Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate; in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to $p$? $ \textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$

Let $A,B,C,D$ be points in the space such that $|AB|=|AC|=3$, $|DB|=|DC|=5$, $|AD|=6$, and $|BC|=2$. Let $P$ be the nearest point of $BC$ to the point $D$, and $Q$ be the nearest point of the plane $ABC$ to the point $D$. What is $|PQ|$? $ \textbf{(A)}\ \frac{1}{\sqrt 2} \qquad\textbf{(B)}\ \frac{3\sqrt 7}{2} \qquad\textbf{(C)}\ \frac{57}{2\sqrt{11}} \qquad\textbf{(D)}\ \frac{9}{2\sqrt 2} \qquad\textbf{(E)}\ 2\sqrt 2 $

Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.

$G$ be a group and $H\le G$. Then which of the following are true? $\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$ $\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$ $\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$ $\textbf{(D)}~\text{None of the above}$

[asy] size(2.5inch); pair A = (-2,0), B = 2dir(150), D = (2,0), C; draw(A..(0,2)..D--cycle); C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B))); draw(A--B--C--D--cycle); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,N); label("$D$",D,E); label("$4$",A--D,S); label("$1$",A--B,E); label("$1$",B--C,SE); //Credit to chezbgone2 for the diagram[/asy] Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length $\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$

Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.

Let $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$ be two partitions of a set $M$ such that $|A_j\cup B_k|\ge n$ for any $j,k\in\{1,2,\ldots,n\}$. Prove that $|M|\ge n^2/2$.

On the table there are written numbers $673, 674, \cdots, 2018, 2019.$ Nibab chooses arbitrarily three numbers $a,b$ and $c$, erases them and writes the number $\frac{\min(a,b,c)}{3}$, then he continues in an analogous way. After Nibab performed this operation $673$ times, on the table remained a single number $k$. Prove that $k\in(0,1).$

Rachel writes down a simple inequality: one $2$-digit number is greater than another. Matt is sitting across from Rachel and peeking at her paper. If Matt, reading upside down, sees a valid inequality between two $2$-digit numbers, compute the number of different inequalities that Rachel could have written. Assume that each digit is either a $1, 6, 8$, or $9$.

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and \[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\] Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value. [i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i]

Prove that if $F(x)$ and $G(x)$ are polynomials with coefficients $0$ and $1$ such that $$F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}$$ holds for some $n > 1$, then one of them can be represented in the form $$ (1 +x + x^2 +...+ x^{k-1}) T(x)$$ for some $k > 1$ where $T(x)$ is a polynomial with coefficients $0$ and $1$. (V Senderov, M Vialiy)

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions which satisfy \[\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),\]for all $a\in\mathbb{R}.$ Given that $f$ has Darboux's Property (intermediate value property), show that functions $f$ and $g$ are continuous and equal to each other. [i]Mathematical Gazette [/i]

a) Prove that you can rearrange the numbers $1, 2, ... , 32$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean. b) Can you rearrange the numbers $1, 2, ... , 100$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?

In a board of $2000\times2001$ squares with integer coordinates $(x,y)$, $0\leq{x}\leq1999$ and $0\leq{y}\leq2000$. A ship in the table moves in the following way: before a move, the ship is in position $(x,y)$ and has a velocity of $(h,v)$ where $x,y,h,v$ are integers. The ship chooses new velocity $(h^\prime,v^\prime)$ such that $h^\prime-h,v^\prime-v\in\{-1,0,1\}$. The new position of the ship will be $(x^\prime,y^\prime)$ where $x^\prime$ is the remainder of the division of $x+h^\prime$ by $2000$ and $y^\prime$ is the remainder of the division of $y+v^\prime$ by $2001$. There are two ships on the board: The Martian ship and the Human trying to capture it. Initially each ship is in a different square and has velocity $(0,0)$. The Human is the first to move; thereafter they continue moving alternatively. Is there a strategy for the Human to capture the Martian, independent of the initial positions and the Martian’s moves? [i]Note[/i]: The Human catches the Martian ship by reaching the same position as the Martian ship after the same move.

Done working on his sand castle design, Joshua sits down and starts rolling a $12$-sided die he found when cleaning the storage shed. He rolls and rolls and rolls, and after $17$ rolls he finally rolls a $1$. Just $3$ rolls later he rolls the first $2\textit{ after}$ that first roll of $1$. $11$ rolls later, Joshua rolls the first $3\textit{ after}$ the first $2$ that he rolled $\textit{after}$ the first $1$ that he rolled. His first $31$ rolls make the sequence \[4,3,11,3,11,8,5,2,12,9,5,7,11,3,6,10,\textbf{1},8,3,\textbf{2},10,4,2,8,1,9,7,12,11,4,\textbf{3}.\] Joshua wonders how many times he should expect to roll the $12$-sided die so that he can remove all but $12$ of the numbers from the entire sequence of rolls and (without changing the order of the sequence), be left with the sequence \[1,2,3,4,5,6,7,8,9,10,11,12.\] What is the expected value of the number of times Joshua must roll the die before he has such a sequence? (Assume Joshua starts from the beginning - do $\textit{not}$ assume he starts by rolling the specific sequence of $31$ rolls above.)