Consider the set $S=\{ 1,2,\ldots ,100\}$ and the family $\mathcal{P}=\{ T\subset S\mid |T|=49\}$. Each $T\in\mathcal{P}$ is labelled by an arbitrary number from $S$. Prove that there exists a subset $M$ of $S$ with $|M|=50$ such that for each $x\in M$, the set $M\backslash\{ x\}$ is not labelled by $x$.

Let $M$ be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point $P=(x,y)\in M$ we call the points $(x-1,y)$, $(x+1,y)$, $(x,y-1)$, $(x,y+1)$ neighbors of $P$. Let $S$ be a finite subset of $M$. A one-to-one mapping $f$ of $S$ onto $S$ is called perfect if $f(P)$ is a neighbor of $P$, for any $P\in S$. Prove that if such a mapping exists, then there exists also a perfect mapping $g:S\to S$ with the additional property $g(g(P))=P$ for $P\in S$.

The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are written on the board. Is it possible to paint half of them red and remaining ones blue, so that the sum of red numbers is divisible by the sum of blue ones?

[u]Round 4[/u] [b]p10.[/b] How many divisors of $10^{11}$ have at least half as many divisors that $10^{11}$ has? [b]p11.[/b] Let $f(x, y) = \frac{x}{y}+\frac{y}{x}$ and $g(x, y) = \frac{x}{y}-\frac{y}{x} $. Then, if $\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)$ can be expressed in the form $a + \frac{b}{c}$, where $a$, $b$,$c$ are nonnegative integers such that $b < c$ and $gcd(b,c) = 1$, find $a + b + \lceil (\log_2 (\log_2 c)\rceil $ [b]p12.[/b] Let $ABC$ be an equilateral triangle, and let$ DEF$ be an equilateral triangle such that $D$, $E$, and $F$ lie on $AB$, $BC$, and $CA$, respectively. Suppose that $AD$ and $BD$ are positive integers, and that $\frac{[DEF]}{[ABC]}=\frac{97}{196}$. The circumcircle of triangle $DEF$ meets $AB$, $BC$, and $CA$ again at $G$, $H$, and $I$, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices $D, E, F, G, H$, and $I$. [u]Round 5 [/u] [b]p13.[/b] Point $X$ is on line segment $AB$ such that $AX = \frac25$ and $XB = \frac52$. Circle $\Omega$ has diameter $AB$ and circle $\omega$ has diameter $XB$. A ray perpendicular to $AB$ begins at $X$ and intersects $\Omega$ at a point $Y$. Let $Z$ be a point on $\omega$ such that $\angle YZX = 90^o$. If the area of triangle $XYZ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$. [b]p14.[/b] Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened? [b]p15.[/b] Let triangle $ABC$ with circumcircle $\Omega$ satisfy $AB = 39$, $BC = 40$, and $CA = 25$. Let $P$ be a point on arc $BC$ not containing $A$, and let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$, respectively. Let $AQ$ and $AR$ meet $\Omega$ again at $S$ and $T$, respectively. Given that the reflection of $QR$ over $BC$ is tangent to $\Omega$ , $ST$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a,b)= 1$. Find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url],Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be an interior point of a triangle of area $T$. Through the point $P$, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let $a$, $b$ and $c$ be the areas of the three triangles. Prove that $ \sqrt { T } = \sqrt { a } + \sqrt { b } + \sqrt { c } $.

Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year. There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Let $C$ and $D$ be different points on the semicircle with diameter $AB$. The lines $AC$ and $BD$ intersect at $E$, and the lines $AD$ and $BC$ intersect at $F$. Prove that the midpoints $X,Y,Z$ of the segments $AB,CD,EF$ respectively are collinear.

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions: [list] [*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$. [*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$. [/list]

Let $\omega$ be the unit circle in the $xy$-plane in $3$-dimensional space. Find all points $P$ not on the $xy$-plane that satisfy the following condition: There exist points $A,B,C$ on $\omega$ such that $$ \angle APB = \angle APC = \angle BPC = 90^\circ.$$

Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$, you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$. Find the number of values of $n$ so that the sum of $1111$ is possible.

Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

Let $ABC$ be a triangle. Let $K$ be a midpoint of $BC$ and $M$ be a point on the segment $AB$. $L=KM \cap AC$ and $C$ lies on the segment $AC$ between $A$ and $L$. Let $N$ be a midpoint of $ML$. $AN$ cuts circumcircle of $\Delta ABC$ in $S$ and $S \neq N$. Prove that circumcircle of $\Delta KSN$ is tangent to $BC$.

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?

Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds. [center][img]http://i.imgur.com/WNp59XW.png[/img][/center]

Only $A$ and $B$ have $n$ friends in a village of $2010$ people. The other $2008$ people have all different numbers of friends. How many possible values of $n$ are there? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None} $

Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.

Let $ABC$ be a triangle. $D$ and $E$ are points on line segments $BC$ and $AC$, respectively, such that $AD=60$, $BD=189$, $CD=36$, $AE=40$, and $CE=50$. What is $AB+DE$? [i]2017 CCA Math Bonanza Tiebreaker Round #2[/i]

What is greater, $31^{11}$ or $17^{14}$ ?

For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $?