Answer the following questions : [b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$ [b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$ [b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.

Find the distinct positive integers $a, b, c,d$, such that the following conditions hold: (1) exactly three of the four numbers are prime numbers; (2) $a^2 + b^2 + c^2 + d^2 = 2018.$

Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$. [asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O); MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, .. 13$. Their scores are $s_1,s_2, ... s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, ... w_5$. Each score $s_i$ is equal to the sums of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_ {s_i} }$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$. [i]Proposed by Jeff Lin[/i]

Let $a,b,c$ be integers larger than $1$. Prove that \[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]

Consider the set: $A = \{1, 2,..., 100\}$ Prove that if we take $11$ different elements from $A$, there are $x, y$ such that $x \neq y$ and $0 < |\sqrt{x} - \sqrt{y}| < 1$

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

Cut the $10$ cm $\times 25$ cm rectangle into two pieces with one straight cut so that they can fit inside the $22.1 $ cm circle without crossing.

Let $f$ be a linear function such that $f(0) = -5$ and $f(f(0)) = -15$. Find the values of $ k \in R$ for which the solutions of the inequality $f(x) \cdot f(k - x) > 0$, lie in an interval of[u][/u] length $2$.

How many subsets containing three different numbers can be selected from the set \[\{ 89,95,99,132, 166,173 \}\] so that the sum of the three numbers is even? $\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$

It is given that one root of $ 2x^2 \plus{} rx \plus{} s \equal{} 0$, with $ r$ and $ s$ real numbers, is $ 3\plus{}2i (i \equal{} \sqrt{\minus{}1})$. The value of $ s$ is: $ \textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \minus{}13 \qquad \textbf{(E)}\ 26$