Found problems: 233
Thaddeus is given a $2013 \times 2013$ array of integers each between $1$ and $2013$, inclusive. He is allowed two operations:
1. Choose a row, and subtract $1$ from each entry.
2. Chooses a column, and add $1$ to each entry.
He would like to get an array where all integers are divisible by $2013$. On how many arrays is this possible?
Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*}
& f(g(h(x))) = h(g(f(x))) = g(x) \\
& g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\
& h(f(g(x))) = g(f(h(x))) = f(x), \\
\end{align*} for all $x\in \{1,2,3,4,5\}$.
Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$. Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\]
Let $ABCDEF$ be a regular hexagon of area $1$. Let $M$ be the midpoint of $DE$. Let $X$ be the intersection of $AC$ and $BM$, let $Y$ be the intersection of $BF$ and $AM$, and let $Z$ be the intersection of $AC$ and $BF$. If $[P]$ denotes the area of a polygon $P$ for any polygon $P$ in the plane, evaluate $[BXC] + [AYF] + [ABZ] - [MXZY]$.
For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
(By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)
The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.
Let $a \star b = ab + a + b$ for all integers $a$ and $b$. Evaluate $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$.
How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.
Let $p > 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo 5.
Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called [i]compositional[/i] if
\[f(g(x)) \equiv x^{p^2} - x\]
in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs.
Let $\omega_1$ and $\omega_2$ be two circles that intersect at points $A$ and $B$. Let line $l$ be tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ such that $A$ is closer to $PQ$ than $B$. Let points $R$ and $S$ lie along rays $PA$ and $QA$, respectively, so that $PQ = AR = AS$ and $R$ and $S$ are on opposite sides of $A$ as $P$ and $Q$. Let $O$ be the circumcenter of triangle $ASR$, and $C$ and $D$ be the midpoints of major arcs $AP$ and $AQ$, respectively. If $\angle APQ$ is $45$ degrees and $\angle AQP$ is $30$ degrees, determine $\angle COD$ in degrees.
Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest
positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
Compute \[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\]
(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A,B,\cdots,Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within $20$ years of each other, then your score is $0$. Otherwise, your score is $\max(3(k-3),0)$ where $k$ is the number of elements in your set.
\[\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}\]
Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.
Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus finds himself back at the point $(0, 0)$. He never visited any point other than $(0, 0)$ more than once, and never visited the point $(0, 0)$ except at the start and end of this sequence of moves.
Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X - Y| = 1$.
[i]Mitchell Lee[/i]
Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.
19. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ ($1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute $\sum_{s\in S}\frac1s$.
20. Let $\mathcal{V}$ be the volume enclosed by the graph
$$x^ {2016} + y^{2016} + z^2 = 2016$$
Find $\mathcal{V}$ rounded to the nearest multiple of ten.
21. Zlatan has $2017$ socks of various colours. He wants to proudly display one sock of each of the
colours, and he counts that there are $N$ ways to select socks from his collection for display. Given this
information, what is the maximum value of $N$?
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.
Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^3+5b^3 \neq 2016$.
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]