HMMT 2015

Let $Q$ be a polynomial

$$Q(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$$

where $a_{0}, \ldots, a_{n}$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

The fraction $\frac{1}{2015}$ has a unique "(restricted) partial fraction decomposition" of the form

$$\frac{1}{2015}=\frac{a}{5}+\frac{b}{13}+\frac{c}{31}$$

where $a, b, c$ are integers with $0 \leq a<5$ and $0 \leq b<13$. Find $a+b$.

Let $p$ be a real number and $c \neq 0$ an integer such that

$$c-0.1<x^{p}\left(\frac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1$$

for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p, c)$.

Compute the number of sequences of integers $\left(a_{1}, \ldots, a_{200}\right)$ such that the following conditions hold.

• $0 \leq a_{1}<a_{2}<\cdots<a_{200} \leq 202$.
• There exists a positive integer $N$ with the following property: for every index $i \in\{1, \ldots, 200\}$ there exists an index $j \in\{1, \ldots, 200\}$ such that $a_{i}+a_{j}-N$ is divisible by 203.

Let $a, b, c$ be positive real numbers such that $a+b+c=10$ and $a b+b c+c a=25$. Let $m=\min \{a b, b c, c a\}$. Find the largest possible value of $m$.

Let $a, b, c, d, e$ be nonnegative integers such that $625 a+250 b+100 c+40 d+16 e=15^{3}$. What is the maximum possible value of $a+b+c+d+e$ ?

Suppose $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ is a 4 -term sequence of real numbers satisfying the following two conditions:

• $a_{3}=a_{2}+a_{1}$ and $a_{4}=a_{3}+a_{2}$;
• there exist real numbers $a, b, c$ such that

$$a n^{2}+b n+c=\cos \left(a_{n}\right)$$

for all $n \in\{1,2,3,4\}$. Compute the maximum possible value of

$$\cos \left(a_{1}\right)-\cos \left(a_{4}\right)$$

over all such sequences $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$.

Find the number of ordered pairs of integers $(a, b) \in\{1,2, \ldots, 35\}^{2}$ (not necessarily distinct) such that $a x+b$ is a "quadratic residue modulo $x^{2}+1$ and 35 ", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

• there exist polynomials $P, Q$ with integer coefficients such that $f(x)^{2}-(a x+b)=\left(x^{2}+1\right) P(x)+$ $35 Q(x)$;
• or more conceptually, the remainder when (the polynomial) $f(x)^{2}-(a x+b)$ is divided by (the polynomial) $x^{2}+1$ is a polynomial with (integer) coefficients all divisible by 35 .

Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in\{1,2, \ldots, N\}^{4}$ (not necessarily distinct) such that for every integer $n, A n^{3}+B n^{2}+2 C n+D$ is divisible by $N$.

Find all ordered 4 -tuples of integers $(a, b, c, d)$ (not necessarily distinct) satisfying the following system of equations: $$$$ \begin{aligned} a^{2}-b^{2}-c^{2}-d^{2} & =c-b-2 \ 2 a b & =a-d-32 \ 2 a c & =28-a-d \ 2 a d & =b+c+31 \end{aligned} $$$$

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0),(2,0),(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown.

Pro selects a point $P$ uniformly at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

Let $A B C$ be a triangle with orthocenter $H$; suppose that $A B=13, B C=14, C A=15$. Let $G_{A}$ be the centroid of triangle $H B C$, and define $G_{B}, G_{C}$ similarly. Determine the area of triangle $G_{A} G_{B} G_{C}$.

Let $A B C D$ be a quadrilateral with $\angle B A D=\angle A B C=90^{\circ}$, and suppose $A B=B C=1, A D=2$. The circumcircle of $A B C$ meets $\overline{A D}$ and $\overline{B D}$ at points $E$ and $F$, respectively. If lines $A F$ and $C D$ meet at $K$, compute $E K$.

