MathNet 2017-part-07 — 172 problems (EN–VI)

Find all pairs $(a, b)$ of positive integers such that $a^{2017}+b$ is a multiple of $a b$.

Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $a b c = 2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute
$$
\sum_{\substack{a b c = 2310 \\ a, b, c \in \mathbb{N}}} (a + b + c)
$$
where $\mathbb{N}$ denotes the positive integers.

A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find $\lfloor 2017 P(2017)\rfloor$.

Determine the largest real number $c$ such that for any $2017$ real numbers $x_{1}, x_{2}, \ldots, x_{2017}$, the inequality
$$
\sum_{i=1}^{2016} x_{i}\left(x_{i}+x_{i+1}\right) \geq c \cdot x_{2017}^{2}
$$
holds.

Consider all ordered pairs of integers $(a, b)$ such that $1 \leq a \leq b \leq 100$ and
$$
\frac{(a+b)(a+b+1)}{a b}
$$
is an integer.
Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ over $(4,84)$. Note that your answer should be an ordered pair.

The Fibonacci sequence is defined as follows: $F_{0}=0$, $F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0 \pmod{127}$ and $F_{m+1} \equiv 1 \pmod{127}$.

Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f: \mathbb{N} \rightarrow \{0,1, \ldots, 16\}$ such that
$$
f(x+17)=f(x) \quad \text{and} \quad f\left(x^{2}\right) \equiv f(x)^{2}+15 \pmod{17}
$$
for all integers $x \geq 1$.

How many ways are there to insert +'s between the digits of $111111111111111$ (fifteen 1's) so that the result will be a multiple of $30$?

There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d\gt 0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?

Sam spends his days walking around the following $2 \times 2$ grid of squares.

| 1 | 2 |
| :--- | :--- |
| 4 | 3 |

Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?

Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability $\frac{1}{2}$. If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability $\frac{1}{2}$). What is the probability that after 2017 seconds her bucket contains exactly 1337 stones?

There are 2017 frogs and 2017 toads in a room. Each frog is friends with exactly 2 distinct toads. Let $N$ be the number of ways to pair every frog with a toad who is its friend, so that no toad is paired with more than one frog. Let $D$ be the number of distinct possible values of $N$, and let $S$ be the sum of all possible values of $N$. Find the ordered pair $(D, S)$.

Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they made during the meeting is a multiple of 4. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.

Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1,2, \ldots, m\}$. Call a subset $S$ of $T$ $\delta$-good if for all $s_{1}, s_{2} \in S$, $s_{1} \neq s_{2}$, $\left|\Delta\left(s_{1}, s_{2}\right)\right| \geq \delta m$, where $\Delta$ denotes symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}$-good set of size $s$.

Compute the number of possible words $w = w_{1} w_{2} \ldots w_{100}$ satisfying:
- $w$ has exactly $50$ $A$'s and $50$ $B$'s (and no other letters).
- For $i = 1, 2, \ldots, 100$, the number of $A$'s among $w_{1}, w_{2}, \ldots, w_{i}$ is at most the number of $B$'s among $w_{1}, w_{2}, \ldots, w_{i}$.
- For all $i = 44, 45, \ldots, 57$, if $w_{i}$ is a $B$, then $w_{i+1}$ must be a $B$.

Let $A$, $B$, $C$, $D$ be four points on a circle in that order. Also, $AB = 3$, $BC = 5$, $CD = 6$, and $DA = 4$. Let diagonals $AC$ and $BD$ intersect at $P$. Compute $\frac{AP}{CP}$.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

Let $S$ be a set of 2017 distinct points in the plane. Let $R$ be the radius of the smallest circle containing all points in $S$ on either the interior or boundary. Also, let $D$ be the longest distance between two of the points in $S$. Let $a, b$ be real numbers such that $a \leq \frac{D}{R} \leq b$ for all possible sets $S$, where $a$ is as large as possible and $b$ is as small as possible. Find the pair $(a, b)$.

Let $ABCD$ be a convex quadrilateral with $AB = 5$, $BC = 6$, $CD = 7$, and $DA = 8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^{2} - PQ^{2}$.

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_{1}, R_{2}, R_{3}, R_{4}$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_{1}=31$ and $R_{2}=24$ and $R_{3}=12$, find $R_{4}$.