$ 5 $ points are situated in the plane so that any three of them form a triangle of area at most $ 1. $ Prove that there is a trapezoid of area at most $ 3 $ which contains all these points ('including' here means that the points can also be on the sides of the trapezoid).

Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.

Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$). ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

Peter Rabbit is hopping along the number line, always jumping in the positive $x$ direction. For his first jump, he starts at $0$ and jumps $1$ unit to get to the number $1.$ For his second jump, he jumps $4$ units to get to the number $5.$ He continues jumping by jumping $1$ unit whenever he is on a multiple of $3$ and by jumping $4$ units whenever he is on a number that is not a multiple of $3.$ What number does he land on at the end of his $100$th jump? $$ \mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301 $$

Let $T=\text{TNFTPP}$. In the binary expansion of \[\dfrac{2^{2007}-1}{2^T-1},\] how many of the first $10,000$ digits to the right of the radix point are $0$'s?

Maria and Joe are jogging towards each other on a long straight path. Joe is running at $10$ mph and Maria at $8$ mph. When they are $3$ miles apart, a fly begins to fly back and forth between them at a constant rate of $15$ mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles, will the fly have traveled when Joe and Maria pass each other?

Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is $ \displaystyle \textbf{(A)}\ \sqrt {\frac {a}{b}} \qquad \textbf{(B)}\ \ \frac {a}{b} \qquad \textbf{(C)}\ \ \frac {a^2}{b^2} \qquad \textbf{(D)}\ \ \frac {a \plus{} b}{2b} \qquad \textbf{(E)}\ \ \frac {a^2 \plus{} 2ab}{b^2 \plus{} 2ab}$ [asy]unitsize(2cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); fill(Arc((1/3,0),2/3,0,180)--reverse(Arc((-2/3,0),1/3,180,360))--reverse(Arc((0,0),1,0,180))--cycle,mediumgray); draw(unitcircle); draw(Arc((-2/3,0),1/3,360,180)); draw(Arc((1/3,0),2/3,0,180)); label("$a$",(-2/3,0)); label("$b$",(1/3,0)); draw((-2/3+1/15,0)--(-1/3,0),EndArrow(4)); draw((-2/3-1/15,0)--(-1,0),EndArrow(4)); draw((1/3+1/15,0)--(1,0),EndArrow(4)); draw((1/3-1/15,0)--(-1/3,0),EndArrow(4));[/asy]

Show how to divide space into (a) congruent tetrahedra, (b) congruent “equifaced” tetrahedra. (A tetrahedron is called equifaced if all its faces are congruent triangles.) (NB Vassiliev)

Points $D$, $E$, $F$ are taken respectively on sides $AB$, $BC$, and $CA$ of triangle $ABC$ so that $AD:DB=BE:CE=CF:FA=1:n$. The ratio of the area of triangle $DEF$ to that of triangle $ABC$ is: $\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad \textbf{(B)}\ \frac{1}{(n+1)^2}\qquad \textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad \textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad \textbf{(E)}\ \frac{n(n-1)}{n+1}$

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$, [list] [*] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and [*] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$ Prove that if $a_1~=~0$ , then $a_{2020}~=~1$

A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is [I]small[/I] if $A, B, C$ are single-digit positive integers. It is [I]full[/I] if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ where $D$, $E$, $F$ lie on $BC$, $AC$, $AB$, respectively. Let $M$ be the midpoint of $BC$. The circumcircle of triangle $AEF$ cuts the line $AM$ at $A$ and $X$. The line $AM$ cuts the line $CF$ at $Y$. Let $Z$ be the point of intersection of $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.

In a trapezoid $ABCD$ with $AB\parallel CD$, the diagonals $AC$ and $BD$ meet at $O$. Let $M$ and $N$ be the centers of the regular hexagons constructed on the sides $AB$ and $CD$ in the exterior of the trapezoid. Prove that $M,O$ and $N$ are collinear.

$15$ ladies and $30$ gentlemen attend a luxurious party. At the start of the party, each one of the ladies shakes hands with a random gentleman. At the end of the party, each of the ladies shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when $m$ is divided by $10000$.

(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that $$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$ (b) Show an example in which equality occurs.

Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$? [i]2015 CCA Math Bonanza Lightning Round #5.3[/i]

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$

Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group? I haven't had much group theory at this point...

Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$

Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.