We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove thatthe points $P, Q, R, S$ are concyclic.

Let $\vartriangle A_0B_0C_0$ be an equilateral triangle with area $1$, and let $A_1$, $B_1$, $C_1$ be the midpoints of $\overline{A_0B_0}$, $\overline{B_0C_0}$, and $\overline{C_0A_0}$, respectively. Furthermore, set $A_2$, $B_2$, $C_2$ as the midpoints of segments $\overline{A_0A_1}$, $\overline{B_0B_1}$, and $\overline{C_0C_1}$ respectively. For $n \ge 1$, $A_{2n+1}$ is recursively defined as the midpoint of $A_{2n}A_{2n-1}$, and $A_{2n+2}$ is recursively defined as the midpoint of $\overline{A_{2n+1}A_{2n-1}}$. Recursively define $B_n$ and $C_n$ the same way. Compute the value of $\lim_{n \to \infty }[A_nB_nC_n]$, where $[A_nB_nC_n]$ denotes the area of triangle $\vartriangle A_nB_nC_n$.

Estimate the value of $$\sum^{2023}_{n=1} \left(1+ \frac{1}{n} \right)^n$$ to $3$ decimal places.

Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993? [asy] int x=101, y=3*floor(x/4); draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x)); int i; for(i=y-2; i<y+4; i=i+1) { dot(dir(360*i/x)); } label("3", dir(360*(y-2)/x), dir(360*(y-2)/x)); label("2", dir(360*(y+1)/x), dir(360*(y+1)/x)); label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]

Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$. Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.

A [i]power[/i] is a positive integer of the form $a^k$, where $a$ and $k$ are positive integers with $k\geq 2$. Let $S$ be the set of positive integers which cannot be expressed as sum of two powers (for example, $4,\ 7,\ 15$ and $27$ are elements of $S$). Determine whether the set $S$ has a finite or infinite number of elements.

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip? [asy] unitsize(12); usepackage("mathptmx"); defaultpen(fontsize(8)+linewidth(.7)); int mod12(int i) {if (i<13) {return i;} else {return i-12;}} void drawgraph(pair sh,string lab) { for (int i=0;i<11;++i) { for (int j=0;j<6;++j) { draw(shift(sh+(i,j))*unitsquare,mediumgray); } } draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8)); draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8)); for (int i=1;i<10;++i) { draw(shift(sh)*((i,-.2)--(i,.2))); } label("8\tiny{\textsc{am}}",sh+(1,-.2),S); for (int i=2;i<9;++i) { label(string(mod12(i+7)),sh+(i,-.2),S); } label("4\tiny{\textsc{pm}}",sh+(9,-.2),S); for (int i=1;i<6;++i) { label(string(30*i),sh+(0,i),2*W); } draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10)); label("$\textbf{("+lab+")}$",sh+(-2.1,6.8),fontsize(10)); } drawgraph((0,0),"A"); drawgraph((15,0),"B"); drawgraph((0,-10),"C"); drawgraph((15,-10),"D"); drawgraph((0,-20),"E"); dotfactor=6; draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9)); dot((1,0)^^(3,3)^^(6,3)^^(8,0)); pair sh = (15,0); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0)); pair sh = (0,-10); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0)); pair sh = (15,-10); draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0)); pair sh = (0,-20); draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0)); [/asy]

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.

Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

Show that for every positive integer $n$ the number $7^n-1$ is not divisible by $6^n-1$.

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and incircle $\omega$. Two points $X$ and $Y$ are chosen on minor arcs $AB$ and $AC$, respectively, of the circumcircle of triangle $\triangle ABC$ such that $XY$ is tangent to $\omega$ at $P$ and $\overline{XY}\perp \overline{AI}$. Let $\omega$ be tangent to sides $AC$ and $AB$ at $E$ and $F$, respectively. Denote the intersection of lines $XF$ and $YE$ as $T$. Prove that if the circumcircles of triangles $\triangle TEF$ and $\triangle ABC$ are tangent at some point $Q$, then lines $PQ$, $XE$, and $YF$ are concurrent. [i]Proposed by Andrew Wen[/i]

Suppose we have $3$ baskets and $4$ distinguishable balls. Each ball is placed into a randomly selected basket. Compute the probability that the basket with the most balls has at least $3$ balls.

Numbers $1, 2, 3, . . . , 100$ are arranged in a circle in some order. A [i]good pair[/i] is a pair of numbers of the same parity, between which there are exactly $3$ numbers. What is the smallest possible number good pairs?

There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments. Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point. [i]An interior point of a line segment is a point of the line segment that is not one of the two endpoints.[/i]

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct, [b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal. [b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational. [i]Marin Tolosi[/i]

Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $

The infinite sequence $a_0,a_1,\ldots$ is given by $a_1=\frac12$, $a_{n+1} = \sqrt{\frac{1+a_n}{2}}$. Determine the infinite product $a_1a_2a_3\cdots$.

The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? $\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$

An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice: $$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$ Determine all poli-pythagorean integers.