Let $A B C D$ be a cyclic quadrilateral with $A B=3, B C=2, C D=2, D A=4$. Let lines perpendicular to $\overline{B C}$ from $B$ and $C$ meet $\overline{A D}$ at $B^{\prime}$ and $C^{\prime}$, respectively. Let lines perpendicular to $\overline{A D}$ from $A$ and $D$ meet $\overline{B C}$ at $A^{\prime}$ and $D^{\prime}$, respectively. Compute the ratio $\frac{\left[B C C^{\prime} B^{\prime}\right]}{\left[D A A^{\prime} D^{\prime}\right]}$, where $[\varpi]$ denotes the area of figure $\varpi$.

Let $I$ be the set of points $(x, y)$ in the Cartesian plane such that

$$x>\left(\frac{y^{4}}{9}+2015\right)^{1 / 4}$$

Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^{2}+y^{2} \leq r^{2}$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<L r^{2}$ for all $r>0$.

In triangle $A B C, A B=2, A C=1+\sqrt{5}$, and $\angle C A B=54^{\circ}$. Suppose $D$ lies on the extension of $A C$ through $C$ such that $C D=\sqrt{5}-1$. If $M$ is the midpoint of $B D$, determine the measure of $\angle A C M$, in degrees.

Let $A B C D E$ be a square pyramid of height $\frac{1}{2}$ with square base $A B C D$ of side length $A B=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $A B C D$ is the center of square $A B C D$ ). The faces $A D E$ and $C D E$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$ ). Find $\tan \alpha$.

Let $S$ be the set of discs $D$ contained completely in the set $\{(x, y): y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^{2}-\frac{3}{4}$. What is the area of the union of the elements of $S$ ?

Let $A B C D$ be a regular tetrahedron with side length 1 . Let $X$ be the point in triangle $B C D$ such that $[X B C]=2[X B D]=4[X C D]$, where $[\varpi]$ denotes the area of figure $\varpi$. Let $Y$ lie on segment $A X$ such that $2 A Y=Y X$. Let $M$ be the midpoint of $B D$. Let $Z$ be a point on segment $A M$ such that the lines $Y Z$ and $B C$ intersect at some point. Find $\frac{A Z}{Z M}$.

Let $\mathcal{G}$ be the set of all points $(x, y)$ in the Cartesian plane such that $0 \leq y \leq 8$ and

$$(x-3)^{2}+31=(y-4)^{2}+8 \sqrt{y(8-y)}$$

There exists a unique line $\ell$ of negative slope tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a unique point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.

Evan's analog clock displays the time $12: 13$; the number of seconds is not shown. After 10 seconds elapse, it is still 12:13. What is the expected number of seconds until 12:14?

Victor has a drawer with 6 socks of 3 different types: 2 complex socks, 2 synthetic socks, and 2 trigonometric socks. He repeatedly draws 2 socks at a time from the drawer at random, and stops if the socks are of the same type. However, Victor is "synthetic-complex type-blind", so he also stops if he sees a synthetic and a complex sock.

What is the probability that Victor stops with 2 socks of the same type? Assume Victor returns both socks to the drawer after each step.

Starting with the number 0, Casey performs an infinite sequence of moves as follows: he chooses a number from $\{1,2\}$ at random (each with probability $\frac{1}{2}$ ) and adds it to the current number. Let $p_{m}$ be the probability that Casey ever reaches the number $m$. Find $p_{20}-p_{15}$.

Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $p$ be the probability that the number of rocks left in the pile after each round is a multiple of 5 . If $p$ is of the form $5^{a} \cdot 31^{b} \cdot \frac{c}{d}$, where $a, b$ are integers and $c, d$ are positive integers relatively prime to $5 \cdot 31$, find $a+b$.

For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1 's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1 's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three 1 's. Compute $g(1)+g(2)+g(3)+\cdots+g(256)$.

Count the number of functions $f: \mathbb{Z} \rightarrow\{$ 'green', 'blue' $\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'.

2015 people sit down at a restaurant. Each person orders a soup with probability $\frac{1}{2}$. Independently, each person orders a salad with probability $\frac{1}{2}$. What is the probability that the number of people who ordered a soup is exactly one more than the number of people who ordered a salad?

