Two circles intersect at points $A\neq B$. A line passing through $A{}$ intersects the circles again at $C$ and $D$. Let $E$ and $F$ be the midpoints of the arcs $\overarc{BC}$ and $\overarc{BD}$ which do not contain $A{}$ and let $M$ be the midpoint of the segment $CD$. Prove that $ME$ and $MF$ are perpendicular.

Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.

Let $P(x)$ be a nonzero quadratic polynomial such that $P(1) = P(2) = 0$. Given that $P(3)^2 = P(4)+P(5)$, find $P(6)$. [i]Proposed by Andy Xu[/i]

For a positive integer $n$, $S_n$ is the set of positive integer $n$-tuples $(a_1,a_2, \cdots ,a_n)$ which satisfies the following. (i). $a_1=1$. (ii). $a_{i+1} \le a_i+1$. For $k \le n$, define $N_k$ as the number of $n$-tuples $(a_1, a_2, \cdots a_n) \in S_n$ such that $a_k=1, a_{k+1}=2$. Find the sum $N_1 + N_2+ \cdots N_{k-1}$.

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy \[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]

Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations \[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]

Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$.

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

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.