HMMT 2016

Let $z$ be a complex number such that $|z|=1$ and $|z-1.45|=1.05$. Compute the real part of $z$.

For which integers $n \in\{1,2, \ldots, 15\}$ is $n^{n}+1$ a prime number?

Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$.

Determine the remainder when

$$\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor$$

is divided by 100 , where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.

An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence

$$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$

for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$. Proposed by: Evan Chen

Call a positive integer $N \geq 2$ "special" if for every $k$ such that $2 \leq k \leq N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100 ?$

Determine the smallest positive integer $n \geq 3$ for which

$$A \equiv 2^{10 n} \left(\bmod 2^{170}\right)$$

where $A$ denotes the result when the numbers $2^{10}, 2^{20}, \ldots, 2^{10 n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A=10241048576$ ).

Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \leq n \leq 50$ such that $n$ divides $\phi^{!}(n)+1$.

For any positive integer $n, S_{n}$ be the set of all permutations of $\{1,2,3, \ldots, n\}$. For each permutation $\pi \in S_{n}$, let $f(\pi)$ be the number of ordered pairs $(j, k)$ for which $\pi(j)>\pi(k)$ and $1 \leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k) \equiv k \pm 1(\bmod n)$. Compute

$$\sum_{\pi \in S_{999}}(-1)^{f(\pi)+g(\pi)}$$

Let $a, b$ and $c$ be positive real numbers such that

$$\begin{aligned} a^{2}+a b+b^{2} & =9 \\ b^{2}+b c+c^{2} & =52 \\ c^{2}+c a+a^{2} & =49 \end{aligned}$$

Compute the value of $\frac{49 b^{2}-33 b c+9 c^{2}}{a^{2}}$.

Dodecagon $Q W A R T Z S P H I N X$ has all side lengths equal to 2 , is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$. What are all possible values of the area of $\triangle S I X$ ?

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $H$ be the orthocenter of $A B C$. Find the distance between the circumcenters of triangles $A H B$ and $A H C$.

In the below picture, $T$ is an equilateral triangle with a side length of 5 and $\omega$ is a circle with a radius of 2 . The triangle and the circle have the same center. Let $X$ be the area of the shaded region, and let $Y$ be the area of the starred region. What is $X-Y$ ?

Let $A B C$ be a triangle with $A B=3, A C=8, B C=7$ and let $M$ and $N$ be the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively. Point $T$ is selected on side $B C$ so that $A T=T C$. The circumcircles of triangles $B A T, M A N$ intersect at $D$. Compute $D C$.

Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2}=36$ lines. Assume that all 36 lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?

Let $A B C$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M, N, P$ be the midpoints of sides $\overline{B C}, \overline{C A}, \overline{A B}$ and let $E, F$ be the tangency points of $\gamma$ with $\overline{C A}$ and $\overline{A B}$, respectively. Let $U$, $V$ be the intersections of line $E F$ with line $M N$ and line $M P$, respectively, and let $X$ be the midpoint of arc $\widehat{B A C}$ of $\Gamma$. Given that $A B=5, A C=8$, and $\angle A=60^{\circ}$, compute the area of triangle $X U V$.

Let $S=\{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y, \leq 2016\}$. Given points $A=\left(x_{1}, y_{1}\right), B=\left(x_{2}, y_{2}\right)$ in $S$, define

$$d_{2017}(A, B)=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2} (\bmod 2017)$$

The points $A=(5,5), B=(2,6), C=(7,11)$ all lie in $S$. There is also a point $O \in S$ that satisfies

$$d_{2017}(O, A)=d_{2017}(O, B)=d_{2017}(O, C)$$

Find $d_{2017}(O, A)$.

For $i=0,1, \ldots, 5$ let $l_{i}$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i \frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_{i}$ is chosen uniformly at random from the intersection of $l_{i}$ with the unit disk. Consider the convex hull of the points $P_{i}$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$ ?

In cyclic quadrilateral $A B C D$ with $A B=A D=49$ and $A C=73$, let $I$ and $J$ denote the incenters of triangles $A B D$ and $C B D$. If diagonal $\overline{B D}$ bisects $\overline{I J}$, find the length of $I J$.

The incircle of a triangle $A B C$ is tangent to $B C$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle A B C$. The $B$-mixtilinear incircle, centered at $O_{B}$, is tangent to lines $B A$ and $B C$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_{C}$, is defined similarly. Suppose that $\overline{D H} \perp \overline{O_{B} O_{C}}, A B=\sqrt{3}$ and $A C=2$. Find $B C$.

For positive integers $n$, let $S_{n}$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_{2}=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_{i}$ contains at least 4 elements?

Starting with an empty string, we create a string by repeatedly appending one of the letters $H, M$, $T$ with probabilities $\frac{1}{4}, \frac{1}{2}, \frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string?

Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $a b$ is not.

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 seven edges.

How many subsets of these seven edges form a connected figure?

Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$, uniformly and at random with replacement. Set

$$M=a+2 b+4 c+8 d+16 e+32 f$$

What is the expected value of the remainder when $M$ is divided by 64 ?

Define the sequence $a_{1}, a_{2} \ldots$ as follows: $a_{1}=1$ and for every $n \geq 2$,

$$a_{n}= \begin{cases}n-2 & \text { if } a_{n-1}=0 \\ a_{n-1}-1 & \text { if } a_{n-1} \neq 0\end{cases}$$

A non-negative integer $d$ is said to be jet-lagged if there are non-negative integers $r, s$ and a positive integer $n$ such that $d=r+s$ and that $a_{n+r}=a_{n}+s$. How many integers in $\{1,2, \ldots, 2016\}$ are jet-lagged?

Kelvin the Frog has a pair of standard fair 8 -sided dice (each labelled from 1 to 8 ). Alex the sketchy Kat also has a pair of fair 8-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their dice, the probability that they get any given sum is equal! Suppose that Alex's two dice have $a$ and $b$ total dots on them, respectively. Assuming that $a \neq b$, find all possible values of $\min \{a, b\}$.

Let $X$ be the collection of all functions $f:\{0,1, \ldots, 2016\} \rightarrow\{0,1, \ldots, 2016\}$. Compute the number of functions $f \in X$ such that

$$\max _{g \in X}\left(\min _{0 \leq i \leq 2016}(\max (f(i), g(i)))-\max _{0 \leq i \leq 2016}(\min (f(i), g(i)))\right)=2015 .$$

Let $V=\{1, \ldots, 8\}$. How many permutations $\sigma: V \rightarrow V$ are automorphisms of some tree? (A graph consists of a some set of vertices and some edges between pairs of distinct vertices. It is connected if every two vertices in it are connected by some path of one or more edges. A tree $G$ on $V$ is a connected graph with vertex set $V$ and exactly $|V|-1$ edges, and an automorphism of $G$ is a permutation $\sigma: V \rightarrow V$ such that vertices $i, j \in V$ are connected by an edge if and only if $\sigma(i)$ and $\sigma(j)$ are.)

Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p, q)$ of nonnegative integers satisfying $p+q \leq 2016$. Kristoff must then give Princess Anna exactly $p$ kilograms of ice. Afterward, he must give Queen Elsa exactly $q$ kilograms of ice. What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which $p$ and $q$ are chosen?

Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^{3}+5 b^{3} \neq 2016$.

For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$ ?

Let $A B C$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle O I A=90^{\circ}$. Given that $A I=97$ and $B C=144$, compute the area of $\triangle A B C$.

Let $n>1$ be an odd integer. On an $n \times n$ chessboard the center square and four corners are deleted. We wish to group the remaining $n^{2}-5$ squares into $\frac{1}{2}\left(n^{2}-5\right)$ pairs, such that the two squares in each pair intersect at exactly one point (i.e. they are diagonally adjacent, sharing a single corner). For which odd integers $n>1$ is this possible?

Find all prime numbers $p$ such that $y^{2}=x^{3}+4 x$ has exactly $p$ solutions in integers modulo $p$.

In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \in\{0,1, \ldots, p-1\}$ and

$$p \text { divides } y^{2}-x^{3}-4 x$$

A nonempty set $S$ is called well-filled if for every $m \in S$, there are fewer than $\frac{1}{2} m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\{1,2, \ldots, 42\}$.

Let $q(x)=q^{1}(x)=2 x^{2}+2 x-1$, and let $q^{n}(x)=q\left(q^{n-1}(x)\right)$ for $n>1$. How many negative real roots does $q^{2016}(x)$ have?

Compute

$$\int_{0}^{\pi} \frac{2 \sin \theta+3 \cos \theta-3}{13 \cos \theta-5} \mathrm{~d} \theta .$$

Fix positive integers $r>s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r-1$ that shares at least $s$ elements with each set in $F$.

Let $A B C$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{B C}, \overline{C A}, \overline{A B}$ at $D, E, F$. Point $P$ lies on $\overline{E F}$ such that $\overline{D P} \perp \overline{E F}$. Ray $B P$ meets $\overline{A C}$ at $Y$ and ray $C P$ meets $\overline{A B}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle A Y Z$ so that $\overline{A Q} \perp \overline{B C}$. Prove that $P, I, Q$ are collinear.

Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^{2}+y^{2}=15$. Compute $|x-y|$.

Sherry is waiting for a train. Every minute, there is a $75 \%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75 \%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?

Let $P R O B L E M Z$ be a regular octagon inscribed in a circle of unit radius. Diagonals $M R, O Z$ meet at $I$. Compute $L I$.

Consider a three-person game involving the following three types of fair six-sided dice.

• Dice of type $A$ have faces labelled $2,2,4,4,9,9$.
• Dice of type $B$ have faces labelled $1,1,6,6,8,8$.
• Dice of type $C$ have faces labelled $3,3,5,5,7,7$.

All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player?

Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of 10 centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of 4 centimeters. Given that the total volume of the snowballs that Anderson constructs cannot exceed the volume of the snowball that Patrick threw, how many snowballs can Anderson construct?

Consider a $2 \times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through once and no two line segments intersect. How many efficient paths are there when $n=2016$ ?

A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems?

For each positive integer $n$ and non-negative integer $k$, define $W(n, k)$ recursively by

$$W(n, k)= \begin{cases}n^{n} & k=0 \\ W(W(n, k-1), k-1) & k>0\end{cases}$$

Find the last three digits in the decimal representation of $W(555,2)$.

Victor has a drawer with two red socks, two green socks, two blue socks, two magenta socks, two lavender socks, two neon socks, two mauve socks, two wisteria socks, and 2000 copper socks, for a total of 2016 socks. He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock. What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step.

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$.

Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when

$$\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)$$

is divided by 50 .

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 seven edges.

Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)

A right triangle has side lengths $a, b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle.

Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F$. Compute the area of the triangle these three lines determine.

Compute $\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)$.

Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$.

Compute the sum of all integers $1 \leq a \leq 10$ with the following property: there exist integers $p$ and $q$ such that $p, q, p^{2}+a$ and $q^{2}+a$ are all distinct prime numbers.

Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?

Let

$$A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}$$

Find the largest integer less than or equal to $\frac{1}{A}$. The following decimal approximation might be useful: $0.6931<\ln (2)<0.6932$, where $\ln$ denotes the natural logarithm function.

Let $A B C$ be a triangle with $A B=13, A C=14$, and $B C=15$. Let $G$ be the point on $A C$ such that the reflection of $B G$ over the angle bisector of $\angle B$ passes through the midpoint of $A C$. Let $Y$ be the midpoint of $G C$ and $X$ be a point on segment $A G$ such that $\frac{A X}{X G}=3$. Construct $F$ and $H$ on $A B$ and $B C$, respectively, such that $F X\|B G\| H Y$. If $A H$ and $C F$ concur at $Z$ and $W$ is on $A C$ such that $W Z \| B G$, find $W Z$.

Tim starts with a number $n$, then repeatedly flips a fair coin. If it lands heads he subtracts 1 from his number and if it lands tails he subtracts 2 . Let $E_{n}$ be the expected number of flips Tim does before his number is zero or negative. Find the pair $(a, b)$ such that

$$\lim _{n \rightarrow \infty}\left(E_{n}-a n-b\right)=0$$

On the Cartesian plane $\mathbb{R}^{2}$, a circle is said to be nice if its center is at the origin $(0,0)$ and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points $A=(20,15)$ and $B=(20,16)$. How many nice circles intersect the open segment $A B$ ? For reference, the numbers $601,607,613,617,619,631,641,643,647,653,659,661,673,677,683$, 691 are the only prime numbers between 600 and 700 .

