Let $ a,b,c,d $ be four non-negative reals such that $ a+b+c+d = 4 $. Prove that $$ a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5} $$

Given a scalene triangle $ \triangle ABC $. $ B', C' $ are points lie on the rays $ \overrightarrow{AB}, \overrightarrow{AC} $ such that $ \overline{AB'} = \overline{AC}, \overline{AC'} = \overline{AB} $. Now, for an arbitrary point $ P $ in the plane. Let $ Q $ be the reflection point of $ P $ w.r.t $ \overline{BC} $. The intersections of $ \odot{\left(BB'P\right)} $ and $ \odot{\left(CC'P\right)} $ is $ P' $ and the intersections of $ \odot{\left(BB'Q\right)} $ and $ \odot{\left(CC'Q\right)} $ is $ Q' $. Suppose that $ O, O' $ are circumcenters of $ \triangle{ABC}, \triangle{AB'C'} $ Show that 1. $ O', P', Q' $ are colinear 2. $ \overline{O'P'} \cdot \overline{O'Q'} = \overline{OA}^{2} $

Given a triangle $ \triangle{ABC} $ and a point $ O $. $ X $ is a point on the ray $ \overrightarrow{AC} $. Let $ X' $ be a point on the ray $ \overrightarrow{BA} $ so that $ \overline{AX} = \overline{AX_{1}} $ and $ A $ lies in the segment $ \overline{BX_{1}} $. Then, on the ray $ \overrightarrow{BC} $, choose $ X_{2} $ with $ \overline{X_{1}X_{2}} \parallel \overline{OC} $. Prove that when $ X $ moves on the ray $ \overrightarrow{AC} $, the locus of circumcenter of $ \triangle{BX_{1}X_{2}} $ is a part of a line.

Find all pairs of integers $ \left(m,n\right) $ such that $ \left(m,n+1\right) = 1 $ and $$ \sum\limits_{k=1}^{n}{\frac{m^{k+1}}{k+1}\binom{n}{k}} \in \mathbb{N} $$

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(f\left(x\right)+y\right) = f\left(x^2-y\right)+4\left(y-2\right)\left(f\left(x\right)+2\right) $$holds for all $ x, y \in \mathbb{R} $

In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$

There are $n$ husbands and wives at a party in the palace. The husbands sit at a round table, and the wives sit at another round tables. The king and queen (not included in the $n$ couples) are going to shake hands with them one by one. Assume that the king starts from a man, and the queen starts from his wife. Consider the following two ways of shaking hands: (i) The king shakes hands with the men one by one clockwise. Each time when the king shakes hands with a man, the queen moves clockwise to his wife and shakes hands with her. Assume that at last when the king gets back to the man he begins with, the queen goes around the table $a$ times. (ii) The queen shakes hands with the women one by one clockwise. Each time when the queen shakes hands with a woman, the king moves clockwise to her husband and shakes hands with him. Assume that at last when the queen gets back to the woman she begins with, the king goes around the table $b$ times. Determine the maximum possible value of $|a-b|$.

Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to \[\frac{(n+1)(n+2)}{2}\]oranges. A triple of oranges $A,B,C$ is good if each $AB,AC$ is some side of some unit regular triangles, and $\angle BAC = 120^{\circ}$. Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.

Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$holds for all positive integers $ m,n $

Given six points $ A, B, C, D, E, F $ such that $ \triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA $ and let $ I $ be the incenter of $ \triangle ABC. $ Prove that the circumcenter of $ \triangle AID, \triangle BIE, \triangle CIF $ are collinear. Proposed by Telv Cohl