Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?

In a table $n\times n$ some unit squares are coloured black and the other unit squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1}=9786453210$. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $M$.

In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)\equal{}0$, $ f(2009)\equal{}2$, etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for $ n\equal{}9,999,999,999$. [i]Yudi Satria, Jakarta[/i]

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no. However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.

The sum $$\sum_{n=0}^{2016\cdot2017^2}2018^n$$ can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$.

Let $ f$ be an integrable real-valued function on the closed interval $ [0, 1]$ such that $$\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\ 1\;\; \text{for}\; m=n. \end{cases} $$ Show that $|f(x)|\geq2^{n}(n+1)$ on a set of positive measure.

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$. Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$.

Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.

In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $ -2 \alpha $. [i]Proposed by Dusan Djukic[/i]

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? [asy] size(100); pair A, B, C, D, E, F; A = (0,0); B = (1,0); C = (2,0); D = rotate(60, A)*B; E = B + D; F = rotate(60, A)*C; draw(Circle(A, 0.5)); draw(Circle(B, 0.5)); draw(Circle(C, 0.5)); draw(Circle(D, 0.5)); draw(Circle(E, 0.5)); draw(Circle(F, 0.5)); [/asy] $\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

Two real numbers $x, y$ are chosen randomly and independently on the interval $(1, r)$ where $r$ is some real number between $1024$ and $2048$. Let $P$ be the probability that $\lfloor \log_2x \rfloor > \lfloor \log_2y \rfloor .$ The value of $P$ is maximized when $r = \frac{p}{q}$ where $p,q$ are relatively prime positive integers. Find $p+q.$