Let $S$ be the set of all 3 -digit numbers with all digits in the set $\{1,2,3,4,5,6,7\}$ (so in particular, all three digits are nonzero). For how many elements $\overline{a b c}$ of $S$ is it true that at least one of the (not necessarily distinct) "digit cycles"

$$\overline{a b c}, \overline{b c a}, \overline{c a b}$$

is divisible by 7 ? (Here, $\overline{a b c}$ denotes the number whose base 10 digits are $a, b$, and $c$ in that order.)

Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?

A group of friends, numbered $1,2,3, \ldots, 16$, take turns picking random numbers. Person 1 picks a number uniformly (at random) in [0, 1], then person 2 picks a number uniformly (at random) in [0, 2], and so on, with person $k$ picking a number uniformly (at random) in $[0, k]$. What is the probability that the 16 numbers picked are strictly increasing?

The complex numbers $x, y, z$ satisfy

$$\begin{aligned} x y z & =-4 \\ (x+1)(y+1)(z+1) & =7 \\ (x+2)(y+2)(z+2) & =-3 \end{aligned}$$

Find, with proof, the value of $(x+3)(y+3)(z+3)$.

Let $P$ be a (non-self-intersecting) polygon in the plane. Let $C_{1}, \ldots, C_{n}$ be circles in the plane whose interiors cover the interior of $P$. For $1 \leq i \leq n$, let $r_{i}$ be the radius of $C_{i}$. Prove that there is a single circle of radius $r_{1}+\cdots+r_{n}$ whose interior covers the interior of $P$.

Let $z=a+b i$ be a complex number with integer real and imaginary parts $a, b \in \mathbb{Z}$, where $i=\sqrt{-1}$ (i.e. $z$ is a Gaussian integer). If $p$ is an odd prime number, show that the real part of $z^{p}-z$ is an integer divisible by $p$.

(Convex) quadrilateral $A B C D$ with $B C=C D$ is inscribed in circle $\Omega$; the diagonals of $A B C D$ meet at $X$. Suppose $A D<A B$, the circumcircle of triangle $B C X$ intersects segment $A B$ at a point $Y \neq B$, and ray $\overrightarrow{C Y}$ meets $\Omega$ again at a point $Z \neq C$. Prove that ray $\overrightarrow{D Y}$ bisects angle $Z D B$. (We have only included the conditions $A D<A B$ and that $Z$ lies on ray $\overrightarrow{C Y}$ for everyone's convenience. With slight modifications, the problem holds in general. But we will only grade your team's solution in this special case.)

For a convex quadrilateral $P$, let $D$ denote the sum of the lengths of its diagonals and let $S$ denote its perimeter. Determine, with proof, all possible values of $\frac{S}{D}$.

$\$$ indy has $\$ 100$ in pennies (worth $\$ 0.01$ each), nickels (worth $\$ 0.05$ each), dimes (worth $\$ 0.10$ each), and quarters (worth $\$ 0.25$ each). Prove that she can split her coins into two piles, each with total value exactly $\$ 50$.

Let $f:[0,1] \rightarrow \mathbb{C}$ be a nonconstant complex-valued function on the real interval $[0,1]$. Prove that there exists $\epsilon>0$ (possibly depending on $f$ ) such that for any polynomial $P$ with complex coefficients, there exists a complex number $z$ with $|z| \leq 1$ such that $|f(|z|)-P(z)| \geq \epsilon$.

Let $\pi$ be a permutation of $\{1,2, \ldots, 2015\}$. With proof, determine the maximum possible number of ordered pairs $(i, j) \in\{1,2, \ldots, 2015\}^{2}$ with $i<j$ such that $\pi(i) \cdot \pi(j)>i \cdot j$.

Let $z=e^{\frac{2 \pi i}{101}}$ and let $\omega=e^{\frac{2 \pi i}{10}}$. Prove that

$$\prod_{a=0}^{9} \prod_{b=0}^{100} \prod_{c=0}^{100}\left(\omega^{a}+z^{b}+z^{c}\right)$$

is an integer and find (with proof) its remainder upon division by 101.

