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*}

Let $S$ be a set. We say $S$ is $D^\ast$[i]-finite[/i] if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a [i]witness[/i] of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have? [i]Proposed by Evan Chen[/i]

Let $p$ be the third-smallest prime number greater than $5$ such that: $\bullet$ $2p + 1$ is prime, and $\bullet$ $5^p \not\equiv 1$ (mod $2p + 1$). Find $p$.

Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$.

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label("$X$",(3.4,1.5)); label("6",(7.6,1.5)); label("1",(7.6,3.5)); label("1",(8.4,4.6)); label("2",(9.4,4.6)); label("2",(10.4,6)); label("3",(8.4,7.4)); label("1",(7.5,7.8)); label("2",(6,8.5)); label("1",(4.7,7.8)); label("1",(4.3,7.5)); label("1",(3.5,6.5)); label("4",(1.8,6.5)); label("1",(-0.5,5.5)); label("2",(0.8,4.5)); label("2",(1.5,3.8)); label("2",(2.8,2.6)); [/asy] $\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}2 \qquad \textbf{(C)}\hspace{.05in}3 \qquad \textbf{(D)}\hspace{.05in}4 \qquad \textbf{(E)}\hspace{.05in}5 $

Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.

Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

Could the three-dimensional space be expressed as the union of disjoint circumferences?

Consider a square of sidelength $ n$ and $ (n\plus{}1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.

Regular hexagon $ABCDEF$ is inscribed in rectangle $PQRS$ with $AB = 1$, A and $B$ on side $PQ$, $C$ on side $QR$, $D$ and $E$ on side $RS$, and $F$ on side $SP$. What is the area of $PQRS$?

Let $ABCD$ be a tetrahedron satisfying i)$\widehat{ACD}+\widehat{BCD}=180^{0}$, and ii)$\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$. Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$.

Let $ABC$ be a triangle, let $O$ be its circumcenter, let $A'$ be the orthogonal projection of $A$ on the line $BC$, and let $X$ be a point on the open ray $AA'$ emanating from $A$. The internal bisectrix of the angle $BAC$ meets the circumcircle of $ABC$ again at $D$. Let $M$ be the midpoint of the segment $DX$. The line through $O$ and parallel to the line $AD$ meets the line $DX$ at $N$. Prove that the angles $BAM$ and $CAN$ are equal.

A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind $12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The [b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called [i]the deviation[/i] of the subset $S$.Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.

A little boy takes a $ 12$ in long strip of paper and makes a Mobius strip out of it by tapping the ends together after adding a half twist. He then takes a $ 1$ inch long train model and runs it along the center of the strip at a speed of $ 12$ inches per minute. How long does it take the train model to make two full complete loops around the Mobius strip? A complete loop is one that results in the train returning to its starting point.