Let $t=2016$ and $p=\ln 2$. Evaluate in closed form the sum

$$\sum_{k=1}^{\infty}\left(1-\sum_{n=0}^{k-1} \frac{e^{-t} t^{n}}{n!}\right)(1-p)^{k-1} p$$

Let $\Delta A_{1} B_{1} C$ be a triangle with $\angle A_{1} B_{1} C=90^{\circ}$ and $\frac{C A_{1}}{C B_{1}}=\sqrt{5}+2$. For any $i \geq 2$, define $A_{i}$ to be the point on the line $A_{1} C$ such that $A_{i} B_{i-1} \perp A_{1} C$ and define $B_{i}$ to be the point on the line $B_{1} C$ such that $A_{i} B_{i} \perp B_{1} C$. Let $\Gamma_{1}$ be the incircle of $\Delta A_{1} B_{1} C$ and for $i \geq 2, \Gamma_{i}$ be the circle tangent to $\Gamma_{i-1}, A_{1} C, B_{1} C$ which is smaller than $\Gamma_{i-1}$. How many integers $k$ are there such that the line $A_{1} B_{2016}$ intersects $\Gamma_{k}$ ?

A particular coin can land on heads (H), on tails ( T ), or in the middle (M), each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times.

For positive integers $a, b, a \uparrow \uparrow b$ is defined as follows: $a \uparrow \uparrow 1=a$, and $a \uparrow \uparrow b=a^{a \uparrow \uparrow(b-1)}$ if $b>1$.

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not \equiv a \uparrow \uparrow 7$ $\bmod n$.

Find the smallest possible area of an ellipse passing through $(2,0),(0,3),(0,7)$, and $(6,0)$.

Among citizens of Cambridge there exist 8 different types of blood antigens. In a crowded lecture hall are 256 students, each of whom has a blood type corresponding to a distinct subset of the antigens; the remaining of the antigens are foreign to them. Quito the Mosquito flies around the lecture hall, picks a subset of the students uniformly at random, and bites the chosen students in a random order. After biting a student, Quito stores a bit of any antigens that student had. A student bitten while Quito had $k$ blood antigen foreign to him/her will suffer for $k$ hours. What is the expected total suffering of all 256 students, in hours?

Katherine has a piece of string that is 2016 millimeters long. She cuts the string at a location chosen uniformly at random, and takes the left half. She continues this process until the remaining string is less than one millimeter long. What is the expected number of cuts that she makes?

Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that

$$2^{m} n-2^{n} m=2^{k}$$

For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because $\varphi(44)+4 \tau(44)=44$. Find the sum of all good positive integers $n$.

How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.)

(Lucas Numbers) The Lucas numbers are defined by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$ for every $n \geq 0$. There are $N$ integers $1 \leq n \leq 2016$ such that $L_{n}$ contains the digit 1 . Estimate $N$. An estimate of $E$ earns $\lfloor 20-2|N-E|\rfloor$ or 0 points, whichever is greater.

(Caos) A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns. An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{4}\right\rfloor$ points.

(Maximal Determinant) In a $17 \times 17$ matrix $M$, all entries are $\pm 1$. The maximum possible value of $|\operatorname{det} M|$ is $N$. Estimate $N$. An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{2}\right\rfloor$ points.

(Self-Isogonal Cubics) Let $A B C$ be a triangle with $A B=2, A C=3, B C=4$. The isogonal conjugate of a point $P$, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines $P A$, $P B, P C$ across the angle bisectors of $\angle A, \angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak{K}(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $P P^{*}$ contains $Q$. Consider: (a) the M'Cay cubic $\mathfrak{K}(O)$, where $O$ is the circumcenter of $\triangle A B C$, (b) the Thomson cubic $\mathfrak{K}(G)$, where $G$ is the centroid of $\triangle A B C$, (c) the Napoleon-Feurerbach cubic $\mathfrak{K}(N)$, where $N$ is the nine-point center of $\triangle A B C$, (d) the Darboux cubic $\mathfrak{K}(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O$ ), (e) the Neuberg cubic $\mathfrak{K}\left(X_{30}\right)$, where $X_{30}$ is the point at infinity along line $O G$, (f) the nine-point circle of $\triangle A B C$, (g) the incircle of $\triangle A B C$, and (h) the circumcircle of $\triangle A B C$.

Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E| / 6}\right\rfloor$ points.