Let $a,b,c$ be reals satisfying\\ $\hspace{2cm} 3ab+2=6b, \hspace{0.5cm} 3bc+2=5c, \hspace{0.5cm} 3ca+2=4a.$\\ \\ Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s}\in \mathbb{Q}$ and $\frac{t}{u}\in \mathbb{Q}$ , in lowest form, find $r+s+t+u$.

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

Find a set of infinite positive integers $ S$ such that for every $ n\ge 1$ and whichever $ n$ distinct elements $ x_1,x_2,\cdots, x_n$ of S, the number $ x_1\plus{}x_2\plus{}\cdots \plus{}x_n$ is not a perfect square.

Let $ABCD$ be a cyclic quadrilateral. A point $P$ moves along the arc $AD$ which does not contain $B$ and $C$. A fixed line $l$, perpendicular to $BC$, meets the rays $BP$, $CP$ at points $B_0$, $C_0$ respectively. Prove that the tangent at $P$ to the circumcircle of triangle $PB_0C_0$ passes through some fixed point.

Let $n$ be a positive integer. An $n-\emph{simplex}$ in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1,\cdots , P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $\mathcal{S}$ we denote by $v(\mathcal{S})$ the volume of $\mathcal{S}$, and we write $C(\mathcal{S})$ for the center of the unique sphere containing all the vertices of $\mathcal{S}$. Suppose that $P$ is a point inside an $n$-simplex $\mathcal{S}$. Let $\mathcal{S}_i$ be the $n$-simplex obtained from $\mathcal{S}$ by replacing its $i^{\text{th}}$ vertex by $P$. Prove that : \[ \sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S}) \]

Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.

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.