Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$

Compute the remainder when $\binom{205}{101}$ is divded by $101 \times 103$. [i]Proposed by Brandon Xu[/i]

For each problem, you will be asked to submit two integers. The first value that you submit represents what you think the correct answer to the problem is. The second value that you submit represents [b]how many teams[/b] you think will submit the correct answer. For example, consider 0. What is $32\div 2 \times 4 +3?$ The correct answer would be $67$. If you think every team will get it right, you should submit the number of teams competing at PUMaC. Therefore, a viable submission for the first entry could be $67$ and $n$ for the second, where $n$ is the number of teams taking the team round. There are $\mathbf{72}$ [b]teams[/b] signed up to take this round at PUMaC: 45 in A division and 27 in B division. You will receive $\left(\min\left(\tfrac{a}{b},\tfrac{b}{a}\right)\right)^2$ points for your guess, where $a$ is the number of teams that correctly answered the question and $b$ is the number of teams you guessed would get it correct (Note that in the case that no teams answer correctly or you guess $0$, you will receive $0$ points).

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

For function $f:\mathbb Z^+ \to \mathbb R$ and coprime positive integers $p,q$ ; define $f_p,f_q$ as $$f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+)$$ $f$ satisfies following conditions. $ $ $ $ $(i)$ $ $ for all $r$ that isn't multiple of $pq$, $f(r)=0$ $ $ $ $ $(ii)$ $ $ $\exists m\in \mathbb Z^+$ $ $ $s.t.$ $ $ $\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x)$ and $f_q(x+m)=f_q(x)$ Prove that if $x\equiv y$ $ $ $(mod m)$, then $f(x)=f(y)$ $ $ ($x, y\in \mathbb Z^+$).

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

Let $ ABC$ be a triangle. Points $ A',B',C'$ are on the sides $ BC, CA, AB,$ respectively such that we have \[ \overline{A'B'} \equal{} \overline{B'C'} \equal{} \overline{C'A'}\] and \[ \overline{AB'} \equal{} \overline{BC'} \equal{} \overline{CA'}.\] Prove that triangle $ ABC$ is equilateral.

Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

There are two $3$-digit numbers which end in $99$. These two numbers are also the product of two integers which differ by $2$. What is the sum of these two numbers?

Let $ABC$ be an acute triangle with altitudes $AD,BE$ and $CF$. Denote $\omega_1,\omega_2$ the circumcircles of $\triangle AEB, \triangle AFC$, respectively. Suppose the line through $A$ parallel to $EF$ intersects $\omega_1$ and $\omega_2$ at $P$ and $Q$, respectively. Show that the circumcenter of $\triangle PQD$ lies on $AD$

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

[b]6.1 [/b] Three people with one double seater motorbike simultaneously headed from city A to city B . How should they act so that time, for which the last of them will get to , was the smallest? Determine this time. Pedestrian speed - 5 km/h, motorcycle speed - 45 km/h, distance from A to B is equal to 60 kilometers . [b]6.2 / 7.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]6.3.[/b] A person's age in $1962$ was one more than the sum of digits of the year of his birth. How old is he? [b]6.4. / 7.3[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]6.5.[/b] Prove that a $201 \times 201$ chessboard can be bypassed by moving a chess knight, visiting each square exactly once. [b]6.6.[/b] Can an integer whose last two digits are odd be the square of another integer? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point. [i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]