The sequences of real numbers $\left\{a_{i}\right\}_{i=1}^{\infty}$ and $\left\{b_{i}\right\}_{i=1}^{\infty}$ satisfy $a_{n+1}=\left(a_{n-1}-1\right)\left(b_{n}+1\right)$ and $b_{n+1}=a_{n} b_{n-1}-1$ for $n \geq 2$, with $a_{1}=a_{2}=2015$ and $b_{1}=b_{2}=2013$. Evaluate, with proof, the infinite sum

$$\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)$$

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0),(2,0),(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown.

The resulting figure has 7 segments of unit length, connecting neighboring lattice points (those lying on or inside $R$ ). Compute the number of paths from $(0,1)$ (the upper left corner) to $(2,0)$ (the lower right corner) along these 7 segments, where each segment can be used at most once.

Let $A B C D E$ be a convex pentagon such that $\angle A B C=\angle A C D=\angle A D E=90^{\circ}$ and $A B=B C=$ $C D=D E=1$. Compute $A E$.

Find the number of pairs of union/intersection operations $\left(\square_{1}, \square_{2}\right) \in\{\cup, \cap\}^{2}$ satisfying the following condition: for any sets $S, T$, function $f: S \rightarrow T$, and subsets $X, Y, Z$ of $S$, we have equality of sets

$$f(X) \square_{1}\left(f(Y) \square_{2} f(Z)\right)=f\left(X \square_{1}\left(Y \square_{2} Z\right)\right),$$

where $f(X)$ denotes the image of $X$ : the set $\{f(x): x \in X\}$, which is a subset of $T$. The images $f(Y)$ (of $Y$ ) and $f(Z)$ (of $Z$ ) are similarly defined.

Consider the function $z(x, y)$ describing the paraboloid

$$z=(2 x-y)^{2}-2 y^{2}-3 y .$$

Archimedes and Brahmagupta are playing a game. Archimedes first chooses $x$. Afterwards, Brahmagupta chooses $y$. Archimedes wishes to minimize $z$ while Brahmagupta wishes to maximize $z$. Assuming that Brahmagupta will play optimally, what value of $x$ should Archimedes choose?

Let $\mathcal{H}$ be the unit hypercube of dimension 4 with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \in$ $\{0,1\}$. (Note that $\mathcal{H}$ has $2^{4}=16$ vertices.) A bug starts at the vertex $(0,0,0,0)$. In how many ways can the bug move to $(1,1,1,1)$ (the opposite corner of $\mathcal{H})$ by taking exactly 4 steps along the edges of $\mathcal{H}$ ?

Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed?

Let $\mathcal{C}$ be a cube of side length 2 . We color each of the faces of $\mathcal{C}$ blue, then subdivide it into $2^{3}=8$ unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3-dimensional cube. What is the probability that its exterior is still completely blue?

Evaluate

$$\sin (\arcsin (0.4)+\arcsin (0.5)) \cdot \sin (\arcsin (0.5)-\arcsin (0.4))$$

where for $x \in[-1,1], \arcsin (x)$ denotes the unique real number $y \in[-\pi, \pi]$ such that $\sin (y)=x$.

Let $a, b, c$ be integers. Define $f(x)=a x^{2}+b x+c$. Suppose there exist pairwise distinct integers $u, v, w$ such that $f(u)=0, f(v)=0$, and $f(w)=2$. Find the maximum possible value of the discriminant $b^{2}-4 a c$ of $f$.

Let $b(x)=x^{2}+x+1$. The polynomial $x^{2015}+x^{2014}+\cdots+x+1$ has a unique "base $b(x)$ " representation $$x^{2015}+x^{2014}+\cdots+x+1=\sum_{k=0}^{N} a_{k}(x) b(x)^{k}$$ where

• $N$ is a nonnegative integer;
• each "digit" $a_{k}(x)$ (for $0 \leq k \leq N$ ) is either the zero polynomial (i.e. $a_{k}(x)=0$ ) or a nonzero polynomial of degree less than $\operatorname{deg} b=2$; and
• the "leading digit $a_{N}(x)$ " is nonzero (i.e. not the zero polynomial).

Find $a_{N}(0)$ (the "leading digit evaluated at 0 ").

Find

$$\sum_{k=0}^{\infty}\left\lfloor\frac{1+\sqrt{\frac{2000000}{4^{k}}}}{2}\right\rfloor$$

where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \in$ $\{1,2,3,4,5\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5 . Find the sum of all possible values of $f(a, b, c, d)$.

Let $P(x)=x^{3}+a x^{2}+b x+2015$ be a polynomial all of whose roots are integers. Given that $P(x) \geq 0$ for all $x \geq 0$, find the sum of all possible values of $P(-1)$.

Find the smallest integer $n \geq 5$ for which there exists a set of $n$ distinct pairs $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ of positive integers with $1 \leq x_{i}, y_{i} \leq 4$ for $i=1,2, \ldots, n$, such that for any indices $r, s \in\{1,2, \ldots, n\}$ (not necessarily distinct), there exists an index $t \in\{1,2, \ldots, n\}$ such that 4 divides $x_{r}+x_{s}-x_{t}$ and $y_{r}+y_{s}-y_{t}$.

Find the maximum possible value of $H \cdot M \cdot M \cdot T$ over all ordered triples $(H, M, T)$ of integers such that $H \cdot M \cdot M \cdot T=H+M+M+T$.

Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points).

Find the least positive integer $N>1$ satisfying the following two properties:

• There exists a positive integer $a$ such that $N=a(2 a-1)$.
• The sum $1+2+\cdots+(N-1)$ is divisible by $k$ for every integer $1 \leq k \leq 10$.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that for any integers $x, y$, we have

$$f\left(x^{2}-3 y^{2}\right)+f\left(x^{2}+y^{2}\right)=2(x+y) f(x-y)$$

Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \cdot f(2016)$ is a perfect square. Find the minimum possible value of $f(1)+f(2)$.

Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is "divisible by $x^{2}+1$ modulo 3 ", or more precisely, either of the following equivalent conditions holds:

• there exist polynomials $P, Q$ with integer coefficients such that $(x+1)^{n}-1=\left(x^{2}+1\right) P(x)+3 Q(x)$;
• or more conceptually, the remainder when (the polynomial) $(x+1)^{n}-1$ is divided by (the polynomial) $x^{2}+1$ is a polynomial with (integer) coefficients all divisible by 3 .

What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that

$$\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$$

and

$$\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ?$$

Define a sequence $a_{i, j}$ of integers such that $a_{1, n}=n^{n}$ for $n \geq 1$ and $a_{i, j}=a_{i-1, j}+a_{i-1, j+1}$ for all $i, j \geq 1$. Find the last (decimal) digit of $a_{128,1}$.

Let $A_{1}, A_{2}, \ldots, A_{2015}$ be distinct points on the unit circle with center $O$. For every two distinct integers $i, j$, let $P_{i j}$ be the midpoint of $A_{i}$ and $A_{j}$. Find the smallest possible value of

$$\sum_{1 \leq i<j \leq 2015} O P_{i j}^{2}$$

$A B C D$ is a cyclic quadrilateral with sides $A B=10, B C=8, C D=25$, and $D A=12$. A circle $\omega$ is tangent to segments $D A, A B$, and $B C$. Find the radius of $\omega$.

Let $r_{1}, \ldots, r_{n}$ be the distinct real zeroes of the equation $$x^{8}-14 x^{4}-8 x^{3}-x^{2}+1=0$$

Evaluate $r_{1}^{2}+\cdots+r_{n}^{2}$.

Let $a, b$ be integers chosen independently and uniformly at random from the set $\{0,1,2, \ldots, 80\}$. Compute the expected value of the remainder when the binomial coefficient $\binom{a}{b}=\frac{a!}{b!(a-b)!}$ is divided by 3 . (Here $\binom{0}{0}=1$ and $\binom{a}{b}=0$ whenever $a<b$.)

Let $w, x, y$, and $z$ be positive real numbers such that

$$\begin{aligned} 0 & \neq \cos w \cos x \cos y \cos z \\ 2 \pi & =w+x+y+z \\ 3 \tan w & =k(1+\sec w) \\ 4 \tan x & =k(1+\sec x) \\ 5 \tan y & =k(1+\sec y) \\ 6 \tan z & =k(1+\sec z) \end{aligned}$$

(Here $\sec t$ denotes $\frac{1}{\cos t}$ when $\cos t \neq 0$.) Find $k$.

Let $A B C$ be a triangle whose incircle has center $I$ and is tangent to $\overline{B C}, \overline{C A}, \overline{A B}$, at $D, E, F$. Denote by $X$ the midpoint of major arc $\widehat{B A C}$ of the circumcircle of $A B C$. Suppose $P$ is a point on line $X I$ such that $\overline{D P} \perp \overline{E F}$. Given that $A B=14, A C=15$, and $B C=13$, compute $D P$.

Find the sum of squares of all distinct complex numbers $x$ satisfying the equation

$$0=4 x^{10}-7 x^{9}+5 x^{8}-8 x^{7}+12 x^{6}-12 x^{5}+12 x^{4}-8 x^{3}+5 x^{2}-7 x+4$$

Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$

A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches (that is, the opening at the top of the cup is 6 inches in diameter). At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by $\frac{1}{n^{2}}$ inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins. As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number $r$. Find $r$.

Let $N$ denote the sum of the decimal digits of $\binom{1000}{100}$. Estimate the value of $N$. If your answer is a positive integer $A$ written fully in decimal notation (for example, 521495223 ), your score will be the greatest integer not exceeding $25 \cdot(0.99)^{|A-N|}$. Otherwise, your score will be zero.

For an integer $n$, let $f(n)$ denote the number of pairs $(x, y)$ of integers such that $x^{2}+x y+y^{2}=n$. Compute the sum

$$\sum_{n=1}^{10^{6}} n f(n)$$

Write your answer in the form $a \cdot 10^{b}$, where $b$ is an integer and $1 \leq a<10$ is a decimal number. If your answer is written in this form, your score will be $\left.\max \left\{0,25-\left\lfloor 100\left|\log _{10}(A / N)\right|\right\rfloor\right)\right\}$, where $N=a \cdot 10^{b}$ is your answer to this problem and $A$ is the actual answer. Otherwise, your score will be zero.

Let $P$ denote the set of all subsets of $\{1, \ldots, 23\}$. A subset $S \subseteq P$ is called good if whenever $A, B$ are sets in $S$, the set $(A \backslash B) \cup(B \backslash A)$ is also in $S$. (Here, $A \backslash B$ denotes the set of all elements in $A$ that are not in $B$, and $B \backslash A$ denotes the set of all elements in $B$ that are not in $A$.) What fraction of the good subsets of $P$ have between 2015 and 3015 elements, inclusive? If your answer is a decimal number or a fraction (of the form $m / n$, where $m$ and $n$ are positive integers), then your score on this problem will be equal to $\max \{0,25-\lfloor 1000|A-N|\rfloor\}$, where $N$ is your answer and $A$ is the actual answer. Otherwise, your score will be zero.

A prime number $p$ is twin if at least one of $p+2$ or $p-2$ is prime and sexy if at least one of $p+6$ and $p-6$ is prime. How many sexy twin primes (i.e. primes that are both twin and sexy) are there less than $10^{9}$ ? Express your answer as a positive integer $N$ in decimal notation; for example, 521495223 . If your answer is in this form, your score for this problem will be $\max \left\{0,25-\left\lfloor\frac{1}{10000}|A-N|\right\rfloor\right\}$, where $A$ is the actual answer to this problem. Otherwise, your score will be zero.