Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.

Which of the following are true? $\textbf{(A)}~\text{For every }n\in\mathbb N,n^3-n\text{ is divisible by }6$ $\textbf{(B)}~\text{For every }n\in\mathbb N,n^7-n\text{ is divisible by }42$ $\textbf{(C)}~\text{Every perfect square is of the form }3m\text{ or }3m+1\text{ for some }n\in\mathbb N$ $\textbf{(D)}~\text{None of the above}$

[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] What is the perimeter of a rectangle if its area is $24$ and one side length is $6$? [b]E2.[/b] John moves 3 miles south, then $2$ miles west, then $7$ miles north, and then $5$ miles east. What is the length of the shortest path, in miles, from John's current position to his original position? [b]E3.[/b] An equilateral triangle $ABC$ is drawn with side length $2$. The midpoints of sides $AB$, $BC$, and $CA$ are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle? [b]E4.[/b] Let triangle $ABC$ have sides $AB = 74$ and $AC = 5$. What is the sum of all possible integral side lengths of BC? [b]E5.[/b] What is the area of quadrilateral $ABCD$ on the coordinate plane with $A(1, 0)$, $B(0, 1)$, $C(1, 3)$, and $D(5, 2)$? [b]E6 / L1.[/b] Let $ABCD$ be a square with side length $30$. A circle centered at the center of $ABCD$ with diameter $34$ is drawn. Let $E$ and $F$ be the points at which the circle intersects side $AB$. What is $EF$? [b]E7 / L2.[/b] What is the area of the quadrilateral bounded by $|2x| + |3y| = 6$? [b]E8.[/b] A circle $O$ with radius $2$ has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length $2$ are erected outwards. Find the area of the union of these triangles and circle $O$. [b]L3.[/b] Right triangle $ABC$ has hypotenuse $AB$. Altitude $CD$ divides $AB$ into segments $AD$ and $DB$, with $AD = 20$ and $DB = 16$. What is the area of triangle $ABC$? [b]L4.[/b] Circle $O$ has chord $AB$. Extend $AB$ past $B$ to a point $C$. A ray from $C$ is drawn, and this ray intersects circle $O$. Let point $D$ be the point of intersection of the ray and the circle that is closest to point $C$. Given $AB = 20$, $BC = 16$, and $OA = \frac{201}{6}$ , find the longest possible length of $CD$. [b]L5.[/b] Consider a circular cone with vertex $A$. The cone's height is $4$ and the radius of its base is $3$. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere. [b]L6.[/b] A disk of radius $\frac12$ is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)? [b]L7.[/b] Let $ABC$ be an equilateral triangle of side length $2$. Let $D$ be the midpoint of $BC$, and let $P$ be a variable point on $AC$. By moving $P$ along $AC$, what is the minimum perimeter of triangle $BDP$? [b]L8.[/b] Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 9$. Let $DEFG$ be a rhombus, where $G$ is on line $BC$ and $A$ is on line $EF$. If $m\angle EFG = 30^o, what is $DE$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$

Let $ m$ and $ n$ be positive integers. Prove that $ 6m | (2m \plus{} 3)^n \plus{} 1$ if and only if $ 4m | 3^n \plus{} 1$

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

The ancient One-Dimensional Empire was located along a straight line. Initially, there were no cities. A total of $n$ different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities. [list=a] [*] For $n=10^6$, give a map from which this deduction can be made. [*] Prove that for $n=10^{13}$, this conclusion cannot be drawn from any map. [/list]

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that: 1: if 2 squares are adjacent then one of them is marked. 2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked. Find the minimun number of squares we most mark.

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$, where $p$ is the largest prime divisor of $n$. Prove that either after 999 seconds or 1000 second the number on the board will be a power of two. [i]A. Voidelevich[/i]

Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?

Let $ n > 1$ be a fixed integer. Define functions $ f_0(x) \equal{} 0,$ $ f_1(x) \equal{} 1 \minus{} \cos(x),$ and for $ k > 0,$ \[ f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x).\] If $ F(x) \equal{} \sum^n_{r\equal{}1} f_r(x),$ prove that [b](a)[/b] $ 0 < F(x) < 1$ for $ 0 < x < \frac{\pi}{n\plus{}1},$ and [b](b)[/b] $ F(x) > 1$ for $ \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.$

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.