[4] Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle{BDF}$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$

We have $2015$ marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take $2$ different colored marbles, then replace it with $2$ marbles with the third color. For example, we take one blue marble and one green marble, and we fill with $2$ red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.

If a number $N$, $N\neq 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is: $\textbf{(A) }\dfrac1R\qquad \textbf{(B) }R\qquad \textbf{(C) }4\qquad \textbf{(D) }\dfrac14\qquad \textbf{(E) }-R$

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

Daniel can hack a finite cylindrical log into $3$ pieces in $6$ minutes. How long would it take him to cut it into $9$ pieces, assuming each cut takes Daniel the same amount of time? [i]2015 CCA Math Bonanza Lightning Round #1.3[/i]

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

Prove that there are infinitely many ordered triples of positive integers $(a,b,c)$ such that the greatest common divisor of $a,b,$ and $c$ is $1$, and the sum $a^2b^2+b^2c^2+c^2a^2$ is the square of an integer.

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

[u]Townspeople and Goons[/u] In the city of Lincoln, there is an empty jail, at least two townspeople and at least one goon. A game proceeds over several days, starting with morning. $\bullet$ Each morning, one randomly selected unjailed person is placed in jail. If at this point all goons are jailed, and at least one townsperson remains, then the townspeople win. If at this point all townspeople are jailed and at least one goon remains, then the goons win. $\bullet$ Each evening, if there is at least one goon and at least one townsperson not in jail, then one randomly selected townsperson is jailed. If at this point there are at least as many goons remaining as townspeople remaining, then the goons win. The game ends immediately after any group wins. [b]p1. [/b]Find the probability that the townspeople win if there are initially two townspeople and one goon. [b]p2.[/b] Find the smallest positive integer $n$ such that, if there are initially $2n$ townspeople and $1$ goon, then the probability the townspeople win is greater than $50\%$. [b]p3.[/b] Find the smallest positive integer $n$ such that, if there are initially $n + 1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1\%$. [b]p4[/b]. Suppose there are initially $1001$ townspeople and two goons. What is the probability that, when the game ends, there are exactly $1000$ people in jail? [b]p5.[/b] Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.) Find the probability that only the Jester wins.

Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$, $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$. Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezoid.

The median of the list \[ n, n \plus{} 3, n \plus{} 4, n \plus{} 5, n \plus{} 6, n \plus{} 8, n \plus{} 10, n \plus{} 12, n \plus{} 15 \]is $ 10$. What is the mean? $ \textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

Given that $0 < a < b < c < d$, which of the following is the largest? $\textbf{(A)}\ \frac{a\plus{}b}{c\plus{}d} \qquad \textbf{(B)}\ \frac{a\plus{} d}{b\plus{} c} \qquad \textbf{(C)}\ \frac{b\plus{} c}{a\plus{}d}\qquad \textbf{(D)}\ \frac{b\plus{} d}{a\plus{} c} \qquad \textbf{(E)}\ \frac{c\plus{} d}{a\plus{}b}$

Find all 2-digit numbers$ n$ having the property: 'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.

The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$. a) Determine the smallest and the largest possible values of $xy + yz - xz$. b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).