Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. $M$ is the midpoint of $BC$. Prove that $M,F,C,E$ are concyclic.

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

$$Problem 2:$$ Points $D$ and $E$ are taken on side $CB$ of triangle $ABC$, with $D$ between $C$ and $E$, such that $\angle BAE =\angle CAD$. If $AC < AB$, prove that $AC.AE < AB.AD$.

The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.

Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called [i]singular[/i] if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there?

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. [img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]

There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$. Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$. Prove that: (a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$. (b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$.

Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that \[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\] for $k \geq 1$, where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?

In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\tfrac{1}{3}$ of all the ninth graders are paired with $\tfrac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy? $ \textbf{(A) } \frac{2}{15} \qquad \textbf{(B) } \frac{4}{11} \qquad \textbf{(C) } \frac{11}{30} \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{11}{15} $

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles. Prove that it is regular.

Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

Let $n$ be an even positive integer. We consider rectangles with integer side lengths $k$ and $k +1$, where $k$ is greater than $\frac{n}{2}$ and at most equal to $n$. Show that for all even positive integers $ n$ the sum of the areas of these rectangles equals $$\frac{n(n + 2)(7n + 4)}{24}.$$

Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers \[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\] to be products of two primes (not necessarily distinct).

What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$

Prove that the sides of any equilateral triangle you can either increase everything or decrease everything by the same amount so that you get a right triangle.

For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie? $\textbf{(A)}\ [1,200] \qquad \textbf{(B)}\ [201,400] \qquad \textbf{(C)}\ [401,600] \qquad \textbf{(D)}\ [601,800] \qquad \textbf{(E)}\ [801,999] $

Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfrac{1}{3}$ of the apples in the second box. Suppose that Eric has $28$ apples in total. How many red apples does Eric have?

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

Let $x_1,x_2,\ldots,x_{2001}$ be positive numbers such that $$x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.$$Prove that $\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999$.

Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to below conversation, $B: $ I'm a truth teller. $C: $ $D$ is a truth teller. $D: $ $B$ and $E$ are not both truth tellers. $E: $ $A$ and $B$ are truth tellers. How many truth tellers are there? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{More information is needed} $

4. A positive integer n is given.Find the smallest $k$ such that we can fill a $3*k$ gird with non-negative integers such that: $\newline$ $i$) Sum of the numbers in each column is $n$. $ii$) Each of the numbers $0,1,\dots,n$ appears at least once in each row.

Carnegie Corporation is trying to promote a new department director from its employees. Carnegie Corporation has $n$ employees each with some unknown real-valued score and wants to pick the $M$-th highest scoring employee to be the new department director. The corporation has a magical machine that, once a day, can be used to compare two employees to see which one has a higher score. Unfortunately, this machine has a magical consequence: after every $k$ uses of this machine, if a new department director has not been chosen by the end of the day, one random employee is fired and a new employee (who has not necessarily the same score) is hired. Assume no two employees have equal scores and scores don't change over time. Machines with a higher constant of $k$ will be more expensive, so management wants to minimize the value of $k$. Devise an algorithm which uses the minimum possible $k$ guaranteeing that, given any $1\leq M \leq n$, we can promote the $M$-th highest scoring employee as the new department director in finitely many days. [b]Scoring:[/b] An algorithm that solves the case for a certain $k$ in terms of $n\geq 2$ and some constant $c\in \mathbb{R}^+$ will be awarded: [list] [*] $10$ pts for any finite $k$ [*] $20$ pts for any $k\leq cn\log(n)$ for some constant $c>0$ [*] $40$ pts for any $k\leq cn$ for some constant $c>1$ [*] $70$ pts for $k\leq n$ [*] $85$ pts for $k\leq n-\lfloor\sqrt{n/2}-\tfrac{1}{2}\rfloor + 1$ [*] $100$ pts for $k\leq n-\lfloor\sqrt{n}-\tfrac{1}{2}\rfloor + 1$ [/list] [i]Proposed by David Tang[/i]

\[\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx\] [i]Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko[/i]

Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $A$ and $E$, what is $|AE|$? $ \textbf{(A)}\ \dfrac 23 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac 32 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $