 2012 Brazil National Olympiad,https://artofproblemsolving.com/community/c5125_2012_brazil_national_olympiad,"Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$." 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,"We call a number pal if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$." 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,"33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete." 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,"Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that: $$ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}$$" 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,"Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?" 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$ 2011 Brazil National Olympiad,https://artofproblemsolving.com/community/c5124_2011_brazil_national_olympiad,"Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove : $|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$" 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,"Find all functions $f$ from the reals into the reals such that $$ f(ab) = f(a+b) $$ for all irrational $a, b$." 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors. 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,"What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: ""The biggest shadow of a figure with the sun at its peak"" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane." 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,"Let $ABCD$ be a convex quadrilateral, and $M$ and $N$ the midpoints of the sides $CD$ and $AD$, respectively. The lines perpendicular to $AB$ passing through $M$ and to $BC$ passing through $N$ intersect at point $P$. Prove that $P$ is on the diagonal $BD$ if and only if the diagonals $AC$ and $BD$ are perpendicular." 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,"Determine all values of $n$ for which there is a set $S$ with $n$ points, with no 3 collinear, with the following property: it is possible to paint all points of $S$ in such a way that all angles determined by three points in $S$, all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited." 2010 Brazil National Olympiad,https://artofproblemsolving.com/community/c5123_2010_brazil_national_olympiad,"Find all pairs $(a, b)$ of positive integers such that $$ 3^a = 2b^2 + 1. $$" 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,"Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?" 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,"Let $ q = 2p+1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$." 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,"There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points $$ (a,b-1),\ (a,b+1),\ (a-1,b),\ (a+1,b)$$ Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations." 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes. 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,"Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point." 2009 Brazil National Olympiad,https://artofproblemsolving.com/community/c5122_2009_brazil_national_olympiad,"Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of $$ {x_1\over x_n+x_1+x_2} + {x_2\over x_1+x_2+x_3} + {x_3\over x_2+x_3+x_4} + \cdots + {x_{n-1}\over x_{n-2}+x_{n-1}+x_n} + {x_n\over x_{n-1}+x_n+x_1}$$" 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,"A positive integer is dapper if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 = 28694\times 7$. Prove that every positive integer is dapper." 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,"Let $ S$ be a set of $ 6n$ points in a line. Choose randomly $ 4n$ of these points and paint them blue; the other $ 2n$ points are painted green. Prove that there exists a line segment that contains exactly $ 3n$ points from $ S$, $ 2n$ of them blue and $ n$ of them green." 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,"Let $ x,y,z$ real numbers such that $ x + y + z = xy + yz + zx$. Find the minimum value of $$ {x \over x^2 + 1} + {y\over y^2 + 1} + {z\over z^2 + 1}$$" 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,"Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$." 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) = b\cdot n$ for all $ n$ positive integer. 2008 Brazil National Olympiad,https://artofproblemsolving.com/community/c5121_2008_brazil_national_olympiad,"The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the Zabrubic word. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a prophetic word of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$." 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,"Let $ f(x) = x^2 + 2007x + 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) = 0$ has at least one real solution." 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,Find the number of integers $ c$ such that $ -2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 + c$ is a multiple of $ 2^{2007}$. 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements. 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,"$ 2007^2$ unit squares are arranged forming a $ 2007\times 2007$ table. Arnold and Bernold play the following game: each move by Arnold consists of taking four unit squares that forms a $ 2\times 2$ square; each move by Bernold consists of taking a single unit square. They play anternatively, Arnold being the first. When Arnold is not able to perform his move, Bernold takes all the remaining unit squares. The person with more unit squares in the end is the winner. Is it possible to Bernold to win the game, no matter how Arnold play?" 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,"Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ= 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$." 2007 Brazil National Olympiad,https://artofproblemsolving.com/community/c5120_2007_brazil_national_olympiad,"Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j - x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"Let $n$ be an integer, $n \geq 3$. Let $f(n)$ be the largest number of isosceles triangles whose vertices belong to some set of $n$ points in the plane without three colinear points. Prove that there exists positive real constants $a$ and $b$ such that $an^{2}< f(n) < bn^{2}$ for every integer $n$, $n \geq 3$." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $$f(xf(y)+f(x)) = 2f(x)+xy$$ for every reals $x,y$." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"A positive integer is bold iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent." 2006 Brazil National Olympiad,https://artofproblemsolving.com/community/c5119_2006_brazil_national_olympiad,"Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is tough when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$. Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$." 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,"A natural number is a palindrome when one obtains the same number when writing its digits in reverse order. For example, $481184$, $131$ and $2$ are palindromes. Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome." 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,"Determine the smallest real number $C$ such that the inequality $$ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} $$ holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$." 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$. 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,"We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not." 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,"Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. Remark. The Fermat point $F$ is also known as the first Fermat point or the first Toricelli point of triangle $ABC$. Floor van Lamoen" 2005 Brazil National Olympiad,https://artofproblemsolving.com/community/c5118_2005_brazil_national_olympiad,"Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that $$ a^x + x \equiv b \pmod c, $$ that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$." 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,"Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus." 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments. 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,"Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?" 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,"Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board. Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers." 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers. 2004 Brazil National Olympiad,https://artofproblemsolving.com/community/c5117_2004_brazil_national_olympiad,"Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$. The set $per(a;b)$ of the periodic points of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$. Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum." 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$. 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,"Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are." 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,"$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel." 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,"Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$." 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,"Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$." 2003 Brazil National Olympiad,https://artofproblemsolving.com/community/c5116_2003_brazil_national_olympiad,A graph $G$ with $n$ vertices is called cool if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool. 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,"Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power." 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length. 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common. 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,"For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$." 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$. 2002 Brazil National Olympiad,https://artofproblemsolving.com/community/c5115_2002_brazil_national_olympiad,Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions. 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,"Show that for any $a,b,c$ positive reals, $$ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} $$" 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,"Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers." 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,"$ABC$ is a triangle $E, F$ are points in $AB$, such that $AE = EF = FB$ $D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$ $AD$ is perpendicular to $CF$. The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$) Determine the ratio $\frac{DB}{DC}$. %Edited!%" 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,"A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)" 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic. 2001 Brazil National Olympiad,https://artofproblemsolving.com/community/c5114_2001_brazil_national_olympiad,"A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either (A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or (B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left. Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$)." 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors. 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,"Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$. Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$." 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,"Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers. Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$." 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light? 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,"Let $ X$ the set of all sequences $ \{a_1, a_2,\ldots , a_{2000}\}$, such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The distance between two members $ a$ and $ b$ of $ X$ is defined as the number of $ i$ for which $ a_i$ and $ b_i$ are different. Find the number of functions $ f : X \to X$ which preserve the distance." 2000 Brazil National Olympiad,https://artofproblemsolving.com/community/c5113_2000_brazil_national_olympiad,Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get? 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,"Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$." 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,"Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits." 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,"How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?" 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,"On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$, then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$. In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$. The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs." 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,There are $n$ football teams in Tumbolia. A championship is to be organised in which each team plays against every other team exactly once. Ever match takes place on a sunday and each team plays at most one match each sunday. Find the least possible positive integer $m_n$ for which it is possible to set up a championship lasting $m_n$ sundays. 1999 Brazil National Olympiad,https://artofproblemsolving.com/community/c5112_1999_brazil_national_olympiad,"Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area." 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime." 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$." 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?" 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win." 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, $$ f(2f(x)) = x + 1998 . $$" 1998 Brazil National Olympiad,https://artofproblemsolving.com/community/c5111_1998_brazil_national_olympiad,"Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer $k$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $kN$ blocks (no matter what $N$ turns out to be)?" 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?" 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$. Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$. Show that $n^2 \leq 4 f(n) < 4 n^3$. Comment(If you have some amount of time, try a best estimative for $f(n)$, such that $f(p)<2p^2$ for prime $p$)." 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$. b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above." 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence ($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$) Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$" 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"Let $f(x)= x^2-C$ where $C$ is a rational constant. Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\ldots\}$ is finite" 1997 Brazil National Olympiad,https://artofproblemsolving.com/community/c5110_1997_brazil_national_olympiad,"$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1. Show that $f$ preserves distances." 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,"Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$." 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,"Does there exist a set of $n > 2, n < \infty$ points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set?" 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,"Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$." 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,"$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$." 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,"There are $n$ boys $B_1, B_2, ... , B_n$ and $n$ girls $G_1, G_2, ... , G_n$. Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners. For example if $(B_1, G_3)$ and $(B_4, G_7)$ are two of the pairs, then it must not be the case that $B_4$ prefers $G_3$ to $G_7$ and $G_3$ prefers $B_4$ to $B_1$." 1996 Brazil National Olympiad,https://artofproblemsolving.com/community/c5109_1996_brazil_national_olympiad,Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x. 1995 Brazil National Olympiad,https://artofproblemsolving.com/community/c5108_1995_brazil_national_olympiad,"$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square." 1995 Brazil National Olympiad,https://artofproblemsolving.com/community/c5108_1995_brazil_national_olympiad,"Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$." 1995 Brazil National Olympiad,https://artofproblemsolving.com/community/c5108_1995_brazil_national_olympiad,"For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which $$ P\left(n\right) 0$ be reals such that $$ a^3=a+1\\ b^6=b+3a $$ Show that $a>b$" 1994 Brazil National Olympiad,https://artofproblemsolving.com/community/c5107_1994_brazil_national_olympiad,"Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$. (Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ ) Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$. Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)" 1994 Brazil National Olympiad,https://artofproblemsolving.com/community/c5107_1994_brazil_national_olympiad,"A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$." 1993 Brazil National Olympiad,https://artofproblemsolving.com/community/c5106_1993_brazil_national_olympiad,"The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$. Find all terms which are perfect squares." 1993 Brazil National Olympiad,https://artofproblemsolving.com/community/c5106_1993_brazil_national_olympiad,"A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$." 1993 Brazil National Olympiad,https://artofproblemsolving.com/community/c5106_1993_brazil_national_olympiad,"Given a circle and its center $O$, a point $A$ inside the circle and a distance $h$, construct a triangle $BAC$ with $\angle BAC = 90^\circ$, $B$ and $C$ on the circle and the altitude from $A$ length $h$." 1993 Brazil National Olympiad,https://artofproblemsolving.com/community/c5106_1993_brazil_national_olympiad,"$ABCD$ is a convex quadrilateral with $$\angle BAC = 30^\circ $$$$\angle CAD = 20^\circ$$$$\angle ABD = 50^\circ$$$$\angle DBC = 30^\circ$$ If the diagonals intersect at $P$, show that $PC = PD$." 1993 Brazil National Olympiad,https://artofproblemsolving.com/community/c5106_1993_brazil_national_olympiad,Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$. 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$ 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1. 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,"Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area." 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,"Let $d(n)=\sum_{01$, $$ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} $$" 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,"Given a set of n elements, find the largest number of subsets such that no subset is contained in any other" 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,"Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$" 1992 Brazil National Olympiad,https://artofproblemsolving.com/community/c5105_1992_brazil_national_olympiad,"In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square." 1991 Brazil National Olympiad,https://artofproblemsolving.com/community/c5104_1991_brazil_national_olympiad,"At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W." 1991 Brazil National Olympiad,https://artofproblemsolving.com/community/c5104_1991_brazil_national_olympiad,"$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$." 1991 Brazil National Olympiad,https://artofproblemsolving.com/community/c5104_1991_brazil_national_olympiad,"Given $k > 0$, the sequence $a_n$ is defined by its first two members and $$ a_{n+2} = a_{n+1} + \frac{k}{n}a_n $$ a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$)." 1991 Brazil National Olympiad,https://artofproblemsolving.com/community/c5104_1991_brazil_national_olympiad,Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's). 1991 Brazil National Olympiad,https://artofproblemsolving.com/community/c5104_1991_brazil_national_olympiad,"$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$. $P_{n+4}$ is the midpoint of $P_nP_{n+1}$. $Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$. $A_n$ is the interior of $Q_n$. Find $\cap_{n \geq 0}A_n$." 1990 Brazil National Olympiad,https://artofproblemsolving.com/community/c5103_1990_brazil_national_olympiad,Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges. 1990 Brazil National Olympiad,https://artofproblemsolving.com/community/c5103_1990_brazil_national_olympiad,"There exists infinitely many positive integers such that $a^3 + 1990b^3 = c^4$." 1990 Brazil National Olympiad,https://artofproblemsolving.com/community/c5103_1990_brazil_national_olympiad,"Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$." 1990 Brazil National Olympiad,https://artofproblemsolving.com/community/c5103_1990_brazil_national_olympiad,"$ABCD$ is a quadrilateral, $E,F,G,H$ are midpoints of $AB,BC,CD,DA$. Find the point P such that $area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)$." 1990 Brazil National Olympiad,https://artofproblemsolving.com/community/c5103_1990_brazil_national_olympiad,"Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$) Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$, show that $F_n(x)=x$ (for any valid x)." 1989 Brazil National Olympiad,https://artofproblemsolving.com/community/c691157_1989_brazil_national_olympiad,"The sides of a triangle $T$, with vertices $(0,0)$,$(3,0)$ and $(0,3)$ are mirrors. Show that one of the images of the triagle $T_1$ with vertices $(0,0)$,$(0,1)$ and $(2,0)$ is the triangle with vertices $(24,36)$,$(24,37)$ and $(26,36)$." 1989 Brazil National Olympiad,https://artofproblemsolving.com/community/c691157_1989_brazil_national_olympiad,Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares. 1989 Brazil National Olympiad,https://artofproblemsolving.com/community/c691157_1989_brazil_national_olympiad,"A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$. Determine, justifying your answer, the set of all possible values for $f$." 1989 Brazil National Olympiad,https://artofproblemsolving.com/community/c691157_1989_brazil_national_olympiad,"A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second. At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A. B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins. Which player has a winning strategy? Suppose now both players has $k$ chips numbered 1 to $k$. Which player has a winning strategy? Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? " 1989 Brazil National Olympiad,https://artofproblemsolving.com/community/c691157_1989_brazil_national_olympiad,"A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron. Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere." 1988 Brazil National Olympiad,https://artofproblemsolving.com/community/c691152_1988_brazil_national_olympiad,Find all primes which are sum of two primes and difference of two primes. 1988 Brazil National Olympiad,https://artofproblemsolving.com/community/c691152_1988_brazil_national_olympiad,"Show that, among all triangles whose vertices are at distances 3,5,7 respectively from a given point P, the ones with largest area have P as orthocenter. (You can suppose, without demonstration, the existence of a triangle with maximal area in this question.)" 1988 Brazil National Olympiad,https://artofproblemsolving.com/community/c691152_1988_brazil_national_olympiad,"Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that $f(x \cdot y) = f(x) + f(y)$ $f(30) = 0$ $f(x)=0$ always when the units digit of $x$ is $7$ " 1988 Brazil National Olympiad,https://artofproblemsolving.com/community/c691152_1988_brazil_national_olympiad,"Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too." 1988 Brazil National Olympiad,https://artofproblemsolving.com/community/c691152_1988_brazil_national_olympiad,"A figure on a computer screen shows $n$ points on a sphere, no four coplanar. Some pairs of points are joined by segments. Each segment is colored red or blue. For each point there is a key that switches the colors of all segments with that point as endpoint. For every three points there is a sequence of key presses that makes the three segments between them red. Show that it is possible to make all the segments on the screen red. Find the smallest number of key presses that can turn all the segments red, starting from the worst case." 1987 Brazil National Olympiad,https://artofproblemsolving.com/community/c691151_1987_brazil_national_olympiad,"$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$." 1987 Brazil National Olympiad,https://artofproblemsolving.com/community/c691151_1987_brazil_national_olympiad,Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$. 1987 Brazil National Olympiad,https://artofproblemsolving.com/community/c691151_1987_brazil_national_olympiad,"Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy." 1987 Brazil National Olympiad,https://artofproblemsolving.com/community/c691151_1987_brazil_national_olympiad,"Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$." 1987 Brazil National Olympiad,https://artofproblemsolving.com/community/c691151_1987_brazil_national_olympiad,$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible? 1986 Brazil National Olympiad,https://artofproblemsolving.com/community/c691150_1986_brazil_national_olympiad,"A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times." 1986 Brazil National Olympiad,https://artofproblemsolving.com/community/c691150_1986_brazil_national_olympiad,Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers. 1986 Brazil National Olympiad,https://artofproblemsolving.com/community/c691150_1986_brazil_national_olympiad,"The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$." 1986 Brazil National Olympiad,https://artofproblemsolving.com/community/c691150_1986_brazil_national_olympiad,"Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number." 1986 Brazil National Olympiad,https://artofproblemsolving.com/community/c691150_1986_brazil_national_olympiad,"A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares." 1985 Brazil National Olympiad,https://artofproblemsolving.com/community/c691149_1985_brazil_national_olympiad,"$a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$." 1985 Brazil National Olympiad,https://artofproblemsolving.com/community/c691149_1985_brazil_national_olympiad,"Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$." 1985 Brazil National Olympiad,https://artofproblemsolving.com/community/c691149_1985_brazil_national_olympiad,A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$. 1985 Brazil National Olympiad,https://artofproblemsolving.com/community/c691149_1985_brazil_national_olympiad,"$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$." 1985 Brazil National Olympiad,https://artofproblemsolving.com/community/c691149_1985_brazil_national_olympiad,"$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$." 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,Find all solutions in positive integers to $(n+1)^k -1 = n!$ 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done? 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,"Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths." 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,"$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?" 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,"$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular." 1984 Brazil National Olympiad,https://artofproblemsolving.com/community/c691148_1984_brazil_national_olympiad,"There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?" 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers. 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,"An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon." 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$. 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color. 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,"Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$." 1983 Brazil National Olympiad,https://artofproblemsolving.com/community/c691147_1983_brazil_national_olympiad,Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$. 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,"The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar." 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,"Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the odd part of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the odd part of $3a_k+1$. Find $a_n$." 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$? 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,"Three numbered tiles are arranged in a tray as shown: Show that we cannot interchange the $1$ and the $3$ by a sequence of moves where we slide a tile to the adjacent vacant space." 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$. 1982 Brazil National Olympiad,https://artofproblemsolving.com/community/c691146_1982_brazil_national_olympiad,Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone. 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,"For which $k$ does the system $x^2 - y^2 = 0, (x-k)^2 + y^2 = 1$ have exactly: (i) two, (ii) three real solutions?" 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$? 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,"Given a sheet of paper and the use of a rule, compass and pencil, show how to draw a straight line that passes through two given points, if the length of the ruler and the maximum opening of the compass are both less than half the distance between the two points. You may not fold the paper." 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,"A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?" 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,"Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$." 1981 Brazil National Olympiad,https://artofproblemsolving.com/community/c691145_1981_brazil_national_olympiad,The centers of the faces of a cube form a regular octahedron of volume $V$. Through each vertex of the cube we may take the plane perpendicular to the long diagonal from the vertex. These planes also form a regular octahedron. Show that its volume is $27V$. 1980 Brazil National Olympiad,https://artofproblemsolving.com/community/c691144_1980_brazil_national_olympiad,"Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$?" 1980 Brazil National Olympiad,https://artofproblemsolving.com/community/c691144_1980_brazil_national_olympiad,Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$. 1980 Brazil National Olympiad,https://artofproblemsolving.com/community/c691144_1980_brazil_national_olympiad,"Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$." 1980 Brazil National Olympiad,https://artofproblemsolving.com/community/c691144_1980_brazil_national_olympiad,"Given $5$ points of a sphere radius $r$, show that two of the points are a distance $\le r \sqrt2$ apart." 1979 Brazil National Olympiad,https://artofproblemsolving.com/community/c582047_1979_brazil_national_olympiad,"Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?" 1979 Brazil National Olympiad,https://artofproblemsolving.com/community/c582047_1979_brazil_national_olympiad,"The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral." 1979 Brazil National Olympiad,https://artofproblemsolving.com/community/c582047_1979_brazil_national_olympiad,The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter. 1979 Brazil National Olympiad,https://artofproblemsolving.com/community/c582047_1979_brazil_national_olympiad,Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it. 1979 Brazil National Olympiad,https://artofproblemsolving.com/community/c582047_1979_brazil_national_olympiad," ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN. The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $\frac{2}{5}$ the area of the parallelogram. ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC. Is (ii) true for all convex quadrilaterals ABCD? " Brazil NMO L2 (OBM) - geometry,https://artofproblemsolving.com/community/c1089421_brazil_nmo_l2_obm__geometry,"A sheet of rectangular $ABCD$ paper, of area $1$, is folded along its diagonal $AC$ and then unfolded, then it is bent so that vertex $A$ coincides with vertex $C$ and then unfolded, leaving the crease $MN$, as shown below. a) Show that the quadrilateral $AMCN$ is a rhombus. b) If the diagonal $AC$ is twice the width $AD$, what is the area of the rhombus $AMCN$? //cdn.artofproblemsolving.com/images/8/7/5/87596f7dc14b51b32eea0f09a20bbdceae3db2f5.png" Brazil NMO L2 (OBM) - geometry,https://artofproblemsolving.com/community/c1089421_brazil_nmo_l2_obm__geometry,"Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$." Brazil NMO L2 (OBM) - geometry,https://artofproblemsolving.com/community/c1089421_brazil_nmo_l2_obm__geometry,"Let $ABC$ be a triangle inscribed in a circle of center $O$ and $P$ be a point on the arc $AB$, that does not contain $C$. The perpendicular drawn fom $P$ on line $BO$ intersects $AB$ at $S$ and $BC$ at $T$. The perpendicular drawn from $P$ on line $AO$ intersects $AB$ at $Q$ and $AC$ at $R$. Prove that: a) $PQS$ is an isosceles triangle b) $PQ^2=QR= ST$" Brazil NMO L2 (OBM) - geometry,https://artofproblemsolving.com/community/c1089421_brazil_nmo_l2_obm__geometry,"The triangle $ABC$ is inscribed in the circle $S$ and $AB a)$." 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,"Find all natural numbers $a>1$, with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$, $a^2-1$. K. Dochev" 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,"Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist? K. Dochev" 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,"On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools. K. Dochev" 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,"Let $\delta_0=\triangle A_0B_0C_0$ be a triangle. On each of the sides $B_0C_0$, $C_0A_0$, $A_0B_0$, there are constructed squares in the halfplane, not containing the respective vertex $A_0,B_0,C_0$ and $A_1,B_1,C_1$ are the centers of the constructed squares. If we use the triangle $\delta_1=\triangle A_1B_1C_1$ in the same way we may construct the triangle $\delta_2=\triangle A_2B_2C_2$; from $\delta_2=\triangle A_2B_2C_2$ we may construct $\delta_3=\triangle A_3B_3C_3$ and etc. Prove that: (a) segments $A_0A_1,B_0B_1,C_0C_1$ are respectively equal and perpendicular to $B_1C_1,C_1A_1,A_1B_1$; (b) vertices $A_1,B_1,C_1$ of the triangle $\delta_1$ lies respectively over the segments $A_0A_3,B_0B_3,C_0C_3$ (defined by the vertices of $\delta_0$ and $\delta_1$) and divide them in ratio $2:1$. K. Dochev" 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true. 1970 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384806_1970_bulgaria_national_olympiad,"In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then: (a) the maximum of $f(X)$ is equal to $9d^2+3+6d$; (b) the minimum of $f(X)$ is equal to $9d^2+3-6d$. K. Dochev and I. Dimovski" 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,A natural number is called triangular if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$. 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,"Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$has no real solutions." 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,"There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points." 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,"It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?" 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,"Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$-gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$, $i=1,2,\ldots,n$. Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$" 1971 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384805_1971_bulgaria_national_olympiad,"In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that (a) $m^2+n^2+p^2\ge18r^2$; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$." 1972 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384804_1972_bulgaria_national_olympiad,"Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$. I. Tonov" 1972 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384804_1972_bulgaria_national_olympiad,"Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$if the following conditions are satisfied: $02.$$Prove that the numbers $a_1$, $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers." 1978 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384741_1978_bulgaria_national_olympiad,"$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$. G. Ganchev" 1978 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384741_1978_bulgaria_national_olympiad,"On the name day of a man there are $5$ people. The men observed that of any $3$ people there are $2$ that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows. N. Nenov, N. Hazhiivanov" 1978 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384741_1978_bulgaria_national_olympiad,"Find the greatest possible real value of $S$ and smallest possible value of $T$ such that for every triangle with sides $a,b,c$ $(a\le b\le c)$ to be true the inequalities: $$S\le\frac{(a+b+c)^2}{bc}\le T.