Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point. Evangelos Psychas (Greece)

Let $C_1,C_2,..., C_n$ , $(n\ge 3)$ be circles having a common point $M$. Three straight lines passing through $M$ intersect again the circles in $A_1, A_2,..., A_n$ ; $B_1,B_2,..., B_n$ and $X_1,X_2,..., X_n$ respectively. Prove that if $$A_1A_2 =A_2A_3 =...=A_{n-1}A_n$$ and $$B_1B_2 =B_2B_3 =...=B_{n-1}B_n$$ then $$X_1X_2 =X_2X_3 =...=X_{n-1}X_n.$$

Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.

Is there a positive integer $N>10^{20}$ such that all its decimal digits are odd, the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal, and it is divisible by each 20-digit number obtained from it by deleting digits? (Neither deleted nor remaining digits must be consecutive.)

A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.

Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to $2018$ times the probability that she gets no questions correct. Compute the number of questions that are on the exam. [i]Proposed by Yannick Yao[/i]

Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

In a triangle $ABC$, points $D$ and $E$ are taken on rays $CB$ and $CA$ respectively so that $CD=CE = \frac{AC+BC}{2}$. Let $H$ be the orthocenter of the triangle, and $P$ be the midpoint of the arc $AB$ of the circumcircle of $ABC$ not containing $C$. Prove that the line $DE$ bisects the segment $HP$.

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.

In a set of $21$ real numbers, the sum of any $10$ numbers is less than the sum of the remaining $11$ numbers. Prove that all the numbers are positive.

Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$, $B = (n,0)$, $C = (n,m)$, $D = (n-1,m)$, $E = (n-1,1)$, $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.

A calculator has two operations $A$ and $B$ and initially shows the number $1$. Operation $A$ turns $x$ into $x+1$ and operation B turns $x$ into $\dfrac{x}{x+1}$. a) Show all the ways we can get the number $\dfrac{20}{23}$. b) For every rational $r \neq 1$, determine if it is possible to get $r$ using only operations $A$ and $B$.

A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.

Given that the expected amount of $1$s in a randomly selected $2021$-digit number is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Hannah Shen[/i]

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

Points $ A_1,A_2,...,A_n$ and $ B_1,B_2,...,B_n$ are given on a plane. Show that the points $ B_i$ can be renumbered in such a way that the angle between vectors $ A_iA_j^{\longrightarrow}$ and $ B_iB_j^{\longrightarrow}$ is acute or right whenever $ i\neq j$.

A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.

For $n > 1$, let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$. Find the maximum value of $\frac{a_n}{n}$.

Given a regular $n$-sided polygon with $n \ge 3$. Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way. Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.

Suppose that $x$ and $y$ are positive reals such that \[x-y^2=3, \qquad x^2+y^4=13.\] Find $x$.

How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?

For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?