Points $P, Q, R$ were marked on the sides $BC, CA, AB$, respectively. Let $a$ be tangent at point $A$ to the circumcircle of triangle $AQR$, $b$ be tangent at point $B$ to the circumcircle of the triangle BPR, $c$ be tangent at point $C$ to the circumscribed circle triangle $CPQ$. Let $X$ be the point of intersection of the lines $b$ and $c, Y$ be the point the intersection of lines $c$ and $a, Z$ is the point of intersection of lines $a$ and $b$. Prove that the lines $AX, BY, CZ$ intersect at one point if and only if the lines $AP, BQ, CR$ intersect at one point.

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that $$|\bigcap_{H'\in S} H'|>1$$

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

If $x$ and $y$ are positive real numbers such that $(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018$: The minimum possible value of $x + y$ is A. $\frac{2017}{\sqrt{2018}}$ B. $\frac{2018}{\sqrt{2019}}$ C. $\frac{2017}{2\sqrt{2018}}$ D. $\frac{2019}{\sqrt{2018}}$ E. $\sqrt{3}$

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]

Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties: [list] [*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$. [*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$. [/list]

Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.

Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$

Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Find the positive integer $n{}$ if $$\left(1-\frac{1}{1+2}\right)\cdot\left(1-\frac{1}{1+2+3}\right)\cdot\ldots\cdot\left(1-\frac{1}{1+2+\ldots+n}\right)=\frac{2021}{6057}.$$

How many neutrons are in $0.025$ mol of the isotope ${ }_{24}^{54}\text{Cr}$? $ \textbf{(A)}\hspace{.05in}1.5\times10^{22} \qquad\textbf{(B)}\hspace{.05in}3.6\times10^{23} \qquad\textbf{(C)}\hspace{.05in}4.5\times10^{23} \qquad\textbf{(D)}\hspace{.05in}8.1\times10^{23} \qquad $

Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.

An experiment consists of performing $n$ independent tests. The $i$-th test is successful with the probability equal to $p_i$. Let $r_k$ be the probability that exactly $k$ tests succeed. Prove that $$\sum_{i=1}^n p_i =\sum_{k=0}^n kr_k.$$

Vasya has $100$ cards of $3$ colors, and there are not more than $50$ cards of same color. Prove that he can create $10\times 10$ square, such that every cards of same color have not common side.

Which of the following numbers has the smallest prime factor? $\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 59\qquad \textbf{(E)}\ 61$

Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.

The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.) [img]https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png[/img]

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

Circle $\omega_1$ has diameter $AB$, and circle $\omega_2$ has center $A$ and intersects $\omega_1$ at points $C$ and $D$. Let $E$ be the intersection of $AB$ and $CD$. Point $P$ is chosen on $\omega_2$ such that $P C = 8$, $P D = 14$, and $P E = 7$. Find the length of $P B$.

Two $1$ inch by $1$ inch squares are cutout from opposite corners of a $7$ inch by $5$ inch piece of paper to form an octagon. What is the distance, in inches, between the two dotted points, both of which lie on corners of the octagon? [asy] size(120); draw((0,1)--(0,5)--(6,5)); draw((1,0)--(7,0)--(7,4)); draw((0,1)--(0,0)--(1,0),dashed); draw((6,5)--(7,5)--(7,4),dashed); draw((0,1)--(1,1)--(1,0)); draw((6,5)--(6,4)--(7,4)); draw((1,1)--(6,4),dashed); dot((1,1),linewidth(5)); dot((6,4),linewidth(5)); label("$?$",(1,1)--(6,4),N); [/asy] $\textbf{(A) }5\qquad\textbf{(B) }\sqrt{34}\qquad\textbf{(C) }5\sqrt2\qquad\textbf{(D) }8\qquad\textbf{(E) }\sqrt{74}$