How many different positive integers divide $10!$ ?

In the below diagram, the three rectangles are similar. Find the area of rectangle $ABCD$. [asy] size(6cm); draw((0,0)--(10,0)--(10,8.16496580928)--(0,8.16496580928)--(0,0)); draw((2,0)--(6,3.26598632371)--(2,8.16496580928)--(-2,4.89897949)--(2,0)); draw((10,3.26598632371)--(6,3.26598632371)--(6,0)); label("$A$",(0,8.16496580928), NW); label("$D$",(0,0), SW); label("$C$",(10,0), SE); label("$B$",(10,8.16496580928), NE); label("$E$",(2,8.16496580928), N); label("$F$",(2,0), S); label("$3$",(1,0),S); label("$3$",(1,8.16496580928),N); label("$12$",(6,8.16496580928),N); [/asy] $\textbf{(A) }75\sqrt{3}\qquad\textbf{(B) }120\sqrt{2}\qquad\textbf{(C) }100\sqrt{3}\qquad\textbf{(D) }180\qquad\textbf{(E) }75\sqrt{6}$

For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$ (S. Manukian, Yerevan)

The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .

It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that: $i)$ sum of lengths of all diameters of all circles is not greater than $2018$ $ii)$ distance between any two circles is greater than $1$

Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None of above} $

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that \[ \frac {z}{z \plus{} n} \equal{} 4i. \]Find $ n$.

For a positive integer $n$, there is a school with $n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is $k$, show that the maximum number of well-formed sets is not greater than $3^{(n+k)/3}$. Here, an empty set and a set with one student is regarded as well-formed as well.

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

Find the number of nonnegative integers $k$, $0 \leq k \leq 2188$, and such that $\binom{2188}{k}$ is divisible by 2188.

Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that: a) the points $A_1$, $B_1$, $C_1$ are collinear; b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$. [i]Proposed by Dominic Yeo, United Kingdom[/i]