Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. [i]Ray Li.[/i]

Let $E: R^n \backslash \{0\} \to R^+$ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and $p \in R^n \backslash \{0\}$ , $E (\lambda p) = \lambda^2 E (p)$). Prove that if the second derivative of $E''(p): R^n \times R^n \to R$ is a non-degenerate bilinear form at any point $p \in R^n \backslash \{0\}$, then $E''(p)$ ($p \in R^n \backslash \{0\}$) is positive definite.

Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.) [asy]unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp); [/asy] $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $

Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and $$\left|P(t)-2019\right| <1.$$ [i]Proposed by Navid Safaei[/i]

The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region. [asy] size(300); defaultpen(linewidth(0.8)); pair A = (0,10),B=origin,C=(12,0),D=(24,0),E=(24,10),F=(12,10),G=extension(C,E,D,F); filldraw(A--C--G--F--cycle,gray(0.7)); draw(A--B--D--E--F^^E--G--D); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,S); label("$D$",D,SE); label("$E$",E,NE); label("$F$",F,N); [/asy]

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

Each square of an $n \times m$ board is assigned a pair of coordinates $(x,y)$ with $1 \le x \le m$ and $1 \le y \le n$. Let $p$ and $q$ be positive integers. A pawn can be moved from the square $(x,y)$ to $(x',y')$ if and only if $|x - x'| = p$ and $|y- y'| = q$. There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done?

Show that every natural number $n\leq2^{1\;000\;000}$ can be obtained first with 1 doing less than $1\;100\;000$ sums; more precisely, there is a finite sequence of natural numbers $x_0,\ x_1,\dots,\ x_k\mbox{ with }k\leq1\;100\;000,\ x_0=1,\ x_k=n$ such that for all $i=1,\ 2,\dots,\ k$ there exist $r,\ s$ with $0\leq{r}\leq{s}<i$ such that $x_i=x_r+x_s$.

Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$.Then prove that \[a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}\]

Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\ 3 \mid f(S)}}} f(S)$$ where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.

It is given the expression $y=\frac{x^2-2x+1}{x^2-2x+2}$, where $x$ is a variable. Prove that: (a) if $x_1$ and $x_2$ are two values of $x$, the $y_1$ and $y_2$ are the respective values of $y$ only if $x_1<x_2$, $y_1<y_2$; (b) when $x$ is varying $y$ attains all possible values for which $0\le y<1$.

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $ x_1 = x = a_1 a_2 ... a_n, $ $ x_2 = a_n a_1 ... a_{n-1}, $ $ x_3 = a_{n-1} a_n a _1 ... a_{n-2} $ , $ x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , $ $ ... , x_n = a_2 a_3 ... a_n a_1$ are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$, then the numbers are $x_1 = 37001, x_2 = 13700, $ $x_3 = 01370(= 1370), x_4 = 00137(= 137), $ $ x_5 = 70013.]$ Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.

In quadrilateral $ABCD$, $AB = CD$, $M$ and $K$ are the midpoints of $BC$ and $AD$.Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$ [i](Proposed by M.Volchkevich)[/i]

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$. Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.

Find all triples $ (x,y,z)$ of integers which satisfy $ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$ $ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$ $ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

In a triangle $ABC$, two angle trisectors of $A$ intersect with $BC$ at $D$ and $E$ respectively so that $B,D,E,C$ comes in order. If we have $BD = 3$, $DE = 1$ and $EC = 2$, find $\angle DAE$.

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.

Broady The Boar is playing a boring board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, ... $2020$ in that order clockwise. Broady is rolling $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive. Broady starts at the point labelled $0$. After each dice roll, Broady moves up the same number of points as the number rolled (point $2020$ is followed by point $0$). For example, if they are at $0$ and roll a $5$, they end up at $5$. If they are at $2019$ and roll a $3$, they end up at $1$. Broady continues rolling until they return to the point labelled $0$. What is the expected number of times they roll the dice? [i]2021 CCA Math Bonanza Lightning Round #2.3[/i]

Does there exist a polynomial $p(x)$ with integer coefficients for which $$p(\sqrt{2})=\sqrt{2}$$ $$p(2\sqrt{2})=2\sqrt{2}+2$$

Consider a convex quadrilateral $ABCD$ with $AB=CB$ and $\angle ABC +2 \angle CDA = \pi$ and let $E$ be the midpoint of $AC$. Show that $\angle CDE =\angle BDA$. Paolo Leonetti