$$" 1978 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384741_1978_bulgaria_national_olympiad,"Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon. Jordan Tabov" 1978 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384741_1978_bulgaria_national_olympiad,"The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. Jordan Tabov" 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,"Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer" 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,"Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent." 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$. 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,"For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation $$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values." 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,"A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar." 1979 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2384480_1979_bulgaria_national_olympiad,"The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $k^3+1\le n$. Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$." 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,"Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits." 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,"(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one. (b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$." 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color. 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,"If $a,b,c$ are arbitrary nonnegative real numbers, prove the inequality $$a^3+b^3+c^3+6abc\ge\frac14(a+b+c)^3$$with equality if and only if two of the numbers are equal and the third one equals zero." 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,"Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$." 1980 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383849_1980_bulgaria_national_olympiad,"Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression." 1981 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383848_1981_bulgaria_national_olympiad,"Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane." 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,"Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$." 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle. 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,"In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$." 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,"If $x_1,x_2,\ldots,x_n$ are arbitrary numbers from the interval $[0,2]$, prove that $$\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\le n^2$$When is the equality attained?" 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,"Find all values of parameters $a,b$ for which the polynomial $$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$." 1982 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383847_1982_bulgaria_national_olympiad,Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$. 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$, then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$." 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$, $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$" 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively. (a) Prove that $AP/AD=BQ/BC$. (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$." 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point." 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?" 1983 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383846_1983_bulgaria_national_olympiad,"Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer." 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,Solve the equation $5^x7^y+4=3^z$ in nonnegative integers. 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,"The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$." 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,"Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number." 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,"Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation $$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal." 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,"Let $01$$for any natural numbers $m$ and $n$." 1984 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383722_1984_bulgaria_national_olympiad,"Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$." 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)" 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real." 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$." 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"Seven points are given in space, no four of which are on a plane. Each of the segments with the endpoints in these points is painted black or red. Prove that there are two monochromatic triangles (not necessarily both of the same color) with no common edge. Does the statement hold for six points?" 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"Let $P$ be a point on the median $CM$ of a triangle $ABC$ with $AC\ne BC$ and the acute angle $\gamma$ at $C$, such that the bisectors of $\angle PAC$ and $\angle PBC$ intersect at a point $Q$ on the median $CM$. Determine $\angle APB$ and $\angle AQB$." 1985 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383713_1985_bulgaria_national_olympiad,"Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit." 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct). 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,"Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$." 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,"A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron." 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,"Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon." 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,"Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$." 1986 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2383687_1986_bulgaria_national_olympiad,Let $0n+1$. Prove that if $a_{n-2}=0$, then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$." 1987 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382160_1987_bulgaria_national_olympiad,"Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$, and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$. Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational." 1987 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382160_1987_bulgaria_national_olympiad,"Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid." 1987 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382160_1987_bulgaria_national_olympiad,"The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares." 1987 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382160_1987_bulgaria_national_olympiad,"Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$." 1987 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382160_1987_bulgaria_national_olympiad,"Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle. (a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal. (b) Find the smallest $n$ with the property from (a)." 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,"Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$has four real roots." 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,"Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$, but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$." 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,"Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?" 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,"Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point." 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color. 1988 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382161_1988_bulgaria_national_olympiad,Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$. 1989 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382128_1989_bulgaria_national_olympiad,"In triangle $ABC$, point $O$ is the center of the excircle touching the side $BC$, while the other two excircles touch the sides $AB$ and $AC$ at points $M$ and $N$ respectively. A line through $O$ perpendicular to $MN$ intersects the line $BC$ at $P$. Determine the ratio $AB/AC$, given that the ratio of the area of $\triangle ABC$ to the area of $\triangle MNP$ is $2R/r$, where $R$ is the circumradius and $r$ the inradius of $\triangle ABC$." 1989 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382128_1989_bulgaria_national_olympiad,"Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$." 1989 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2382128_1989_bulgaria_national_olympiad,"Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer." 1991 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2016274_1991_bulgaria_national_olympiad,"Prove that for every prime number $p\ge5$, (a) $p^3$ divides $\binom{2p}p-2$; (b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$." 1991 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2016274_1991_bulgaria_national_olympiad,"Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$, $$f(x)f''(x)\le f'(x)^2.$$" 1991 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2016274_1991_bulgaria_national_olympiad,"On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$. (a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$. (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit." 1991 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2016274_1991_bulgaria_national_olympiad,"White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker." 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$. The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$. Prove that the measure of the angle $HMC$ doesn’t depend on the position of the point $C_1$. (Ivan Tonov)" 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"Prove that there exists $1904$-element subset of the set $\{1,2,\ldots,1992\}$, which doesn’t contain an arithmetic progression consisting of $41$ terms. (Ivan Tonov)" 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied: (i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$. (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$. (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. (Ivan Gochev, Hristo Minchev)" 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. (Plamen Koshlukov)" 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. (Plamen Koshlukov)" 1992 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2005112_1992_bulgaria_national_olympiad,"There are given one black box and $n$ white boxes ($n$ is a random natural number). White boxes are numbered with the numbers $1,2,\ldots,n$. In them are put $n$ balls. It is allowed the following rearrangement of the balls: if in the box with number $k$ there are exactly $k$ balls, that box is made empty - one of the balls is put in the black box and the other $k-1$ balls are put in the boxes with numbers: $1,2,\ldots,k-1$. (Ivan Tonov)" 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$." 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"Find the side length of the smallest equilateral triangle in which three discs of radii $2,3,4$ can be placed without overlap." 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"The quadratic polynomials $f$ and $g$ with real coefficients are such that if $g(x)$ is an integer for some $x>0$, then so is $f(x)$. Prove that there exist integers $m,n$ such that $f(x)=mg(x)+n$ for all $x$." 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"Sequence $\{a_n\}$ it define $a_1=1$ and $$a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$$for all $n\ge 1$ Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$" 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$." 1996 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c2381894_1996_bulgaria_national_olympiad,"A square table of size $7\times 7$ with the four corner squares deleted is given. What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table? Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [asy] size(3.5cm); usepackage(""amsmath""); MP(""\text{Figure }1."", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy]" 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Consider the polynomial $P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$ where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$ (a) Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$ (b) Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$" 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Let $M$ be the centroid of $\Delta ABC$ Prove the inequality $\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$ (a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$ (b) for any $\Delta ABC$" 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Let $n$ and $m$ be natural numbers such that $m+ i=a_ib_i^2$ for $i=1,2, \cdots n$ where $a_i$ and $b_i$ are natural numbers and $a_i$ is not divisible by a square of a prime number. Find all $n$ for which there exists an $m$ such that $\sum_{i=1}^{n}a_i=12$" 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$. Prove that $ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$." 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Given a triangle $ABC$. Let $M$ and $N$ be the points where the angle bisectors of the angles $ABC$ and $BCA$ intersect the sides $CA$ and $AB$, respectively. Let $D$ be the point where the ray $MN$ intersects the circumcircle of triangle $ABC$. Prove that $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$." 1997 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4571_1997_bulgaria_national_olympiad,"Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called disjoint if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples." 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions. 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,"The polynomials $P_n(x,y), n=1,2,... $ are defined by $$P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)$$ Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $." 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,"On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$." 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,"Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation: $$\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n$$ has at most one real nonzero root." 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,"let m and n be natural numbers such that: $3m|(m+3)^n+1$ Prove that $\frac{(m+3)^n+1}{3m}$ is odd" 1998 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4572_1998_bulgaria_national_olympiad,"The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$. (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$." 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes. 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,"Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$." 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,"The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number." 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations. 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,"The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC." 1999 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4573_1999_bulgaria_national_olympiad,Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$. 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called good if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary." 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$." 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p - 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k - 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$." 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"Find all polynomials $P(x)$ with real coefficients such that $$P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.$$" 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent." 2000 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4574_2000_bulgaria_national_olympiad,"Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$." 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,"Consider the sequence $\{a_n\}$ such that $a_0=4$, $a_1=22$, and $a_n-6a_{n-1}+a_{n-2}=0$ for $n\ge2$. Prove that there exist sequences $\{x_n\}$ and $\{y_n\}$ of positive integers such that $$ a_n=\frac{y_n^2+7}{x_n-y_n} $$ for any $n\ge0$." 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic. 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,"Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive ""blocks"" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the ""blocks"" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$" 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,"Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties: (i) the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times; (ii) the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times." 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,"Find all real values $t$ for which there exist real numbers $x$, $y$, $z$ satisfying : $3x^2 + 3xz + z^2 = 1$ , $3y^2 + 3yz + z^2 = 4$, $x^2 - xy + y^2 = t$." 2001 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4575_2001_bulgaria_national_olympiad,"Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$. Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$." 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Let $a_1, a_2... $ be an infinite sequence of real numbers such that $a_{n+1}=\sqrt{{a_n}^2+a_n-1}$. Prove that $a_1 \notin (-2,1)$ Proposed by Oleg Mushkarov and Nikolai Nikolov " 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ Proposed by Alexander Ivanov" 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$. What is the minimal number of connecting segments among the points, such that for each $n$-plet of points we can find four points, which are all connected to each other? Proposed by Alexander Ivanov and Emil Kolev" 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$. Proposed by Georgi Ganchev" 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms. Proposed by Oleg Mushkarov and Nikolai Nikolov" 2002 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4576_2002_bulgaria_national_olympiad,"Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}1$ such that there exist a uniform set $A$ with $n$ elements." 2003 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4577_2003_bulgaria_national_olympiad,"Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are equal integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime." 2003 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4577_2003_bulgaria_national_olympiad,"Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root." 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,"Let $ I$ be the incenter of triangle $ ABC$, and let $ A_1$, $ B_1$, $ C_1$ be arbitrary points on the segments $ (AI)$, $ (BI)$, $ (CI)$, respectively. The perpendicular bisectors of $ AA_1$, $ BB_1$, $ CC_1$ intersect each other at $ A_2$, $ B_2$, and $ C_2$. Prove that the circumcenter of the triangle $ A_2B_2C_2$ coincides with the circumcenter of the triangle $ ABC$ if and only if $ I$ is the orthocenter of triangle $ A_1B_1C_1$." 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,"For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, where $P(n)$ and $Q(n)$ are relatively prime. a) Prove that $P(67)$ is not divisible by 3; b) Find all possible $n$, for which $P(n)$ is divisible by 3." 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\displaystyle \frac 2{5}n$ tourists. 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,"In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times." 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,"Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$." 2004 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4578_2004_bulgaria_national_olympiad,"Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}+b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that a) if $ m+n > p$ then $ k = p$; b) if $ m+n\leq p$ then $ k\geq m+n-1$." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"Determine all triples $\left( x,y,z\right)$ of positive integers for which the number $$ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} $$ is an integer ." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that : (a) the points $C,T,Y,I$ are concyclic. (b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$, find $\widehat{EMF}$." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"For positive integers $t,a,b,$a $(t,a,b)$-game is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$." 2005 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4579_2005_bulgaria_national_olympiad,"Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide." 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$. Aleksandar Ivanov" 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ $$f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}$$ a) Prove that $f(2x)=4f(x)$ for all $x>0$; b) Find all such functions. Nikolai Nikolov, Oleg Mushkarov " 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times. Aleksandar Ivanov, Emil Kolev " 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. Aleksandar Ivanov" 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point. Emil Kolev " 2006 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4580_2006_bulgaria_national_olympiad,"Consider a point $O$ in the plane. Find all sets $S$ of at least two points in the plane such that if $A\in S$ ad $A\neq O$, then the circle with diameter $OA$ is in $S$. Nikolai Nikolov, Slavomir Dinev" 2007 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4581_2007_bulgaria_national_olympiad,"The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$." 2007 Bulgaria National Olympiad,https://artofproblemsolving.com/community/c4581_2007_bulgaria_national_olympiad,"Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i