A $ 100 \times 100$ square floor consisting of $ 10000$ squares is to be tiled by rectangular $ 1 \times 3$ tiles, fitting exactly over three squares of the floor. $ (a)$ If a $ 2 \times 2$ square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles. $ (b)$ If, instead, a $ 2 \times 2$ square is removed from the corner, prove that such a tiling is not possble.

Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of \[x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) \]

Find all pairs of primes $(p,q)$ such that $6pq$ divides $$p^3+q^2+38$$

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call the figure shown in the picture consisting of five unit squares a $\emph{plus}$, and each rectangle consisting of two such squares a $\emph{minus}$. Does there exist an odd integer $n$ with the property that a square with side length $n$ can be dissected into pluses and minuses? Justify your answer. [img] https://wiki-images.artofproblemsolving.com//6/6a/18-1-6.png [/img]

Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that \[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \] for every real number $x$ and every positive integer $n$.

Let $p > 3$ be a prime such that $p\equiv 3 \pmod 4.$ Given a positive integer $a_0$ define the sequence $a_0, a_1, \ldots $ of integers by $a_n = a^{2^n}_{n-1}$ for all $n = 1, 2,\ldots.$ Prove that it is possible to choose $a_0$ such that the subsequence $a_N , a_{N+1}, a_{N+2}, \ldots $ is not constant modulo $p$ for any positive integer $N.$

Several positive numbers each not exceeding 1 are written on the circle. Prove that one can divide the circle into three arcs so that the sums of numbers on any two arcs differ by no more than 1. (If there are no numbers on an arc, the sum is equal to zero.) [i](6 points)[/i]

A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.

Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying $$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$ [b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $ [b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $

There are $n$ boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of $n$ can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than $n$ dances?

In cyclic quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $E$, while segments $AC$ and $BD$ intersect at $F$. Point $P$ is on ray $EF$ such that angles $BPE$ and $CPE$ are congruent. Prove that angles $APB$ and $DPC$ are also equal.

Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? $\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$

In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.

Let $M$ be the midpoint of side $BC$ in $\triangle ABC$, and $P \neq B$ is such that the quadrilateral $ABMP$ is cyclic and the circumcircle of $\triangle BPC$ is tangent to the line $AB$. If $E$ is the second common point of the line $BP$ and the circumcircle of $\triangle ABC$, determine the ratio $BE: BP$.

[u]Round 5[/u] [b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p14.[/b] Maisy chooses a random set $(x, y)$ that satisfies $$x^2 + y^2 -26x -10y \le 482.$$ The probability that $y>0$ can be expressed as $\frac{A\pi -B\sqrt{C}}{D \pi}$. Find $A+B +C +D$. [color=#f00]Due to the problem having a typo, all teams who inputted answers received points[/color] [b]p15.[/b] $6$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). [u]Round 6[/u] [b]p16.[/b] Find the number of $3$ by $3$ grids such that each square in the grid is colored white or black and no two black squares share an edge. [b]p17.[/b] Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 25$, and $AC = 15$. Let $D$ be the point on BC such that $CD = 4$. Let $E$ be the foot of the altitude from $A$ to $BC$. Let $F$ be the intersection of $AE$ with the circle of radius $7$ centered at $A$ such that $F$ is outside of triangle $ABC$. $DF$ can be expressed as $\sqrt{m}$, where $m$ is a positive integer. Find $m$. [b]p18.[/b] Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a $9$ year prison sentence. Given that neither of them talk, they both face a $3$ year sentence. If both of them talk, they both will serve a $6$ year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is $\frac{11}{36}$ , find the expected duration of Bill’s sentence in months. [u]Round 7[/u] [b]p19.[/b] Rectangle $ABCD$ has point $E$ on side $\overline{CD}$. Point $F$ is the intersection of $\overline{AC}$ and $\overline{BE}$. Given that the area of $\vartriangle AFB$ is $175$ and the area of $\vartriangle CFE$ is $28$, find the area of $ADEF$. [b]p20.[/b] Real numbers $x, y$, and $z$ satisfy the system of equations $$5x+ 13y -z = 100,$$ $$25x^2 +169y^2 -z2 +130x y= 16000,$$ $$80x +208y-2z = 2020.$$ Find the value of $x yz$. [color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points. [/color] [b]p21.[/b] Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n$, he can move to $n +1$, $n +2$, or $n +4$. In howmany different ways can he move to the number $10$? [u]Round 8[/u] [b]p22.[/b] A sequence $a_1,a_2,a_3, ...$ of positive integers is defined such that $a_1 = 4$, and for each integer $k \ge 2$, $$2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.$$ Given that $a_6 = 488$, find $a_2 +a_3 +a_4 +a_5$. [b]p23.[/b] $\overline{PQ}$ is a diameter of circle $\omega$ with radius $1$ and center $O$. Let $A$ be a point such that $AP$ is tangent to $\omega$. Let $\gamma$ be a circle with diameter $AP$. Let $A'$ be where $AQ$ hits the circle with diameter $AP$ and $A''$ be where $AO$ hits the circle with diameter $OP$. Let $A'A''$ hit $PQ$ at $R$. Given that the value of the length $RA'$ is is always less than $k$ and $k$ is minimized, find the greatest integer less than or equal to $1000k$. [b]p24.[/b] You have cards numbered $1,2,3, ... ,100$ all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make ${100 \choose 2}$ total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a fixed prime number. Jomland has $p$ cities labelled $0,1,\dots,p-1$. Navi is a traveller and JomAirlines only has flights between two cities with labels $a$ and $b$ (flights are available in both directions) iff there exist positive integers $x$ and $y$ such that \[ \begin{cases} a \equiv x^2 + 2025xy + y^2\pmod{p}\\ b \equiv 20x^2 + xy + 25y^2\pmod{p} \end{cases} \] Prove that: i) There exist infinitely many primes $p$ such that there exist $2$ cities where Navi cannot start from one city and get to the other through a sequence of flights; ii) There exist infinitely many primes $p$ such that for any $2$ cities, Navi can start from one city and get to the other through a sequence of flights. [i](Proposed by Ivan Chan Guan Yu)[/i]

Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

In $\triangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. A circle centered at $D$ with radius $AD$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Show that $\triangle AQP\sim\triangle ABC$.

How many 4-digit numbers greater than 1000 are there that use the four digits of 2012? $\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}7 \qquad \textbf{(C)}\hspace{.05in}8 \qquad \textbf{(D)}\hspace{.05in}9 \qquad \textbf{(E)}\hspace{.05in}12 $

For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

Let $a,b,c$ be the lengths of the sides of a triangle, let $p=(a+b+c)/2$, and $r$ be the radius of the inscribed circle. Show that $$\frac{1}{(p-a)^2}+ \frac{1}{(p-b)^2}+\frac{1}{(p-c)^2} \geq \frac{1}{r^2}.$$