Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone? [i]Author: Ray Li[/i]

A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?

Given regular pentagon $ABCDE$, a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A$. The number of degrees in minor arc $AD$ is $\textbf{(A)}\ 72 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 144$ [asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]

Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.

Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

Let $n\geqslant 0$ be an integer, and let $a_0,a_1,\dots,a_n$ be real numbers. Show that there exists $k\in\{0,1,\dots,n\}$ such that $$a_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k$$ for all real numbers $x\in[0,1]$.

If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties: [list] [*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and [*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$. [/list] [i]Viktor Simjanoski, Macedonia[/i]

Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then \[\angle DBC+\angle DPC = \angle DQC.\]

Let $ABC$ be an acute triangle and $D$ be the foot of the altiutude from $A$ on $BC$, $E$ and $F$ are the midpoints of $BD$ and $DC$ respectively. $O$ and $Q$ are the circumcenters of the triangles $\vartriangle BF$ and $\vartriangle ACE$ respectively. $P$ is the intersection point of $OE$ and $QF$, show that $PB = PC$.

Let $ABC$ be an acute triangle, where $AB$ is the smallest side and let $D$ be the midpoint of $AB$. Let $P$ be a point in the interior of the triangle $ABC$ such that $\angle CAP = \angle CBP = \angle ACB$. From the point $P$, we draw perpendicular lines on $BC$ and $AC$ where the intersection point with $BC$ is $M$, and with $AC$ is $N$ . Through the point $M$ we draw a line parallel to $AC$, and through $N$ parallel to $BC$. These lines intercept at the point $K$. Prove that $D$ is the center of the circumscribed circle for the triangle $MNK$.

Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube? Note: Two colorings are considered the same if one can be obtained from the other by rotation.

Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the step several times until she ends up with all $1$'s on the whiteboard. Then the game is over, and the final score is calculated. Let $M, m$ be the maximum and minimum final score that can be possibly obtained respectively. Find $M-m$.

Consider a sequence of positive integers $x_n$ such that: \[(\text{A})\ x_{2n+1}=4x_n+2n+2 \] \[(\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n \] for all $n\ge 0$. Prove that \[(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n \] for all $n\ge 0$.

For a nonisosceles triangle $ABC$, consider the altitude from vertex $A$ and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex $A$.

Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that \[45 < x_{1000} < 45. 1.\]

Let $p$ be a polynomial with degree less than $4$ such that $p(x)$ attains a maximum at $x = 1$. If $p(1) = p(2) = 5$, find $p(10)$.

Let $\alpha=0.d_{1}d_{2}d_{3} \cdots$ be a decimal representation of a real number between $0$ and $1$. Let $r$ be a real number with $\vert r \vert<1$. [list=a][*] If $\alpha$ and $r$ are rational, must $\sum_{i=1}^{\infty} d_{i}r^{i}$ be rational? [*] If $\sum_{i=1}^{\infty} d_{i}r^{i}$ and $r$ are rational, $\alpha$ must be rational? [/list]

Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$.

Some polygon can be divided into two equal parts by three different ways. Is it certainly valid that this polygon has an axis or a center of symmetry?

The equation $ x^3\minus{}ax^2\plus{}bx\minus{}c\equal{}0$ has three (not necessarily different) positive real roots. Find the minimal possible value of $ \frac{1\plus{}a\plus{}b\plus{}c}{3\plus{}2a\plus{}b}\minus{}\frac{c}{b}$.