In equiangular octagon $CAROLINE$, $CA = RO = LI = NE = \sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, that total area of the six triangular regions. Then $K=\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.

For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$?

Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.

Two permutations $a_1,a_2,\dots,a_{2010}$ and $b_1,b_2,\dots,b_{2010}$ of the numbers $1,2,\dots,2010$ are said to [i]intersect[/i] if $a_k=b_k$ for some value of $k$ in the range $1\le k\le 2010$. Show that there exist $1006$ permutations of the numbers $1,2,\dots,2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.

Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer. Show that $p = q$.

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let \[S_n= 2\sum_{k=1}^n \sin(2kU\pi).\] Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$. [i]Bulgaria[/i]

Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$? $\text{(A)}\ \frac 14 \qquad \text{(B)}\ \frac 13 \qquad \text{(C)}\ \frac 12 \qquad \text{(D)}\ \frac 23\qquad \text{(E)}\ \frac 34$

$\triangle ABC$ has the property that $\angle ACB = 90^{\circ}$. Let $D$ and $E$ be points on $AB$ such that $D$ is on ray $BA$, $E$ is on segment $AB$, and $\angle DCA = \angle ACE$. Let the circumcircle of $\triangle CDE$ hit $BC$ at $F \ne C$, and $EF$ hit $AC$ and $DC$ at $P$ and $Q$, respectively. If $EP = FQ$, then the ratio $\frac{EF}{PQ}$ can be written as $a+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$. [i]Proposed by Kevin Zhao[/i]

Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.

In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$ is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.

In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$ and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\angle PDF$ in degrees?

Find all positive intger solutions of $3^x+29=2^y$.

In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.

Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

In triangle $ ABC,H,I,O$ are orthocenter, incenter and circumcenter, respectively. $ CI$ cuts circumcircle at $ L$. If $ AB\equal{}IL$ and $ AH\equal{}OH$, find angles of triangle $ ABC$.

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

Let $\omega_1,\omega_2$ be two circles centered at $O_1$ and $O_2$ and lying outside each other. Points $C_1$ and $C_2$ lie on these circles in the same semi plane with respect to $O_1O_2$. The ray $O_1C_1$ meets $\omega _2$ at $A_2,B_2$ and $O_2C_2$ meets $\omega_1$ at $A_1,B_1$. Prove that $\angle A_1O_1B_1=\angle A_2O_2B_2$ if and only if $C_1C_2||O_1O_2$.

We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$? [i]2020 CCA Math Bonanza Lightning Round #2.1[/i]

Arianna and Brianna play a game in which they alternate turns writing numbers on a paper. Before the game begins, a referee randomly selects an integer $N$ with $1 \leq N \leq 2019$, such that $i$ has probability $\frac{i}{1 + 2 + \dots + 2019}$ of being chosen. First, Arianna writes $1$ on the paper. On any move thereafter, the player whose turn it is writes $a+1$ or $2a$, where $a$ is any number on the paper, under the conditions that no number is ever written twice and any number written does not exceed $N$. No number is ever erased. The winner is the person who first writes the number $N$. Assuming both Arianna and Brianna play optimally, the probability that Brianna wins can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n.$ [i]Proposed by Edward Wan[/i]