A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$. $(\text{a})$ Find $2004$ perfect powers in arithmetic progression. $(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle. Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points. [i]Kevin Cong[/i]

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

In the triangle $ABC$, the segment $CL$ is the angle bisector. The $C$-exscribed circle with center at the point $ I_c$ touches the side of the $AB$ at the point $D$ and the extension of sides $CA$ and $CB$ at points $P$ and $Q$, respectively. It turned out that the length of the segment $CD$ is equal to the radius of this exscribed circle. Prove that the line $PQ$ bisects the segment $I_CL$.

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n$ be distinct positive real numbers. For any $(i,j)$ in a country consisting of cities $C_1,C_2,\cdots,C_n$, there is a two-way flight between $C_i$ and $C_j$ that costs $a_i+a_j$.A traveler travels between cities of this country such that every time they pay a strictly higher cost than their previous flight. Find the maximum number of flight this traveler could take.

For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.

Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.

A class with $2N$ students took a quiz, on which the possible scores were $0,1,\dots,10.$ Each of these scores occurred at least once, and the average score was exactly $7.4.$ Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4.$

On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that: $$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$ [i]Cristinel Mortici[/i]

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Suppose a continuous function $f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]$ and differentiable on $(-\frac{\pi}{4},\frac{\pi}{4})$. Then, there exists a point $x_0\in (-\frac{\pi}{4},\frac{\pi}{4})$ such that $$|f'(x_0)|\leq 1+f(x_0)^2$$

At the Mountain School, Micchell is assigned a [i]submissiveness rating[/i] of $3.0$ or $4.0$ for each class he takes. His [i]college potential[/i] is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of $3.975$. Unfortunately, he needs a college potential of at least $3.995$ to get into the [url=http://en.wikipedia.org/wiki/Accepted#Plot]South Harmon Institute of Technology[/url]. Otherwise, he becomes a rock. Assuming he receives a submissiveness rating of $4.0$ in every class he takes from now on, how many more classes does he need to take in order to get into the South Harmon Institute of Technology? [i]Victor Wang[/i]

Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that, \begin{align*} AB+BC \leq 2AC \end{align*}

A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.

Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three different boxes. Show that there is a box such that all the balls in all the other boxes have the same color.

The George Washington Bridge is $2016$ meters long. Sally is standing on the George Washington Bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\dfrac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$. Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares.

For a positive integer, $m$, answer the following questions. 1) Show that $2^{m+1}+1$ is a prime number, when $2^{m+1}+1$ is a factor of $3^{2^m}+1$. 2) Is converse of 1) true?

Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.

Prove that there exist integers $a$, $b$, $c$ with $1 \le a < b < c \le 25$ and \[ S(a^6+2014) = S(b^6+2014) = S(c^6+2014) \] where $S(n)$ denotes the sum of the decimal digits of $n$. [i]Proposed by Evan Chen[/i]

An arbitrary point $D$ was marked on the median $BM$ of the triangle $ABC$. It is known that the point $DE\parallel AB$ and $CE \parallel BM$. Prove that $BE = AD$

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.” After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made. Can the students come up with a strategy that guarantees a “passed” mark to each of them? [i] Denis Afrizonov [/i]

There were $n> 1$ aborigines living on an island, each of them telling only the truth or only lying, and each having at least one friend among the others. The new governor asked each aborigine whether there are more truthful aborigines or liars among his friends, or an equal number of both. Each aborigine answered that there are more liars than truthful aborigines among his friends. The governor then ordered one of the aborigines to be executed for being a liar and asked each of the remaining $n- 1$ aborigines the same question again. This time each aborigine answered that there are more truthful aborigines than liars among his friends. Determine whether the executed aborigine was truthful or a liar, and whether there are more truthful aborigines or liars remaining on the island.

Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal. a) Prove that and also indicate how the points $P$ and $Q$ can be constructed. b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .