Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: \[\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}\]

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Let $ABCD$ be a convex quadrilateral with $\angle B < \angle A < 90^{o}$. Let $I$ be the midpoint of $AB$ and $S$ the intersection of $AD$ and $BC$. Let $R$ be a variable point inside the triangle $SAB$ such that $\angle ASR = \angle BSR$. On the straight lines $AR, BR$ , take the points $E, F$, respectively so that $BE , AF$ are parallel to $RS$. Suppose that $EF$ intersects the circumcircle of triangle $SAB$ at points $H, K$. On the segment $AB$, take points $M , N$ such that $\angle AHM =\angle BHI$ , $\angle BKN = \angle AKI$. a) Prove that the center $J$ of the circumcircle of triangle $SMN$ lies on a fixed line. b) On $BE, AF$ , take the points $P, Q$ respectively so that $CP$ is parallel to $SE$ and $DQ$ is parallel to $SF$. The lines $SE, SF$ intersect the circle $(SAB)$, respectively, at $U, V$. Let $G$ be the intersection of $AU$ and $BV$. Prove that the median of vertex $G$ of the triangle $GPQ$ always passes through a fixed point .

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

Circles $\omega_1$, $\omega_2$, and $\omega_3$ are centered at $M$, $N$, and $O$, respectively. The points of tangency between $\omega_2$ and $\omega_3$, $\omega_3$ and $\omega_1$, and $\omega_1$ and $\omega_2$ are tangent at $A$, $B$, and $C$, respectively. Line $MO$ intersects $\omega_3$ and $\omega_1$ again at $P$ and $Q$ respectively, and line $AP$ intersects $\omega_2$ again at $R$. Given that $ABC$ is an equilateral triangle of side length $1$, compute the area of $PQR$.

Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.

Let $A_1A_2 \cdots A_{10}$ be a regular decagon such that $[A_1A_4]=b$ and the length of the circumradius is $R$. What is the length of a side of the decagon? $ \textbf{a)}\ b-R \qquad\textbf{b)}\ b^2-R^2 \qquad\textbf{c)}\ R+\dfrac b2 \qquad\textbf{d)}\ b-2R \qquad\textbf{e)}\ 2b-3R $

Find the number of three-digit numbers such that its first two digits are each divisible by its third digit.

For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$. What is the expected value of $X(P)$ if each permutation is equally likely?

Let $ n$ be a positive integer, and consider the matrix $ A \equal{} (a_{ij})_{1\leq i,j\leq n}$ where $ a_{ij} \equal{} 1$ if $ i\plus{}j$ is prime and $ a_{ij} \equal{} 0$ otherwise. Prove that $ |\det A| \equal{} k^2$ for some integer $ k$.

Let $a,b$ be real numbers. Prove the inequality \[ 2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).\]

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 [i]fibonatic[/i] when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not [i]fibonatic[/i] integers.

Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that for any real numbers $x,y$ with $y\neq 0$ we have $$f(f(x)+y)f\left(\frac{1}{y}\right)=xf\left(\frac{1}{y}\right) + 1.$$ [i]Proposed by Marius Cerlat[/i]

Rosa dos Ventos, Aurora Boreal and Manuela do Norte organized a competition between them last weekend, consisting of several athletics events. The winner in each test obtained $x$ points, the second placed $y$ points and the third placed $z$ points ($x,y,z \in N$ and $x >y>z$). The final result of the competition, obtained by adding up the scores in each event, was Rosa had $22$ points, Manuela had $9$ points, Aurora had $9 $ points. In how many tests did they compete and who came second in the high jump knowing that the Manuela won the $100$ meters and no one gave up in any race? [hide=official wording]Rosa dos Ventos, a Aurora Boreal e a Manuela do Norte organizaram no passado fim de semana uma competi¸c˜ao entre elas, consistindo em v´arias provas de atletismo. A vencedora em cada prova obteve x pontos, a segunda classificada y pontos e a terceira classificada z pontos (x,y,z ∈ IN e x >y>z). O resultado final da competi¸c˜ao, obtido por soma das pontua¸c˜oes em cada prova, foi Rosa 22 pontos Manuela 9 pontos Aurora 9 pontos Em quantas provas competiram e quem ficou em segundo lugar no salto em altura sabendo que a Manuela ganhou os 100 metros e que ningu´em desistiu em nenhuma prova?[/hide]

Determine all positive integers $n > 1$ for which $n + D(n)$ is a power of $10$, where $D(n)$ denotes the largest integer divisor of $n$ satisfying $D(n) < n$.

The European zoos with at least two types of species are separated in two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A,B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. What is the least $k$ for which we can color the cages in the zoos (each cage only has all animals of one species) such that no zoo has cages of only one color (with every animal across all zoos having to be colored in the same color)? For the maximal value of $k$, find all possibilities (zoos and species), for which this maximum is achieved.

Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$ for all x, y positive reals.

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

Given a non negative real $a$ and a sequence $(u_n)$ defined by \[ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} \] a) Prove that for $a=0$, the sequence is convergent and find its limit. b) For $a\in [0,1]$, prove that the sequence if convergent.

Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$

Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = min \{k \in N | \phi^k (n)=1 \}$ . For example, $\phi^1 (13) = \phi(13) = 12$ $\phi^2 (13) = \phi (\phi (13)) = 4$ $\phi^3 (13) = \phi(\phi(\phi(13))) = 2$ $\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$ so $\theta (13) = 4$. Let $f(r) = \theta (13^r)$. Determine $f(2012)$.