Find the number of pairs of integers $(a,b)$ with $0 \le a,b \le 2019$ where $ax \equiv b \pmod{2020}$ has exactly $2$ integer solutions $0 \le x \le 2019$. [i]Proposed by Richard Chen[/i]

Let $ABCD$ be a square with side length $43$ and points $X$ and $Y$ lies on sides $AD$ and $BC$ respectively such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $20 : 23$ . Find the maximum possible length of $XY$.

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.

Prove that there is exactly a function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{\ge 0} $ satisfying the following two properties: $ \text{(i)} x\in\mathbb{R}_{> 0}\implies \left( f(x)+f(f(x)) =4018020x \wedge f(x)>0 \right) $ $ \text{(ii)} 0=f(0)+f(f(0)) $

Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a [i]grade[/i] is a nonnegative decimal number with finitely many digits after the decimal point. Given a grade with digits $a_1a_2 \dots a_m.b_1b_2 \dots b_n$, Joe first rounds the number to the nearest $10^{-n+1}$th place. He then repeats the procedure on the new number, rounding to the nearest $10^{-n+2}$th, then rounding the result to the nearest $10^{-n+3}$th, and so on, until he obtains an integer. For example, he rounds the number $2014.456$ via $2014.456 \to 2014.46 \to 2014.5 \to 2015$. There exists a rational number $M$ such that a grade $x$ gets rounded to at least $90$ if and only if $x \ge M$. If $M = \tfrac pq$ for relatively prime integers $p$ and $q$, compute $p+q$. [i]Proposed by Yang Liu[/i]

Let $I$ be the center of the inscribed circle of the triangle $ABC$. The inscribed circle is tangent to sides $BC$ and $AC$ at points $K_1$ and $K_2$ respectively. Using a ruler and a compass, find the center of excircle for triangle $CK_1K_2$ which is tangent to side $CK_2$, in at most $4$ steps (each step is to draw a circle or a line). (Hryhorii Filippovskyi, Volodymyr Brayman)

Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). [i]Proposed by Raymond Feng[/i]

Farmer James wishes to cover a circle with circumference $10\pi$ with six different types of colored arcs. Each type of arc has radius $5$, has length either $\pi$ or $2\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions: [list][*] Any two adjacent arcs are of different colors. [*] Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. [/list] Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another. [i]Proposed by James Lin.[/i]

Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$. (a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots. (b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.

For every $ x,y\in \Re ^{\plus{}}$, the function $ f: \Re ^{\plus{}} \to \Re$ satisfies the condition $ f\left(x\right)\plus{}f\left(y\right)\equal{}f\left(x\right)f\left(y\right)\plus{}1\minus{}\frac{1}{xy}$. If $ f\left(2\right)<1$, then $ f\left(3\right)$ will be $\textbf{(A)}\ 2/3 \\ \textbf{(B)}\ 4/3 \\ \textbf{(C)}\ 1 \\ \textbf{(D)}\ \text{More information needed} \\ \textbf{(E)}\ \text{There is no } f \text{ satisfying the condition above.}$

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

Let $ABC$ be an acute triangle and let $X$ and $Y$ be points on the segments $AB$ and $AC$ such that $BX = CY$. If $I_{B}$ and $I_{C}$ are centers of inscribed circles in triangles $ABY$ and $ACX$, and $T$ is the second intersection point of the circumcircles of $ABY$ and $ACX$, show that: $$\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.$$ [i]Proposed by Nikola Velov[/i]

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

Is it possible to divide the plane into polygons so that each polygon is transformed into itself under some rotation by $360/7$ degrees about some point? All sides of these polygons must be greater than $1$ cm. (A polygon is the part of a plane bounded by one non-self-intersect-ing closed broken line, not necessarily convex.) (A. Andjans, Riga)

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $ 13$ visible numbers have the greatest possible sum. What is that sum? [asy]unitsize(.8cm); pen p = linewidth(.8pt); draw(shift(-2,0)*unitsquare,p); label("1",(-1.5,0.5)); draw(shift(-1,0)*unitsquare,p); label("2",(-0.5,0.5)); label("32",(0.5,0.5)); draw(shift(1,0)*unitsquare,p); label("16",(1.5,0.5)); draw(shift(0,1)*unitsquare,p); label("4",(0.5,1.5)); draw(shift(0,-1)*unitsquare,p); label("8",(0.5,-0.5));[/asy]$ \textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$

Given that $p$ and $p^4 + 34$ are both prime numbers, compute $p$.

Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$ Find the maximum value of the expression $$x^3+2y$$

Let $p$ and $q$ be real numbers such that the quadratic equation $x^2 + px + q = 0$ has two distinct real solutions $x_1$ and $x_2$. Suppose $|x_1-x_2|=1$, $|p-q|=1$. Prove that $p, q, x_1, x_2$ are all integers.

Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.