Let $\vartriangle ABC$ be acute, with incircle $\Gamma$ and incenter $ I$. $\Gamma$ touches sides $AB$, $BC$ and $AC$ at $Z$, $X$ and $Y$, respectively. Let $D$ be the intersection of $XZ$ with $CI$ and $L$ the intersection of $BI$ with $XY$. Suppose $D$ and $L$ are outside of $\vartriangle ABC$. Prove that $A$, $D$, $Z$, $I$, $Y$, and $ L$ lie on a circle.

Daniel counts the number of ways he can form a positive integer using the digits $1, 2, 2, 3$, and $4$ (each digit at most once). Edward counts the number of ways you can use the letters in the word "$BANANAS$" to form a six-letter word (it doesn't have to be a real English word). Fernando counts the number of ways you can distribute nine identical pieces of candy to three children. By their powers combined, they add each of their values to form the number that represents the meaning of life. What is the meaning of life? (Hint: it's not $42$.)

[b]p1. [/b] In basketball, teams can score $1, 2$, or $3$ points each time. Suppose that Duke basketball have scored $8$ points so far. What is the total number of possible ways (ordered) that they have scored? For example, $(1, 2, 2, 2, 1)$,$(1, 1, 2, 2, 2)$ are two different ways. [b]p2.[/b] All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$. [b]p3.[/b] Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$. [b]p4.[/b] What is the smallest positive integer greater than $9$ such that when its left-most digit is erased, the resulting number is one twenty-ninth of the original number? [b]p5. [/b] Jonathan is operating a projector in the cartesian plane. He sets up $2$ infinitely long mirrors represented by the lines $y = \tan(15^o)x$ and $y = 0$, and he places the projector at $(1, 0)$ pointed perpendicularly to the $x$-axis in the positive $y$ direction. Jonathan furthermore places a screen on one of the mirrors such that light from the projector reflects off the mirrors a total of three times before hitting the screen. Suppose that the coordinates of the screen is $(a, b)$. Find $10a^2 + 5b^2$. [b]p6.[/b] Dr Kraines has a cube of size $5 \times 5 \times 5$, which is made from $5^3$ unit cubes. He then decides to choose $m$ unit cubes that have an outside face such that any two different cubes don’t share a common vertex. What is the maximum value of $m$? [b]p7.[/b] Let $a_n = \tan^{-1}(n)$ for all positive integers $n$. Suppose that $$\sum_{k=4}^{\infty}(-1)^{\lfloor \frac{k}{2} \rfloor +1} \tan(2a_k)$$ is equals to $a/b$ , where $a, b$ are relatively prime. Find $a + b$. [b]p8.[/b] Rishabh needs to settle some debts. He owes $90$ people and he must pay \$ $(101050 + n)$ to the $n$th person where $1 \le n \le 90$. Rishabh can withdraw from his account as many coins of values \$ $2021$ and \$ $x$ for some fixed positive integer $x$ as is necessary to pay these debts. Find the sum of the four least values of $x$ so that there exists a person to whom Rishabh is unable to pay the exact amount owed using coins. [b]p9.[/b] A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$ [b]p10.[/b] In the triangle $ABC$, $AB = 585$, $BC = 520$, $CA = 455$. Define $X, Y$ to be points on the segment $BC$. Let $Z \ne A$ be the intersection of $AY$ with the circumcircle of $ABC$. Suppose that $XZ$ is parallel to $AC$ and the circumcircle of $XYZ$ is tangent to the circumcircle of $ABC$ at $Z$. Find the length of $XY$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$, let $H$ be its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Prove that: [b](a)[/b] $|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma}$ where $\alpha, \beta, \gamma$ are angles of the triangle $ABC;$ [b](b)[/b] $O \equiv H$ if and only if $ABC$ is equilateral.

Let $D$ be the midpoint of the side $[BC]$ of the triangle $ABC$ with $AB \ne AC$ and $E$ the foot of the altitude from $BC$. If $P$ is the intersection point of the perpendicular bisector of the segment line $[DE]$ with the perpendicular from $D$ onto the the angle bisector of $BAC$, prove that $P$ is on the Euler circle of triangle $ABC$.

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

Let $\{x\}$ denote the fractional part of $x$. For example, $\{5.5\}=0.5$. Find the smallest prime $p$ such that the inequality \[\sum_{n=1}^{p^2}\left\{\dfrac{n^p}{p^2}\right\}>2016\] holds.

Two bodies simultaneously emerge from points $A$ and $B$ towards each other. Every time each body finds itself at point $A$ or point $B$, it changes the direction of its movement to the opposite (the absolute values of the velocities of both bodies are constant). A body that initially leaves point $A$ travels the path $AB$ in $101$ seconds. A body that initially left point $B$ travels the same path $AB$ in $201$ seconds. After $2 \cdot 101 \cdot 201$ seconds, each body will be at its starting point. How many times during this time will the first body overtake the second?

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Prove that for each prime $ P =9k+1$ ,exist natural n such that $P|n^3-3n+1$.

The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.

If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property: If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even). Prove this.

One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly $12$ feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around $15$ feet. He then measures $2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet? [asy] size(400); defaultpen(linewidth(0.8)); draw((0,-3)--(0,3)); int d=8; for(int i=0;i<=5;i=i+1) { draw(circle(7/2+d*i,3/2)); } draw((5*d+7,-3)--(5*d+7,3)); draw((0,0)--(2,0),Arrows(size=7)); draw((5,0)--(2+d,0),Arrows(size=7)); draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7)); label("$2$",(1,0),S); label("$12$",((7+d)/2,0),S); label("$15$",((7+3*d)/2,0),S); [/asy]

The number 555,555,555,555 factors into eight distinct prime factors, each with a multiplicity of 1. What are the three largest prime factors of 555,555,555,555?

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$

How many integer values satisfy $|x|<3\pi$? $\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.

On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.

A square-shaped field is divided into $n$ rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of $n$, if the sum of the perimeters of the farms is equal to $100$ times of the perimeter of the field? $ \textbf{(A)}\ 10000 \qquad\textbf{(B)}\ 20000 \qquad\textbf{(C)}\ 50000 \qquad\textbf{(D)}\ 100000 \qquad\textbf{(E)}\ 200000 $

Let \( n \) be a positive integer. Determine the largest possible value of \( k \) with the following property: there exists a bijective function \( \phi: [0, 1] \to [0, 1]^k \) and a constant \( C > 0 \) such that, for all \( x, y \in [0, 1] \), \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: \( \| \cdot \| \) denotes the Euclidean norm, that is, \( \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} \).

Let $abc$ be real numbers satisfying $ab+bc+ca=1$. Show that $\frac{|a-b|}{|1+c^2|}$ + $\frac{|b-c|}{|1+a^2|}$ $>=$ $\frac{|c-a|}{|1+b^2|}$

What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order?