Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.

Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]

Let $0<a<b$.Prove the following inequaliy. \[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.

Problem-2: Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that $\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.

Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments $T$ and $T$' yesterday such that everyone who visited the library yesterday was in the library at $T$ or $T'$ (or both).

p1. Find an integer that has the following properties: a) Every two adjacent digits in the number are prime. b) All prime numbers referred to in item (a) above are different. p2. Determine all integers up to $\sqrt{50+\sqrt{n}}+\sqrt{50-\sqrt{n}}$ p3. The following figure shows the path to form a series of letters and numbers “OSN2015”. Determine as many different paths as possible to form the series of letters and numbers by following the arrows. [img]https://cdn.artofproblemsolving.com/attachments/6/b/490a751457871184a506c2966f8355f20cebbd.png[/img] p4. Given an acute triangle $ABC$ with $L$ as the circumcircle. From point $A$, a perpendicular line is drawn on the line segment $BC$ so that it intersects the circle $L$ at point $X$. In a similar way, a perpendicular line is made from point $B$ and point $C$ so that it intersects the circle $L$, at point $Y$ and point $Z$, respectively. Is arc length $AY$ = arc length $AZ$ ? p5. The students of class VII.3 were divided into five groups: $A, B, C, D$ and $E$. Each group conducted five science experiments for five weeks. Each week each group performs an experiment that is different from the experiments conducted by other groups. Determine at least two possible trial schedules in week five, based on the following information: $\bullet$ In the first week, group$ D$ did experiment $4$. $\bullet$ In the second week, group $C$ did the experiment $5$. $\bullet$ In the third week, group $E$ did the experiment $5$. $\bullet$ In the fourth week, group $A$ did experiment $4$ and group $D$ did experiment $2$.

Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .

Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties: (i) it has a unit, $1$, and its order is $p$, a prime number; (ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$. Prove that $A$ is commutative. [i]Ion Savu[/i]

For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

A cyclic quadrilateral $ABCD$ is given. The lines $AB$ and $DC$ meet at point $E$, and the lines $BC$ and $AD$ meet at point $F$. Let $I$ be the incenter of triangle $AED$, and a ray with origin $F$ be perpendicular to the bisector of angle AID. In which ratio this ray dissects the angle $AFB$?

Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.

Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$ b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$

[u]Round 1[/u] [b]p1.[/b] Alice and Bob played $25$ games of rock-paper-scissors. Alice played rock $12$ times, scissors $6$ times, and paper $7$ times. Bob played rock $13$ times, scissors $9$ times, and paper $3$ times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.) [b]p2.[/b] On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were $30$ eruptions. How many volcanoes could erupt this year? [b]p3.[/b] A tangle is a sequence of digits constructed by picking a number $N\ge 0$ and writing the integers from $0$ to $N$ in some order, with no spaces. For example, $010123459876$ is a tangle with $N = 10$. A palindromic sequence reads the same forward or backward, such as $878$ or $6226$. The shortest palindromic tangle is $0$. How long is the second-shortest palindromic tangle? [b]p4.[/b] Balls numbered $1$ to $N$ have been randomly arranged in a long input tube that feeds into the upper left square of an $8 \times 8$ board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from $1$ to $N$ in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number $N$ for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. [img]https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png[/img] [b]p5.[/b] A $2018 \times 2018$ board is covered by non-overlapping $2 \times 1$ dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? [img]https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png[/img] [u]Round 2[/u] [b]p6.[/b] Seventeen teams participated in a soccer tournament where a win is worth $1$ point, a tie is worth $0$ points, and a loss is worth $-1$ point. Each team played each other team exactly once. At least $\frac34$ of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament. [b]p7.[/b] The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are $2018$ influential secret societies. Show that it is possible to form a council of at most $11$ people such that each influential secret society has at least one member on the council. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?

Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

The following three squares are inscribed within each other such that they all share the same center, and the largest and smallest squares have parallel sides. If the largest square has side length $17$ and the middle square has side length $13,$ the side length of the smallest square can be expressed in the form $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/c/e/86948ff8c3941fa125784a1ca0d53ac769b169.png[/img] [/center]

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$ a) Prove that $a_1+a_2+a_3$ is even. b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.

Let $a,b,c$ be positive reals. Prove that \[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \] [i]Calvin Deng.[/i]

(B.Frenkin) An inscribed and circumscribed $ n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $ n$.

Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$.

Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.

In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.