Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties: • $ \mathcal{A} \cup \mathcal{B} \equal{} \{1,2,3,4,5,6,7,8,9,10,11,12\}$, • $ \mathcal{A} \cap \mathcal{B} \equal{} \emptyset$, • The number of elements of $ \mathcal{A}$ is not an element of $ \mathcal{A}$, • The number of elements of $ \mathcal{B}$ is not an element of $ \mathcal{B}$. Find $ N$.

How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$? $ \textbf{(A) \ }198\qquad \textbf{(B) \ } 199 \qquad \textbf{(C) \ } 398 \qquad \textbf{(D) \ }399\qquad \textbf{(E) \ } 597$

What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$? [i]2016 CCA Math Bonanza Lightning #1.1[/i]

Find the maximum number of colors used in coloring integers $n$ from $49$ to $94$ such that if $a, b$ (not necessarily different) have the same color but $c$ has a different color, then $c$ does not divide $a+b$.

Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

The $260$ volumes of the [i]Encyclopedia Galactica[/i] are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the position of an even numbered volume with that of an odd numbered volume. Find the minimum number of such transpositions sufficient to get the books back into ordinary sequential order, regardless of the starting positions of the books. (Find the minimum number of transpositions in the worst-case scenario.)

In a Greek village of $100$ inhabitants in the beginning there lived $12$ Olympians and $88$ humans, they were the first generation. The Olympians are $100\%$ gods while humans are $0\%$ gods. In each generation they formed $50$ couples and each couple had $2$ children, who form the next generation (also consisting of $100$ members). From the second generation onwards, every person’s percentage of godness is the average of the percentages of his/her parents’ percentages. (For example the children of $25\%$ and $12.5\% $gods are $18.75\%$ gods.) a) Which is the earliest generation in which it is possible that there are equally many $100\%$ gods as $ 0\%$ gods? b) At most how many members of the fifth generation are at least 25% gods?

A man walks $x$ miles due west, turns $150^\circ$ to his left and walks 3 miles in the new direction. If he finishes a a point $\sqrt{3}$ from his starting point, then $x$ is $\text{(A)} \ \sqrt 3 \qquad \text{(B)} \ 2\sqrt{5} \qquad \text{(C)} \ \frac 32 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

A nonempty set $S$ is called $\emph{well-filled}$ if for every $m \in S$, there are fewer than $\frac 12 m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\left\{ 1,2,\dots,42 \right\}$.

Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$. Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

We color numbers $1,2,3,...,20$ in two colors, blue and yellow, such that both colors are used (not all numbers are colored in one color). Determine number of ways we can color those numbers, such that product of all blue numbers and product of all yellow numbers have greatest common divisor $1$.

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

There is one nonzero digit in every cell of $2017\times 2017 $ table. On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$. [hide=Example]Example for $2\times2$. If table is |1|4| |3|7|. Then numbers on board are $14,37,13,47$[/hide]

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$.

A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. (a) If $\theta = 90^o$, determine all locations where the point could end up. (b) If $\theta = 45^o$, prove that for every location $ L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (c) Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (d) Prove that when $\theta$ is irrational, for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L.$

There are $30$ teams in NBA and every team play $82$ games in the year. Bosses of NBA want to divide all teams on Western and Eastern Conferences (not necessary equally), such that number of games between teams from different conferences is half of number of all games. Can they do it?

Balls of $3$ colours — red, blue and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy] $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~\displaystyle\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\displaystyle\frac{4}{5}\qquad\textbf{(D)}~\displaystyle\frac{11}{13}\qquad\textbf{(E)}~\displaystyle\frac{7\sqrt{3}}{13}$

Let $ABCD$ and $AEFG$ be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point $P$. Prove that $P$ lies on line $BE$.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is: $ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $