HMMT 2009

If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$.

Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\log _{2}(S)$ ?

If $\tan x+\tan y=4$ and $\cot x+\cot y=5$, compute $\tan (x+y)$.

Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$.

Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$.

Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2} n$ for any integer $n$. Suppose

$$\frac{\sin \theta}{x}=\frac{\cos \theta}{y}$$

and

$$\frac{\cos ^{4} \theta}{x^{4}}+\frac{\sin ^{4} \theta}{y^{4}}=\frac{97 \sin 2 \theta}{x^{3} y+y^{3} x}$$

Compute $\frac{x}{y}+\frac{y}{x}$.

Simplify the product

$$\prod_{m=1}^{100} \prod_{n=1}^{100} \frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}$$

Express your answer in terms of $x$.

If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$.

Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $\left|z_{a} z_{b}+z_{c} z_{d}\right|$ where $\{a, b, c, d\}=\{1,2,3,4\}$.

Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle?

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.

Circle $B$ has radius $6 \sqrt{7}$. Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$. Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $X Y$ is also tangent to $A$. Find the area contained by the boundary of $L$.

Let $A B C$ be a triangle in the coordinate plane with vertices on lattice points and with $A B=1$. Suppose the perimeter of $A B C$ is less than 17 . Find the largest possible value of $1 / r$, where $r$ is the inradius of $A B C$.

In $\triangle A B C, D$ is the midpoint of $B C, E$ is the foot of the perpendicular from $A$ to $B C$, and $F$ is the foot of the perpendicular from $D$ to $A C$. Given that $B E=5, E C=9$, and the area of triangle $A B C$ is 84 , compute $|E F|$.

Triangle $A B C$ has side lengths $A B=231, B C=160$, and $A C=281$. Point $D$ is constructed on the opposite side of line $A C$ as point $B$ such that $A D=178$ and $C D=153$. Compute the distance from $B$ to the midpoint of segment $A D$.

Let $A B C$ be a triangle with $A B=16$ and $A C=5$. Suppose the bisectors of angles $\angle A B C$ and $\angle B C A$ meet at point $P$ in the triangle's interior. Given that $A P=4$, compute $B C$.

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $A B$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $A B=7, B C=2$, and $A D=3$, compute $B P$.

How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing?

Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers?

In how many ways can you rearrange the letters of "HMMTHMMT" such that the consecutive substring "HMMT" does not appear?

How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ satisfy $f(f(x))=f(x)$ for all $x \in\{1,2,3,4,5\}$ ?

Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute

$$\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)}$$

How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ?

Paul fills in a $7 \times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?

There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

The squares of a $3 \times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelings are possible?

Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutive elements?

Let $f$ be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$ ?

The differentiable function $F: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $F(0)=-1$ and

$$\frac{d}{d x} F(x)=\sin (\sin (\sin (\sin (x)))) \cdot \cos (\sin (\sin (x))) \cdot \cos (\sin (x)) \cdot \cos (x)$$

Find $F(x)$ as a function of $x$.

Compute $e^{A}$ where $A$ is defined as

$$\int_{3 / 4}^{4 / 3} \frac{2 x^{2}+x+1}{x^{3}+x^{2}+x+1} d x$$

Let $P$ be a fourth degree polynomial, with derivative $P^{\prime}$, such that $P(1)=P(3)=P(5)=P^{\prime}(7)=0$. Find the real number $x \neq 1,3,5$ such that $P(x)=0$.

Compute

$$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}$$

Let $p_{0}(x), p_{1}(x), p_{2}(x), \ldots$ be polynomials such that $p_{0}(x)=x$ and for all positive integers $n$, $\frac{d}{d x} p_{n}(x)=p_{n-1}(x)$. Define the function $p(x):[0, \infty) \rightarrow \mathbb{R} x$ by $p(x)=p_{n}(x)$ for all $x \in[n, n+1]$. Given that $p(x)$ is continuous on $[0, \infty)$, compute

$$\sum_{n=0}^{\infty} p_{n}(2009)$$

A line in the plane is called strange if it passes through $(a, 0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points?

Compute

$$\int_{1}^{\sqrt{3}} x^{2 x^{2}+1}+\ln \left(x^{2 x^{2 x^{2}+1}}\right) d x$$

let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^{2}$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$.

Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate

$$\int_{0}^{2 \pi} \frac{1}{a+b \cos (\theta)} d \theta$$

Express your answer in terms of $a$ and $b$.

Suppose $N$ is a 6-digit number having base-10 representation $\underline{a} \underline{b} \underline{c} \underline{d} \underline{e} \underline{f}$. If $N$ is $6 / 7$ of the number having base-10 representation $\underline{d} \underline{e} \underline{f} \underline{a} \underline{b} \underline{c}$, find $N$.

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.

Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

A kite is a quadrilateral whose diagonals are perpendicular. Let kite $A B C D$ be such that $\angle B=$ $\angle D=90^{\circ}$. Let $M$ and $N$ be the points of tangency of the incircle of $A B C D$ to $A B$ and $B C$ respectively. Let $\omega$ be the circle centered at $C$ and tangent to $A B$ and $A D$. Construct another kite $A B^{\prime} C^{\prime} D^{\prime}$ that is similar to $A B C D$ and whose incircle is $\omega$. Let $N^{\prime}$ be the point of tangency of $B^{\prime} C^{\prime}$ to $\omega$. If $M N^{\prime} \| A C$, then what is the ratio of $A B: B C$ ?

How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing?

How many ways can you tile the white squares of following $2 \times 24$ grid with dominoes? (A domino covers two adjacent squares, and a tiling is a non-overlapping arrangement of dominoes that covers every white square and does not intersect any black square.)

A torus (donut) having inner radius 2 and outer radius 4 sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $x-y$ plane is the table, the torus is formed by revolving the circle in the $x-z$ plane centered at $(3,0,1)$ with radius 1 about the $z$ axis. The spherical ball has its center on the $z$-axis and rests on either the table or the donut.)

Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$.

Let $F_{n}$ be the Fibonacci sequence, that is, $F_{0}=0, F_{1}=1$, and $F_{n+2}=F_{n+1}+F_{n}$. Compute $\sum_{n=0}^{\infty} F_{n} / 10^{n}$.

The incircle $\omega$ of equilateral triangle $A B C$ has radius 1. Three smaller circles are inscribed tangent to $\omega$ and the sides of $A B C$, as shown. Three smaller circles are then inscribed tangent to the previous circles and to each of two sides of $A B C$. This process is repeated an infinite number of times. What is the total length of the circumferences of all the circles?

How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ?

Let $T$ be a right triangle with sides having lengths 3,4 , and 5 . A point $P$ is called awesome if $P$ is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?

(a) [6] Let $P$ be a graph with one vertex $v_{n}$ for each positive integer $n$. If $a<b$, then an edge connects vertices $v_{a}$ and $v_{b}$ if and only if $\frac{b}{a}$ is a prime number. What is the chromatic number of $P$ ? Prove your answer.

A graph is finite if it has a finite number of vertices. (a) $[6]$ Let $G$ be a finite graph in which every vertex has degree $k$. Prove that the chromatic number of $G$ is at most $k+1$.

A $k$-clique of a graph is a set of $k$ vertices such that all pairs of vertices in the clique are adjacent. (a) [4] Find a graph with chromatic number 3 that does not contain any 3 -cliques.

The size of a finite graph is the number of vertices in the graph. (a) [15] Show that, for any $n>2$, and any positive integer $N$, there are finite graphs with size at least $N$ and with chromatic number $n$ such that removing any vertex (and all its incident edges) from the graph decreases its chromatic number.

For any set of graphs $G_{1}, G_{2}, \ldots, G_{n}$ all having the same set of vertices $V$, define their overlap, denoted $G_{1} \cup G_{2} \cup \cdots \cup G_{n}$, to be the graph having vertex set $V$ for which two vertices are adjacent in the overlap if and only if they are adjacent in at least one of the graphs $G_{i}$. (a) $[\mathbf{1 0}]$ Let $G$ and $H$ be graphs having the same vertex set and let $a$ be the chromatic number of $G$ and $b$ the chromatic number of $H$. Find, in terms of $a$ and $b$, the largest possible chromatic number of $G \cup H$. Prove your answer.

Let $n$ be a positive integer. Let $V_{n}$ be the set of all sequences of 0 's and 1's of length $n$. Define $G_{n}$ to be the graph having vertex set $V_{n}$, such that two sequences are adjacent in $G_{n}$ if and only if they differ in either 1 or 2 places. For instance, if $n=3$, the sequences $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$ are mutually adjacent, but $(1,0,0)$ is not adjacent to $(0,1,1)$. Show that, if $n+1$ is not a power of 2 , then the chromatic number of $G_{n}$ is at least $n+2$.

Two colorings are distinct if there is no way to relabel the colors to transform one into the other. Equivalently, they are distinct if and only if there is some pair of vertices which are the same color in one coloring but different colors in the other. For what pairs $(n, k)$ of positive integers does there exist a finite graph with chromatic number $n$ which has exactly $k$ distinct good colorings using $n$ colors?

What are the chromatic numbers of each of the three graphs shown below? Draw a coloring having the minimum number of colors for each. Label the vertices with symbols to indicate the color of each vertex. (For example, you may mark a vertex "R" if you wish to indicate that it is colored red.)

In a connected graph, it is possible to reach any vertex from any other vertex by following the edges. A tree is a connected graph with $n$ vertices and $n-1$ edges for some positive integer $n$. Suppose $n \geq 2$. What is the chromatic number of a tree having $n$ vertices? Prove your answer.

