Which of the following are true? $\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$ $\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$ $\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$ $\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

Let $ABCD$ be a cyclic quadrilateral with center $O$ with $AB > CD$ and $BC > AD$. Let $M$ and $N$ be the midpoint of sides $AD$ and $BC$, respectively, and let $X$ and $Y$ be on $AB$ and $CD$, respectively, such that $AX \cdot CY = BX \cdot DY = 20000$, and $AX \le CY$. Let lines $AD$ and $BC$ hit at $P$, and let lines $AB$ and $CD$ hit at $Q$. The circumcircles of $\triangle MNP$ and $\triangle XYQ$ hit at a point $R$ that is on the opposite side of $CD$ as $O$. Let $R_1$ be the midpoint of $PQ$ and $B$, $D$, and $R$ be collinear. Let $O_1$ be the circumcenter of $\triangle BPQ$. Let the lines $BO_1$ and $DR_1$ intersect at a point $I$. If $BP \cdot BQ = 823875$, $AB=429$, and $BC=495$, then $IR=\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a,c) = 1$. Find the value of $a+b+c$. [i]Proposed by Kevin Zhao[/i]

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$ .

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD, AB$ has length $1,$ and $CD$ has length $41.$ Let points $X$ and $Y$ lie on sides $AD$ and $BC,$ respectively, such that $XY$ is parallel to $AB$ and $CD,$ and $XY$ has length $31.$ Let $m$ and $n$ be two relatively prime positive integers such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $\tfrac{m}{n}.$ Find $m+2n.$

Last Sunday at noon the date on the calendar was 15 (April 15, 2007). What will be the date on the calendar one million minutes after that time?

Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have \[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]

Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$

Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.

[list=a] [*]Determine all real numbers $x{}$ satisfying $\lfloor x\rfloor^2-x=-0.99$. [*]Prove that if $a\leqslant -1$, the equation $\lfloor x\rfloor^2-x=a$ does not have real solutions. [/list]

Consider the addition $\begin{tabular}{cccc} & O & N & E \\ + & T & W & O \\ \hline F & O & U & R \\ \end{tabular}$ where different letters represent different nonzero digits. What is the smallest possible value of the four-digit number $FOUR$?

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? [asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy] $\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$

Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$ .

Find the minimum value of the following definite integral. $ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$

Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

[b]a)[/b] Let $ B,A $ be two subsets of a finite group $ G $ such that $ |A|+|B|>|G| . $ Show that $ G=AB. $ [b]b)[/b] Show that the cyclic group of order $ n+1 $ is the product of the sets $ \{ 0,1,2,\ldots ,m \} $ and $ \{ m,m+1,m+2,\ldots ,n\} , $ where $ 0,1,2,\ldots n $ are residues modulo $ n+1 $ and $ m\le n. $

Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.

A [i]form [/i] is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An [i]angle [/i] in a [i]form [/i] consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a [i]form [/i] of an [i]angle [/i] is the number of boxes it contains. Find the maximum number of [i]angles [/i] of size $11$ in a form of size $400$. [url=http://www.oma.org.ar/enunciados/omr20.htm]source[/url]

Let $F$ be the collection of subsets of a set with $n$ elements such that no element of $F$ is a subset of another of its elements. Prove that $$|F|\le\binom n{\lfloor n/2\rfloor}.$$

Given a natural number $k> 1$. Prove that if through any edge of the graph $G$ passes less than $[e(k-1)! - 1]$ simple cycles, then the vertices of this graph can be colored with $k$ colors in the correct way.

Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?

Find all triples $(p,q,n)$ that satisfy \[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\] where $p,q$ are odd primes and $n$ is an positive integer.

Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.