Determine the largest number of steps for $\gcd(k,76)$ to terminate over all choices of $0 < k < 76$, using the following algorithm for gcd. Give your answer in the form $(n,k)$ where $n$ is the maximal number of steps and $k$ is the $k$ which achieves this. If multiple $k$ work, submit the smallest one. \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{gcd}(a,b)$: \\ 2: $\qquad$ \textbf{IF} $a = 0$ \textbf{RETURN} $b$ \\ 3: $\qquad$ \textbf{ELSE RETURN} $\text{gcd}(b \bmod a,a)$ \end{tabular}

What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

For how many real numbers $x$ is $\sqrt{-(x+1)^2}$ a real number? $\textbf{(A) }\text{none}\qquad\textbf{(B) }\text{one}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{a finite number greater than two}\qquad \textbf{(E) }\text{infinitely many}$

[u]Linear? What's The Problem?[/u] A function $f(x_1, x_2,..., x_n)$ is said to be linear in each of its variables if it is a polynomial such that no variable appears with power higher than one in any term. For example, $1 + x + xy$ is linear in $x$ and $y$, but $1 + x^2$ is not. Similarly, $2x + 3yz$ is linear in $x$, $y$, and $z$, but $xyz^2$ is not. [b]p8.[/b] A function $f(x,y)$ is linear in $x$ and in $y$. $f(x,y) =\frac{1}{xy}$ for $x,y \in \{3, 4\}$. What is $f(5,5)$? [b]p9.[/b] A function $f(x, y,z)$ is linear in $x$, $y$, and $z$ such that $f(x,y, z) = \frac{1}{xyz}$ for $x,y,z \in \{3,4\}$. What is $f(5, 5, 5)$? [b]p10.[/b] A function $f(x_1, x_2,..., x_n)$ is linear in each of the $x_i$ and $f(x_1, x_2,..., x_n)= \frac{1}{x_1x_2...x_n}$ when $x_i \in \{3,4\}$ for all $ i$. In terms of $n$, what is $f(5,5,...,5)$?

Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.

The longer base of trapezoid $ABCD$ is $AB$, while the shorter base is $CD$. Diagonal $AC$ bisects the interior angle at $A$. The interior bisector at $B$ meets diagonal $AC$ at $E$. Line $DE$ meets segment $AB$ at $F$. Suppose that $AD = FB$ and $BC = AF$. Find the interior angles of quadrilateral $ABCD$, if we know that $\angle BEC = 54^o$.

Let $ABCD$ be a convex quadrilateral, $I_1$ and $I_2$ be the incenters of triangles $ABC$ and $DBC$ respectively. The line $I_1I_2$ intersects the lines $AB$ and $DC$ at points $E$ and $F$ respectively. Given that $AB$ and $CD$ intersect in $P$, and $PE=PF$, prove that the points $A$, $B$, $C$, $D$ lie on a circle.

Points $R, S$ and $T$ are vertices of an equilateral triangle, and points $X, Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? [asy] pair SS,R,T,X,Y,Z; SS = (2,2*sqrt(3)); R = (0,0); T = (4,0); X = (2,0); Y = (1,sqrt(3)); Z = (3,sqrt(3)); dot(SS); dot(R); dot(T); dot(X); dot(Y); dot(Z); label("$S$",SS,N); label("$R$",R,SW); label("$T$",T,SE); label("$X$",X,S); label("$Y$",Y,NW); label("$Z$",Z,NE);[/asy] $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 20 $

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.

Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\ x + y+ z = 2b \\ x^2 + y^2-z^2 = b^2 \end{cases}$ in $C$

Let $S = \{1, 2, \cdots , n\}, n \in N$. Find the number of functions $f : S \to S$ with the property that $x + f(f(f(f(x)))) = n + 1$ for all $x \in S$?

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary tangential polygon. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.