You are given a deck of $n \cdot m$ different cards where $n$ and $m$ are fixed numbers both between $10^{99}$ and $10^{100}$. You perform an [i]$m$-perfect shuffle[/i] for some times. In a single $m$-perfect shuffle, you divide the deck into $m$ piles with $n$ consecutive cards in each pile. You take one card from each pile, in order of the piles, for $n$ times to form the new deck. (The $m$-perfect shuffle is deterministic) For example, if the cards are labeled 12345678 where $n=4$ and $m=2$, you divide the deck into 1234 and 5678, and after one $2$-perfect shuffle you get 15263748. In another example, if the cards are labeled 123456789 where $n=3$ and $m=3$, you divide the deck into 123, 456, and 789, and after one $3$-perfect shuffle you get 147258369. Find an algorithm that, in at most $k$ steps, outputs the smallest positive number of $m$-perfect shuffle after which the deck is exactly the same as the original deck. In each step, you can do one arithmetic operation in $\{+, -, *, /, \bmod\}$, do one comparison, break out of a loop, or store one number to a specific location of an array. You can use the following precomputed numbers of steps in your solution: [list] [*] Checking if $a$ divides $b$ for any two integers $a$ and $b$ takes 2 steps because you need to compute $b \bmod a$ then compare with $0$. [*]A loop over $k$ iterations takes $2k$ steps because you need to increment the loop index by $1$ $k$ times and check the loop guard $k$ times. [*]Simulating one "$m$-perfect shuffle" takes $7nm$ steps because there is one loop index increment, four arithmetic operations, and one store in each iteration of the loop. [/list] [b]Scoring:[/b] An algorithm that completes in at most $k$ steps will be awarded: [list] [*] 1 pt for $k > 10^{10^{10^{10}}}$ [*] 10 pts for $k = 10^{10^{10^{10}}}$ [*] 20 pts for $k = 10^{420}$ [*] 30 pts for $k = 10^{360}$ [*] 50 pts for $k = 10^{240}$ [*] 70 pts for $k = 10^{202}$ [*] 80 pts for $k = 10^{201}$ [*] 95 pts for $k = 10^{120}$ [*] 98 pts for $k = 10^{102}$ [*] 100 pts for $k = 10^{101}$ [/list] [i]Proposed by Mingkuan Xu[/i]

Ten coins are placed in a circle, showing “heads” (the tails are down). Two moves are allowed: (a) turn over four consecutively placed coins; (b) turn over four coins placed as $XX OXX$ ($X$ is one of the coins to be turned over, $O$ is not touched). Is it possible to have all ten coins showing “tails” after a finite sequence of such moves? (A Tolpygo)

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?

Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column. a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares. b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue. [i]Adriana & Lucian Dragomir[/i]

Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a+b+c+d$.

Maria was a brilliant mathematician who found the following property about her year of birth: if $f$ is a function defined in the set of natural numbers $N = \{0, 1, 2, 3, 4, 5,...\}$ such that $f(1) = 1335$ and $f(n+1) = f(n)-2n+43$ for all $n \in N$, then his year of birth is the maximum value that $f(n)$ can reach when $n$ takes values in $N$. Determine the year of birth of Mary.

Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC}$, respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two non square rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD$. Find $\frac{AE}{EB} + \frac{EB}{AE}$.

Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.

Let $r,g,b$ be non negative integers and $\Gamma$ be a connected graph with $r+g+b+1$ vertices. Its edges are colored in red green and blue. It turned out that $\Gamma $ contains A spanning tree with exactly $r$ red edges. A spanning tree with exactly $g$ green edges. A spanning tree with exactly $b$ blue edges. Prove that $\Gamma$ contains a spanning tree with exactly $r$ red edges, $g$ green edges and $b$ blue edges.

Pikachu, Charmander, and Vulpix are three of the four equally-skilled players in a Pokemon bracket tournament. Because they are equally skilled, whenever any two of the players battle, they are equally likely to win. In the bracket tournament, the four players are randomly paired into two rounds, each round consisting of two players. The winners of the first two rounds then play each other in the final round. The winner of the final match ranks first; the loser of the final round ranks second; and the two losers of the previous rounds jointly rank third. What is the probability that Charmander plays Vulpix in a round, but ranks lower than Pikachu? $\textbf{(A) }\dfrac1{24}\qquad\textbf{(B) }\dfrac18\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) } \dfrac38 \qquad \textbf{(E) } \dfrac12$

Given the equation $\frac{ax^2-24x+b}{x^2-1} = x$. Find all the real numbers $a$ and $b$ for which you have two real solutions whose sum is equal to $12$.

Find the least positive integer $n$ greater than $1$ such that $n^3 -n^2$ is divisible by $7^2 \times 11$. [i]Proposed by Jacob Xu[/i]

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of \[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]

Let $\Omega$ be the circumcircle of triangle $ABC$ such that the tangents to $\Omega$ in points $B$ and $C$ intersect in $P$. The squares $ABB_1B_2$ and $ACC_1C_2$ are constructed on the sides $AB$ and $AC$ in the exterior of triangle $ABC$, such that the lines $B_1B_2$ and $C_1C_2$ intersect in point $Q$. Prove that $P$, $A$, and $Q$ are collinear.

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space? [i]Alexey Tolpygo[/i]

Find the coordinates of the vertices of a square $ABCD$, knowing that $A$ is on the line $y -2x -6 = 0$, $C$ at $x = 0$ and $B$ is the point $(a, 0)$ , being $a = \log_{2/3}(16/81)$.

A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touches $\omega$ externally and meets the sides of the angle at points $A$ and $B$. Prove that the perimeters of all triangles $ABC$ are equal.

In a class of $14$ boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers $0, 1, 2, 3, 4, 5$ and $6$. Proves that there are two colleagues with the same first name and the same last name

The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.