Let $a,b,c>0$ and $a+b+c+abc=4$. Prove that $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).$$ [i](Zhuge Liang)[/i]

$ABC$ is any triangle. Tangent at $C$ to circumcircle ($O$) of $ABC$ meets $AB$ at $M$. Line perpendicular to $OM$ at $M$ intersects $BC$ at $P$ and $AC$ at $Q$. P.T. $MP=MQ$.

A trapezoid has side lengths $24$, $25$, $26$, and $27$ in some order. Find its area.

$A$ is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of $A$ also belongs to $A$.

Given integers \( m, n \geq 2 \), the points \( A_1, A_2, \dots, A_n \) are chosen independently and uniformly at random on a circle of circumference \( 1 \). That is, for each \( i = 1, \dots, n \), for any \( x \in (0,1) \), and for any arc \( \mathcal{C} \) of length \( x \) on the circle, we have $\mathbb{P}(A_i \in \mathcal{C}) = x$. What is the probability that there exists an arc of length \( \frac{1}{m} \) on the circle that contains all the points \( A_1, A_2, \dots, A_n \)?

There exists a unique circle that is both tangent to the parabola $y=x^2$ at two points and tangent to the curve $x=\sqrt{\tfrac{y^3}{1-y}}.$ Compute the radius of this circle.

Let $x, y, z$ be the real numbers that satisfies the following. $(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1$ Find the minimum value of $x^4+y^4+z^4$.

A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] [i]Proposed by Holden Mui[/i]

Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

In triangle $ABC$, $\angle BAC = 94^{\circ},\ \angle ACB = 39^{\circ}$. Prove that \[ BC^2 = AC^2 + AC\cdot AB\].

Let $ABCD$ be a square with side length $1$. If point $E$ is on $BC$, point $F$ is on $DC$, and triangle $AEF$ is equilateral, compute the side length of triangle $AEF$. (Note: if your answer has a square root inside a square root, you have not fully simplified your answer.)

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

Let $x, y, z$ be positive real numbers such that $\frac{1}{1 + x}+\frac{1}{1 + y}+\frac{1}{1 + z}= 2$. Prove that $8xyz \le 1$.

Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$. [i]Proposed by Andrew Wu[/i]

Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal $N$ they may show such a trick? [i]K. Knop, O. Leontieva[/i]

Describe all convex, inscriptible polygons which have the property that however we choose three distinct vertexes of of one of them, those vertexes form an isosceles triangle. [i]Gheorghe Iurea[/i]

Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties: i) $ H$ has parallel opposite sides. ii) Any 3 vertices of $ H$ can be covered with a strip of width 1. Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$. Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.

Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

$m,n$ are positive intergers and $x,y,z$ positive real numbers such that $0 \leq x,y,z \leq 1$. Let $m+n=p$. Prove that: $0 \leq x^p+y^p+z^p-x^m*y^n-y^m*z^n-z^m*x^n \leq 1$