There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?

Define $A(n)$ as the average of all positive divisors of the positive integer $n$. Find the sum of all solutions to $A(n)=42$. [i] Proposed by Yannick Yao [/i]

The triangle $ABC$ is isosceles with apex at $A{}$ and $M,N,P$ are the midpoints of the sides $BC,CA,AB$ respectively. Let $Q{}$ and $R{}$ be points on the segments $BM$ and $CM$ such that $\angle BAQ =\angle MAR.$ The segment $NP{}$ intersects $AQ,AR$ at $U,V$ respectively. The point $S{}$ is considered on the ray $AQ$ such that $SV$ is the angle bisector of $\angle ASM.$ Similarly, the point $T{}$ lies on the ray $AR$ uch that $TU$ is the angle bisector of $\angle ATM.$ Prove that one of the intersection points of the circles $(NUS)$ and $(PVT)$ lies on the line $AM.$ [i]Proposed by Flavian Georgescu[/i]

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions: (1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$. (2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$. Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.

Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$.

Let $n \geq 4$ be an integer. $a_1, a_2, \ldots, a_n \in (0, 2n)$ are $n$ distinct integers. Prove that there exists a subset of the set $\{a_1, a_2, \ldots, a_n \}$ such that the sum of its elements is divisible by $2n.$

Let $ABC$ a triangle and let $I$ be the center of its inscribed circle. Let $D$ be the symmetric point of $I$ with respect to $AB$ and $E$ be the symmetric point of $I$ with respect to $AC$. Show that the circumcircles of the triangles $BID$ and $CIE$ are eachother tangent.

Let $n$ be a positive integer and let $z_1,z_2,\dots,z_n$ be positive integers such that for $j=1,2,\dots,n$ the inequalites $z_j \le j$ hold and $z_1+z_2+\dots+z_n$ is even. Prove that the number $0$ occurs among the values \[z_1 \pm z_2 \pm \dots \pm z_n,\] where $+$ or $-$ can be chosen independently for each operation. [i](Walther Janous)[/i]

Let $\mathbb Q_{\geq 0}$ be the non-negative rational numbers, $f: \mathbb Q_{\geq 0} \to \mathbb Q_{\geq 0}$ such that $f(z+1) = f(z)+1$, $f(1/z) = f(z)$ for $z\neq 0$, and $f(0) = 0.$ Define a sequence $P_n$ of non-negative integers recursively via $$P_0 = 0,\quad P_1 = 1,\quad P_n = 2 P_{n-1}+P_{n-2}$$ for every $n \geq 2$. Find $f\left(\frac{P_{20}}{P_{24}}\right).$ [i]Proposed by Robert Trosten[/i]

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

A square is divided into $n^4$ fields like a chessboard. $n^3$ game pieces are placed on these squares placed, on each at most one. There are the same number of stones in each row. Besides, the whole arrangement symmetrical to one of the diagonals of the square; this diagonal is called $d$. Prove that: a) If $n$ is odd, then there is at least one stone on $d$. b) If $n$ is even, then there is an arrangement of the type described, in which there is no stone on $d$.

The product of the $ 9$ factors $ \left (1\minus{}\frac{1}{2} \right ) \left (1\minus{}\frac{1}{3} \right ) \left (1\minus{}\frac{1}{4} \right ) \ldots \left (1\minus{}\frac{1}{10} \right )\equal{}$ \[ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{10}{11} \qquad \textbf{(E)}\ \frac{11}{2} \]

In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.

Let a geometric progression with $n$ terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is $\textbf{(A) }\frac{1}{s}\qquad\textbf{(B) }\frac{1}{r^ns}\qquad\textbf{(C) }\frac{s}{r^{n-1}}\qquad\textbf{(D) }\frac{r^n}{s}\qquad \textbf{(E) }\frac{r^{n-1}}{s}$

Let $a_{1},a_{2},...,a_{n}$ be positive real numbers with $a_{1}\leq a_{2}\leq ... a_{n}$ such that the arithmetic mean of $a_{1}^{2},...,a_{n}^{2}$ is 1. If the arithmetic mean of $a_{1}, a_{2},...,a_{n}$ is $m$. Prove that if $a_{i}\leq$ m for some $1 \leq i \leq n$, then $n(m-a_{i})^2\leq n-i$

Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions? $(\textbf{A})\: 1.6\qquad(\textbf{B}) \: 1.8\qquad(\textbf{C}) \: 2.0\qquad(\textbf{D}) \: 2.2\qquad(\textbf{E}) \: 2.4$

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Suppose $a_1,a_2,\cdots,a_8$ are eight distinct integers from $\{1,2,\cdots,16,17\}$. Show that there is an integer $k > 0$ such that the equation $a_i - a_j = k$ has at least three different solutions. Also, find a specific set of 7 distinct integers from $\{1,2,\ldots,16,17\}$ such that the equation $a_i - a_j = k$ does not have three distinct solutions for any $k > 0$.

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? $ \textbf{(A)}\ 1-\frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ \frac{2\sqrt{3}-3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(D)}\ \frac{1}{4} \qquad\textbf{(E)}\ \frac{\sqrt{3}(2-\sqrt{2})}{4} $

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.