Let $n \geq 3$ be a positive integer. A triangulation of a convex $n$-gon is a set of $n-3$ of its diagonals which do not intersect in the interior of the polygon. Along with the $n$ sides, these diagonals separate the polygon into $n-2$ disjoint triangles. Any triangulation can be viewed as a graph: the vertices of the graph are the corners of the polygon, and the $n$ sides and $n-3$ diagonals are the edges. For a fixed $n$-gon, different triangulations correspond to different graphs. Prove that all of these graphs have the same chromatic number.

Let $G$ be a finite graph in which every vertex has degree less than or equal to $k$. Prove that the chromatic number of $G$ is less than or equal to $k+1$.

A $k$-clique of a graph is a set of $k$ vertices such that all pairs of vertices in the clique are adjacent. The clique number of a graph is the size of the largest clique in the graph. Does there exist a graph which has a clique number smaller than its chromatic number?

(a) [5] If a single vertex (and all its incident edges) is removed from a finite graph, show that the graph's chromatic number cannot decrease by more than 1.

Compute

$$1 \cdot 2^{2}+2 \cdot 3^{2}+3 \cdot 4^{2}+\cdots+19 \cdot 20^{2}$$

Given that $\sin A+\sin B=1$ and $\cos A+\cos B=3 / 2$, what is the value of $\cos (A-B)$ ?

Find all pairs of integer solutions $(n, m)$ to

$$2^{3^{n}}=3^{2^{m}}-1$$

Simplify: $i^{0}+i^{1}+\cdots+i^{2009}$.

In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.)

Let $A B C$ be a right triangle with hypotenuse $A C$. Let $B^{\prime}$ be the reflection of point $B$ across $A C$, and let $C^{\prime}$ be the reflection of $C$ across $A B^{\prime}$. Find the ratio of $\left[B C B^{\prime}\right]$ to $\left[B C^{\prime} B^{\prime}\right]$.

How many perfect squares divide $2^{3} \cdot 3^{5} \cdot 5^{7} \cdot 7^{9}$ ?

Which is greater, $\log _{2008}(2009)$ or $\log _{2009}(2010)$ ?

An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces. How many vertices does it have?

Let $a, b$, and $c$ be real numbers. Consider the system of simultaneous equations in variables $x$ and $y$ :

$$\begin{aligned} a x+b y & =c-1 \\ (a+5) x+(b+3) y & =c+1 \end{aligned}$$

Determine the value(s) of $c$ in terms of $a$ such that the system always has a solution for any $a$ and $b$.

There are 2008 distinct points on a circle. If you connect two of these points to form a line and then connect another two points (distinct from the first two) to form another line, what is the probability that the two lines intersect inside the circle?

Bob is writing a sequence of letters of the alphabet, each of which can be either uppercase or lowercase, according to the following two rules:

• If he had just written an uppercase letter, he can either write the same letter in lowercase after it, or the next letter of the alphabet in uppercase.
• If he had just written a lowercase letter, he can either write the same letter in uppercase after it, or the preceding letter of the alphabet in lowercase.

For instance, one such sequence is $a A a A B C D d c b B C$. How many sequences of 32 letters can he write that start at (lowercase) $a$ and end at (lowercase) $z$ ? (The alphabet contains 26 letters from $a$ to $z$.)

How many ordered quadruples $(a, b, c, d)$ of four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$ satisfy $b<a, b<c$, and $d<c$ ?

Compute

$$\sum_{k=1}^{2009} k\left(\left\lfloor\frac{2009}{k}\right\rfloor-\left\lfloor\frac{2008}{k}\right\rfloor\right)$$

Stan has a stack of 100 blocks and starts with a score of 0 , and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.) (b) Stan adds the product of the two piles' sizes, $a b$, to his score.

The game ends when there are only 1-block stacks left. What is the expected value of Stan's score at the end of the game?

A spider is making a web between $n>1$ distinct leaves which are equally spaced around a circle. He chooses a leaf to start at, and to make the base layer he travels to each leaf one at a time, making a straight line of silk between each consecutive pair of leaves, such that no two of the lines of silk cross each other and he visits every leaf exactly once. In how many ways can the spider make the base layer of the web? Express your answer in terms of $n$.

How many positive integers $n \leq 2009$ have the property that $\left\lfloor\log _{2}(n)\right\rfloor$ is odd?

If $n$ is a positive integer such that $n^{3}+2 n^{2}+9 n+8$ is the cube of an integer, find $n$.

Shelly writes down a vector $v=(a, b, c, d)$, where $0<a<b<c<d$ are integers. Let $\sigma(v)$ denote the set of 24 vectors whose coordinates are $a, b, c$, and $d$ in some order. For instance, $\sigma(v)$ contains $(b, c, d, a)$. Shelly notes that there are 3 vectors in $\sigma(v)$ whose sum is of the form $(s, s, s, s)$ for some $s$. What is the smallest possible value of $d$ ?

A positive integer is called jubilant if the number of 1's in its binary representation is even. For example, $6=110_{2}$ is a jubilant number. What is the 2009th smallest jubilant number?

Simplify $$2 \cos ^{2}(\ln (2009) i)+i \sin (\ln (4036081) i)$$

A circle having radius $r_{1}$ centered at point $N$ is tangent to a circle of radius $r_{2}$ centered at $M$. Let $l$ and $j$ be the two common external tangent lines to the two circles. A circle centered at $P$ with radius $r_{2}$ is externally tangent to circle $N$ at the point at which $l$ coincides with circle $N$, and line $k$ is externally tangent to $P$ and $N$ such that points $M, N$, and $P$ all lie on the same side of $k$. For what ratio $r_{1} / r_{2}$ are $j$ and $k$ parallel?

The roots of $z^{6}+z^{4}+z^{2}+1=0$ are the vertices of a convex polygon in the complex plane. Find the sum of the squares of the side lengths of the polygon.

Compute, in terms of $n$,

$$\sum_{k=0}^{n}\binom{n-k}{k} 2^{k}$$

Note that whenever $s<t,\binom{s}{t}=0$.

Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices?

Define the sequence $\left\{x_{i}\right\}_{i \geq 0}$ by $x_{0}=2009$ and $x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}$ for all $n \geq 1$. Compute the value of $\sum_{n=0}^{2009} 2^{n} x_{n}$.

Circle $\Omega$ has radius 5. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 6 . A unit circle $\omega$ is tangent to chord $A B$ at point $T$. Given that $\omega$ is also internally tangent to $\Omega$, find $A T \cdot B T$.

The vertices of a regular hexagon are labeled $\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\theta$ ), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge?

The average of a set of distinct primes is 27 . What is the largest prime that can be in this set?

Let $f$ be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is 1 . For any $n \in \mathbb{N}, f(n)$ is a multiple of 85 . Find the smallest possible degree of $f$.

How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$ ? (That is, how many lines pass through three of the lattice points $\left(a_{1}, \ldots, a_{n}\right)$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\} ?$ ) Express your answer in terms of $n$.

Circle $\Omega$ has radius 13. Circle $\omega$ has radius 14 and its center $P$ lies on the boundary of circle $\Omega$. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 24 and is tangent to $\omega$ at point $T$. Find $A T \cdot B T$.

Let $m$ be a positive integer. Let $d(n)$ denote the number of divisors of $n$, and define the function

$$F(x)=\sum_{n=1}^{105^{m}} \frac{d(n)}{n^{x}}$$

Define the numbers $a(n)$ to be the positive integers for which

$$F(x)^{2}=\sum_{n=1}^{105^{2 m}} \frac{a(n)}{n^{x}}$$

for all real $x$. Express $a\left(105^{m}\right)$ in terms of $m$.

Descartes's Blackjack: How many integer lattice points (points of the form $(m, n)$ for integers $m$ and $n$ ) lie inside or on the boundary of the disk of radius 2009 centered at the origin?

If your answer is higher than the correct answer, you will receive 0 points. If your answer is $d$ less than the correct answer, your score on this problem will be the larger of 0 and $25-\lfloor d / 10\rfloor$.

Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \ldots$ stands for $\sum_{n=0}^{\infty} \frac{A_{n}}{(n+1)!}$, where each $A_{n}$ is an integer between 0 and $n$, inclusive. Chebby has an infinite number of cards labeled $\mathbf{0}, \mathbf{1}, \mathbf{2}, \ldots$. He begins by putting cards $\mathbf{0}$ and $\mathbf{1}$ into a hat and drawing randomly to determine $A_{1}$. The card assigned $A_{1}$ does not get reused. Chebby then adds in card 2 and draws for $A_{2}$, and continues in this manner to determine the random number. At each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby's final number?

Your score on this problem will be the larger of 0 and $\lfloor 25(1-d)\rfloor$, where $d$ is the positive difference between your answer and the correct answer.

Euler's Bridge: The following figure is the graph of the city of Konigsburg in 1736 - vertices represent sections of the cities, edges are bridges. An Eulerian path through the graph is a path which moves from vertex to vertex, crossing each edge exactly once. How many ways could World War II bombers have knocked out some of the bridges of Konigsburg such that the Allied victory parade could trace an Eulerian path through the graph? (The order in which the bridges are destroyed matters.) Your score on this problem will be the larger of 0 and $25-\lfloor d / 10\rfloor$, where $d$ is the positive difference between your answer and the correct answer.