To be continuous at $ x \equal{} \minus{} 1$, the value of $ \frac {x^3 \plus{} 1}{x^2 \minus{} 1}$ is taken to be: $ \textbf{(A)}\ \minus{} 2 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \infty \qquad\textbf{(E)}\ \minus{} \frac {3}{2}$

Consider a $2 \times n$ grid where each cell is either black or white, which we attempt to tile with $2 \times 1$ black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer $N$ where $N$ takes the value $n$ with probability $\frac{1}{2^n}$. We then take a $2 \times N$ grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.

In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.

Let $N \geq 3$ be an integer, and let $a_0, \dots, a_{N-1}$ be pairwise distinct reals so that $a_i \geq a_{2i}$ for all $i$ (indices are taken $\bmod~ N$). Find all possible $N$ for which this is possible. [i]Proposed by Sutanay Bhattacharya[/i]

Let $A$ and $B$ be integers with an odd sum. Show that every integer can be written in the form $x^2 -y^2 +Ax+By$, where $x,y$ are integers.

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 16$

A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away?

Let $f(n) = \sum_{gcd(k,n)=1,1\le k\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_1 p^{e_2}_2 ... p^{e_k}_k$, find $\sum^k_{i=1} p_ie_i$.

Compute $$\lim_{x\rightarrow1}\frac{x^3-1}{x-1}.$$

Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.

Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?

Answer this question with a positive integer $1$ through $1000$. A positive integer ``answer" has been randomly selected from 1 to 1000, inclusive; if your selected integer is less than or equal to the ``answer", you will gain $\lfloor 20\left(\frac{x}{a}\right)^2 \rfloor$ points, where $x$ is your number and $a$ is the ``answer". If you select an integer greater than the ``answer", you will not gain any points. [i]Lightning 5.4[/i]

Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?

In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

Find all injective functions $ f:\mathbb{N} \to \mathbb{N} $ such that $$ f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2 $$ holds for all $ a,b \in \mathbb{N} $. Note that $ f^{k}\left(n\right) $ means $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $

Find all prime numbers $p$, $q$ and $r$ such that the fourth power of any of them minus one is divisible by the product of the other two. (Author: V. Senderov)

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? $ \textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3} $

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin　University entrance exam/Medicine [/i]

Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square. Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.

[u]Round 1[/u] [b]p1.[/b] Alec rated the movie Frozen $1$ out of $5$ stars. At least how many ratings of $5$ out of $5$ stars does Eric need to collect to make the average rating for Frozen greater than or equal to $4$ out of $5$ stars? [b]p2.[/b] Bessie shuffles a standard $52$-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card? [b]p3.[/b] Find the value of $121 \cdot 1020304030201$. [u]Round 2[/u] [b]p4.[/b] Find the smallest positive integer $c$ for which there exist positive integers $a$ and $b$ such that $a \ne b$ and $a^2 + b^2 = c$ [b]p5.[/b] A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle. [b]p6.[/b] There are $10$ monsters, each with $6$ units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters? [u]Round 3[/u] [b]p7.[/b] It is known that $2$ students make up $5\%$ of a class, when rounded to the nearest percent. Determine the number of possible class sizes. [b]p8.[/b] At $17:10$, Totoro hopped onto a train traveling from Tianjin to Urumuqi. At $14:10$ that same day, a train departed Urumuqi for Tianjin, traveling at the same speed as the $17:10$ train. If the duration of a one-way trip is $13$ hours, then how many hours after the two trains pass each other would Totoro reach Urumuqi? [b]p9.[/b] Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.) [u]Round 4[/u] [b]p10.[/b] Compute the smallest positive integer with at least four two-digit positive divisors. [b]p11.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC = 10$ and $AD = 18$. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$. [b]p12.[/b] How many length ten strings consisting of only $A$s and Bs contain neither "$BAB$" nor "$BBB$" as a substring? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934037p26256063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that each of $20$ students has made a choice of anywhere from $0$ to $6$ courses from a total of $6$ courses offered. Prove or disprove : there are $5$ students and $2$ courses such that all $5$ have chosen both courses or all $5$ have chosen neither course.

The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A $clique$ is a set of cities such that every 2 cities in the set are connected by a direct flight. A $cluque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A $claque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque. Tuymaada 2017 Q3 Juniors

Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$. [i]Proposed by Andy Xu[/i]