Using the letters $ A$, $ M$, $ O$, $ S$, and $ U$, we can form $ 120$ five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $ USAMO$ occupies position $ \textbf{(A)}\ 112 \qquad \textbf{(B)}\ 113 \qquad \textbf{(C)}\ 114 \qquad \textbf{(D)}\ 115 \qquad \textbf{(E)}\ 116$

Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?

Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right triangle), cutting it along the altitude to its hypotenuse and randomly discarding one of the two pieces once again, and continues doing this forever. As the number of iterations of this process approaches infinity, the total length of the cuts made in the paper approaches a real number $l$. Compute $[\mathbb{E}(l)]^2$, that is, the square of the expected value of $l$. [i]Proposed by Matthew Kroesche[/i]

From a two-digit number $ N$ we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then: $ \textbf{(A)}\ {N}\text{ cannot end in 5}\qquad\\ \textbf{(B)}\ {N}\text{ can end in any digit other than 5}\qquad \\ \textbf{(C)}\ {N}\text{ does not exist}\qquad \\ \textbf{(D)}\ \text{there are exactly 7 values for }{N}\qquad \\ \textbf{(E)}\ \text{there are exactly 10 values for }{N}$

Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.

For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 21$

Let $a,b,c>0.$ If $\frac 1a,\frac 1b,\frac 1c$ are in arithmetic progression, and if $a^2+b^2,b^2+c^2,c^2+a^2$ are in geometric progression, show that $a=b=c.$

$(1+11+21+31+41)+(9+19+29+39+49)=$ $\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 \qquad \text{(E)}\ 250$

Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$

Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$. Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.

Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.

A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$. Show that its area, disregarding sign, is $$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$ where $\Delta$ is the discriminant of the matrix $$M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.$$

In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? [asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy] $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.

Given $1000$ squares on the plane with their sides parallel to the coordinate axes. Let $M$ be the set of those squares centres. Prove that you can mark some squares in such a way, that every point of $M$ will be contained not less than in one and not more than in four marked squares

Let $ABC$ be a triangle with symmedian point $K$, and let $\theta = \angle AKB-90^{\circ}$. Suppose that $\theta$ is both positive and less than $\angle C$. Consider a point $K'$ inside $\triangle ABC$ such that $A,K',K,$ and $B$ are concyclic and $\angle K'CB=\theta$. Consider another point $P$ inside $\triangle ABC$ such that $K'P\perp BC$ and $\angle PCA=\theta$. If $\sin \angle APB = \sin^2 (C-\theta)$ and the product of the lengths of the $A$- and $B$-medians of $\triangle ABC$ is $\sqrt{\sqrt{5}+1}$, then the maximum possible value of $5AB^2-CA^2-CB^2$ can be expressed in the form $m\sqrt{n}$ for positive integers $m,n$ with $n$ squarefree. Compute $100m+n$. [i]Proposed by Vincent Huang[/i]

Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$. a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above. b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.

Prove that there exists an infinite sequence of positive integers $ a_1, a_2, ... $ such that for all positive integers $ i $, \\ i) $ a_{i + 1} $ is divisible by $ a_{i} $.\\ ii) $ a_i $ is not divisible by $ 3 $.\\ iii) $ a_i $ is divisible by $ 2^{i + 2} $ but not $ 2^{i + 3} $.\\ iv) $ 6a_i + 1 $ is a prime power.\\ v) $ a_i $ can be written as the sum of the two perfect squares.

For every n = 2; 3; : : : , we put $$A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$ Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that: \[ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) \]

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

Let three circles $ \Gamma_1, \Gamma_2, \Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $ P$ in the plane, outside the three circles, construct six points $ A_1, B_1, A_2, B_2, A_3, B_3$ as follows: For each $ i \equal{} 1, 2, 3$, $ A_i, B_i$ are distinct points on the circle $ \Gamma_i$ such that the lines $ PA_i$ and $ PB_i$ are both tangents to $ \Gamma_i$. Call the point $ P$ exceptional if, from the construction, three lines $ A_1B_1, A_2 B_2, A_3 B_3$ are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.

Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$? [asy] size(12cm); draw((5,10)--(5,6.7),dashed+gray+linewidth(.5)); draw((5,3)--(5,5.3),dashed+gray+linewidth(.5)); filldraw((1.5,3)--(8.5,3)--(10,0)--(0,0)--cycle,lightgray); draw((0,0)--(10,0)--(5,10)--cycle,linewidth(1.3)); dot((0,0)); dot((5,10)); dot((10,0)); label(scale(.8)*"$11$", (5,6.5),S); dot((17.5,0)); dot((27.5,0)); dot((22.5,10)); draw((22.5,1.3)--(22.5,0),dashed+gray+linewidth(.5)); draw((22.5,2.5)--(22.5,3.6),dashed+gray+linewidth(.5)); draw((17.5,0)--(27.5,0)--(22.5,10)--cycle,linewidth(1.3)); filldraw((19.3,3.6)--(25.7,3.6)--(22.5,10)--cycle,lightgray); label(scale(.8)*"$5$", (22.5,1.9)); draw((5,10)--(22.5,10),dashed+gray+linewidth(.5)); draw((10,0)--(17.5,0),dashed+gray+linewidth(.5)); draw((13.75,4.3)--(13.75,0),dashed+gray+linewidth(.5)); draw((13.75,5.7)--(13.75,10),dashed+gray+linewidth(.5)); label(scale(.8)*"$h$", (13.75,5)); label(scale(.7)*"$A$", (0,0), S); label(scale(.7)*"$C$", (10,0), S); label(scale(.7)*"$B$", (5,10), N); label(scale(.7)*"$A$", (17.5,0), S); label(scale(.7)*"$C$", (27.5,0), S); label(scale(.7)*"$B$", (22.5,10), N); [/asy] $\textbf{(A) } 14.6 \qquad \textbf{(B) } 14.8 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 15.2 \qquad \textbf{(E) } 15.4$

The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 9/2 \qquad \textbf{(C)}\ 27/2 \qquad \textbf{(D)}\ \frac{9\sqrt3}{2} \qquad \textbf{(E)}\ \text{none of these}$