Found problems: 85335
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that
\[ A^2B+BA^2=2ABA \]
Prove there exists $k\in \mathbb{N}$ such that
\[ (AB-BA)^k=\mathbf{0}_n\]
Here $\mathbf{0}_n$ is the null matrix of order $n$.
Find the number of ordered integer triplets $ x, y, z $ with absolute value less than or equal to 100 such that $ 2x^2 + 3y^2 + 3z^2 + 2xy + 2xz - 4yz < 5 $.
Let $ABCD$ be a cyclic quadrilateral (a quadrilateral which can be inscribed in a circle). Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = \frac{C}{FD}$. Let $P$ be the point on the segment $EF$ such that $\frac{PE}{PF} = \frac{AB}{CD}$. Prove that the ratio between the areas of triangle $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
Equilateral triangle $DEF$ is inscribed inside equilateral triangle $ABC$ such that $DE$ is perpendicular to $BC$. Let $x$ be the area of triangle $ABC$ and $y$ be the area of triangle $DEF$. Compute $\tfrac{x}{y}$.
John has a standard four-sided die. Each roll, he gains points equal to the value of the roll multiplied by the number of times he has now rolled that number; for example, if his first rolls were $3,3,2,3$, he would have $3+6+2+9=20$ points. Find the expected number of points John will have after rolling the die 25 times.
Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.)
[i]Proposed by David Altizio[/i]
Given that $0 < a < b < c < d$, which of the following is the largest?
$\textbf{(A)}\ \frac{a\plus{}b}{c\plus{}d} \qquad
\textbf{(B)}\ \frac{a\plus{} d}{b\plus{} c} \qquad
\textbf{(C)}\ \frac{b\plus{} c}{a\plus{}d}\qquad
\textbf{(D)}\ \frac{b\plus{} d}{a\plus{} c} \qquad
\textbf{(E)}\ \frac{c\plus{} d}{a\plus{}b}$
Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.
For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$?
[i]Adapted by Dan Schwarz from A.M.M.[/i]
Find the all $(m,n)$ integer pairs satisfying $m^4+2n^3+1=mn^3+n$.
A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$,
$$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$, showing all necessary details in your derivation.
A child has a set of $96$ distinct blocks. Each block is one of $2$ materials ([i]plastic, wood[/i]), $3$ sizes ([i]small, medium, large[/i]), $4$ colors ([i]blue, green, red, yellow[/i]), and $4$ shapes ([i]circle, hexagon, square, triangle[/i]). How many blocks in the set are different from the "[i]plastic medium red circle[/i]" in exactly two ways? (The "[i]wood medium red square[/i]" is such a block.)
$ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 39 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 62 $
We consider $5 \times 5$ tables containing a real number in each of the $25$ cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is [i]balanced [/i] if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell is called [i]small [/i] if the number in that cell is strictly smaller than the number in the cell in the very middle of the table. What is the least number of small cells that a balanced table can have?
Find the last three digits of the number $2003^{{2002}^{2001}}$.
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
For the equation $ \frac {1 \plus{} x}{1 \minus{} x} \equal{} \frac {N \plus{} 1}{N}$ to be true where $ N$ is positive, $ x$ can have:
$ \textbf{(A)}\ \text{any positive value less than }1 \qquad\textbf{(B)}\ \text{any value less than }1$
$ \textbf{(C)}\ \text{the value zero only} \qquad\textbf{(D)}\ \text{any non \minus{} negative value} \qquad\textbf{(E)}\ \text{any value}$
For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 21\qquad
\textbf{(C)}\ 27\qquad
\textbf{(D)}\ 33\qquad
\textbf{(E)}\ 34$
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is
$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 36$
Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.
[i]Proposed by Tony Kim[/i]
Let $AB$ be a segment of length $2$. The locus of points $P$ such that the $P$-median of triangle $ABP$ and its reflection over the $P$-angle bisector of triangle $ABP$ are perpendicular determines some region $R$. Find the area of $R$.
Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.
Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.
Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$.
(Dan Schwarz)
Let $A = \{1,2,3, . . . ,11\}$. How many subsets $B$ of $A$ are there, such that for each $n\in \{1,2, . . . ,8\}$, if $n$ and $n+2$ are in $B$ then at least one of the numbers $ n+1$ and $n+3$ is also in $B$?