Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy \[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\] and \[xyz=\sqrt{3} + i\sqrt{5}.\] If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then \[(x_1x_2+y_1y_2+z_1z_2)^2\] can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b.$

The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants: ${{ \textbf{(A)}\ \text{I and II} \qquad\textbf{(B)}\ \text{II and III} \qquad\textbf{(C)}\ \text{I and III} \qquad\textbf{(D)}\ \text{III and IV} }\qquad\textbf{(E)}\ \text{I and IV} } $

Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.

Find all functions $f : R \to R$ satisfying the following conditions (a) $f(1) = 1$, (b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$ (c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$ Trần Nam Dũng

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.

For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$?

The flag of the United Nations consists of a polar map of the world, with the North Pole as its center, extending to approximately $45^\circ$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^\circ$ S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?

Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. \[ 2^m p^2 + 1 = q^5 \]

Estimate the product of all the nonzero digits in the decimal expansion of $2020!$. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\max\Big(0, \Big\lfloor 15-0.02\cdot\Big\lvert \log_{10}\Big(\frac{A}{E}\Big)\Big\rvert \Big\rfloor\Big).\] [i]Proposed by Alex Li[/i]

Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as: $\bullet$ $f(1) = 1$. $\bullet$ $f(n) = f(n/2)$ if $n$ is even. $\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$. $\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$. a) Compute $f(2^{2020}-1)$. b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$.

There are six empty slots corresponding to the digits of a six-digit number. Claire and William take turns rolling a standard six-sided die, with Claire going first. They alternate with each roll until they have each rolled three times. After a player rolls, they place the number from their die roll into a remaining empty slot of their choice. Claire wins if the resulting six-digit number is divisible by $6$, and William wins otherwise. If both players play optimally, compute the probability that Claire wins.

For a real number $ k\ge 2, $ solve the equation $ \frac{\{x\}[x]}{x} =k. $

The closure (interior and boundary) of a convex quadrangle is covered by four closed discs centered at each vertex of the quadrangle each. Show that three of these discs cover the closure of the triangle determined by their centers.

$\Delta ABC$ is a right-angled triangle, $\angle C = 90^{\circ}$. Draw a circle centered at $B$ with radius $BC$. Let $D$ be a point on the side $AC$, and $DE$ is tangent to the circle at $E$. The line through $C$ perpendicular to $AB$ meets line $BE$ at $F$. Line $AF$ meets $DE$ at point $G$. The line through $A$ parallel to $BG$ meets $DE$ at $H$. Prove that $GE = GH$.

Triangle $ABC$ is isoceles with $AB = AC$. Point $D$ lies on $AB$ such that the inradius of $ADC$ and the inradius of $BDC$ both equal $\frac{3-\sqrt3}{2}$ . The inradius of $ABC$ equals $1$. What is the length of $BD$?

You come across an ancient mathematical manuscript. It reads, "To find out whether a number is divisible by seventeen, take the number formed by the last two digits of the number, subtract the number formed by the third- and fourth-to-last digits of the number, add the number formed by the fifth- and sixth-to-last digits of the number and so on. The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen." What is the sum of the five smallest bases the ancient culture might have been using? (Note that "seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)

Determine all the pairs of positive integers $(a,b),$ such that $$14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2,$$ where $\varphi(n)$ is Euler's totient function.

(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal. (b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.

Let $z_1, z_2, ..., z_{2020}$ be the roots of the polynomial $z^{2020} + z^{2019} +...+ z + 1$. Compute $$\sum^{2020}_{i=1} \frac{1}{1 -z^{2020}_i}.$$

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\triangle ABC.$

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.

For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$ respectively. a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$; b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is similar with the triangle $A_2B_2C_2$.