Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.

Solve with full solution: \[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]

A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.

$\sqrt{8}+\sqrt{18}=$ $\textbf{(A)}\ \sqrt{20} \qquad \textbf{(B)}\ 2(\sqrt{2}+\sqrt{3}) \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 5\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{13}$

Compute all real values of $b$ such that, for $f(x) = x^2+bx-17, f(4)=f'(4)$.

In triangle $ABC$ $(\angle A\neq 90^\circ)$, let $O$, $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$, $N$ are the midpoints of $BC$, $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$. Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$. Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$. [i]Proposed by Mehran Talaei[/i]

Let $[ABCD]$ be a convex quadrilateral with $AB = 2, BC = 3, CD = 7$ and $\angle B = 90^o$, for which there is a inscribed circle. Determine the radius of this circle. [img]https://1.bp.blogspot.com/-sDKOdmceJlY/X4KaJxi8AoI/AAAAAAAAMk8/7UkTzaWqQSkdqb0N_-r0CZZjD-OGZknSACLcBGAsYHQ/s260/2017%2Bportugal%2Bp5.png[/img]

Suppose we have sequences $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ and the function $f(x)=\tfrac{1}{x}$ such that for all $n$ we have: [list] [*]$a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n))$ [*]$a_{n+2} = f(1-a_n) - f(1+a_n)$ [*]$b_{n+2} = f(1-b_n) - f(1+b_n)$ [/list] Given that $a_0=\tfrac{1}{6}$ and $b_0=\tfrac{1}{7},$ then $b_5=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the sum of the prime factors of $mn.$

A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \] If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.

There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?

Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $ 29$, $ 23$, $ 21$, and $ 17$ are obtained. One of the original integers is: $ \textbf{(A)}\ 19 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 29 \qquad \textbf{(E)}\ 17$

Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$

Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

Let $A = \left\{1,2,3,\cdots 72 \right\}$.Prove that you can choose $36$ element from $A$ such that the sum of those $36$ elements is equal with the sum of other $36$

[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$? [b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win. [b]p3.[/b] What is the sum of the measures of the exterior angles of a regular $2023$-gon in degrees? [b]p4.[/b] If $N$ is a base $10$ digit of $90N3$, what value of $N$ makes this number divisible by $477$? [b]p5.[/b] What is the rightmost non-zero digit of the decimal expansion of $\frac{1}{2^{2023}}$ ? [b]p6.[/b] if graphs of $y = \frac54 x + m$ and $y = \frac32 x + n$ intersect at $(16, 27)$, what is the value of $m + n$? [b]p7.[/b] Bryan is hitting the alphabet keys on his keyboard at random. If the probability he spells out ABMC at least once after hitting $6$ keys is $\frac{a}{b^c}$ , for positive integers $a$, $b$, $c$ where $b$, $c$ are both as small as possible, find $a+b+c$. Note that the letters ABMC must be adjacent for it to count: AEBMCC should not be considered as correctly spelling out ABMC. [b]p8.[/b] It takes a Daniel twenty minutes to change a light bulb. It takes a Raymond thirty minutes to change a light bulb. It takes a Bryan forty-five minutes to change a light bulb. In the time that it takes two Daniels, three Raymonds, and one and a half Bryans to change $42$ light bulbs, how many light bulbs could half a Raymond change? Assume half a person can work half as productively as a whole person. [b]p9.[/b] Find the value of $5a + 4b + 3c + 2d + e$ given $a, b, c, d, e$ are real numbers satisfying the following equations: $$a^2 = 2e + 23$$ $$b^2 = 10a - 34$$ $$c^2 = 8b - 23$$ $$d^2 = 6c - 14$$ $$e^2 = 4d - 7.$$ [b]p10.[/b] How many integers between $1$ and $1000$ contain exactly two $1$’s when written in base $2$? [b]p11.[/b] Joe has lost his $2$ sets of keys. However, he knows that he placed his keys in one of his $12$ mailboxes, each labeled with a different positive integer from $1$ to $12$. Joe plans on opening the $2$ mailbox labeled $1$ to see if any of his keys are there. However, a strong gust of wind blows by, opening mailboxes $11$ and $12$, revealing that they are empty. If Joe decides to open one of the mailboxes labeled $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ , or $10$, the probability that he finds at least one of his sets of keys can be expressed as $\frac{a}{b}$, where a and b are relatively prime positive integers. Find the sum $a + b$. Note that a single mailbox can contain $0$, $1$, or $2$ sets of keys, and the mailboxes his sets of keys were placed in are determined independently at random. [b]p12.[/b] As we all know, the top scientists have recently proved that the Earth is a flat disc. Bob is standing on Earth. If he takes the shortest path to the edge, he will fall off after walking $1$ meter. If he instead turns $90$ degrees away from the shortest path and walks towards the edge, he will fall off after $3$ meters. Compute the radius of the Earth. [b]p13.[/b] There are $999$ numbers that are repeating decimals of the form $0.abcabcabc...$ . The sum of all of the numbers of this form that do not have a $1$ or $2$ in their decimal representation can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Find $a + b$. [b]p14.[/b] An ant is crawling along the edges of a sugar cube. Every second, it travels along an edge to another adjacent vertex randomly, interested in the sugar it notices. Unfortunately, the cube is about to be added to some scalding coffee! In $10$ seconds, it must return to its initial vertex, so it can get off and escape. If the probability the ant will avoid a tragic doom can be expressed as $\frac{a}{3^{10}}$ , where $a$ is a positive integer, find $a$. Clarification: The ant needs to be on its initial vertex in exactly $10$ seconds, no more or less. [b]p15.[/b] Raymond’s new My Little Pony: Friendship is Magic Collector’s book arrived in the mail! The book’s pages measure $4\sqrt3$ inches by $12$ inches, and are bound on the longer side. If Raymond keeps one corner in the same plane as the book, what is the total area one of the corners can travel without ripping the page? If the desired area in square inches is $a\pi+b\sqrt{c}$ where $a$, $b$, and $c$ are integers and $c$ is squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product $\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)$

Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that \[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\] What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$

Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.

Determine all natural numbers $n$ so that numbers $1, 2,... , n$ can be placed on the circumference of a circle and for each natural number $s$ with $1\le s \le \frac12n(n+1)$ , there is a circular arc which has the sum of all numbers in that arc to be $s$.