MathNet 2015-part-07 — 185 problems (EN–VI)

Let $A$, $B$, $C$ be points in a line on this order, with $AB = BC$. A car travels from $A$ to $B$ at $40$ kilometers per hour and then from $B$ to $C$ at $60$ kilometers per hour, without stopping in between. What is the overall speed of the trip from $A$ to $C$?

For positive integers $n$, prove that $\operatorname{gcd}(6 n+1,15 n+2)=1$.

Is there a polynomial $P(n)$ with integer coefficients such that $P(2)=4$ and $P(P(2))=7$? Prove your answer.

Are there integers $a, b, c, d$ which satisfy $a^{4}+b^{4}+c^{4}+2016=10 d$?

A line in the Cartesian plane is called stable if it passes through at least two points $(x, y)$ such that $x$ and $y$ are rational numbers. Prove or disprove: every point lies on some stable line.

Let $ABC$ be a triangle and $P$ a point inside it. Rays $BP$ and $CP$ meet $AC$ and $AB$ at $Y$ and $X$, respectively. Prove that if $AP$ bisects $BC$ then $XY \parallel BC$.

There are three boxes of stones. Each hour, Sisyphus moves a stone from one box to another. For each transfer of a stone, he receives from Zeus a number of coins equal to the number of stones in the box from which the stone is drawn minus the number of stones in the recipient box, with the stone Sisyphus just carried not counted. If this number is negative, Sisyphus pays the corresponding amount (and can pay later if he is broke).
After 1000 years, all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?

Yan and Jacob play the following game. Yan shows Jacob a weighted 4-sided die labelled $1$, $2$, $3$, $4$, with weights $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{7}$, $\frac{1}{42}$, respectively. Then, Jacob specifies $4$ positive real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ such that $x_{1}+x_{2}+x_{3}+x_{4}=1$. Finally, Yan rolls the die, and Jacob earns $10+\log \left(x_{k}\right)$ dollars if the die shows $k$ (note this may be negative). Which $x_{i}$ should Jacob pick to maximize his expected payoff?

(Here log is the natural logarithm, which has base $e \approx 2.718$.)

Let $a, b, c$ be positive reals. Prove that
$$
\frac{a^{3}}{b c}+\frac{b^{3}}{c a}+\frac{c^{3}}{a b} \geq a+b+c
$$

Let $\mathcal{P}$ be a regular $17$-gon; we draw in the $\binom{17}{2}$ diagonals and sides of $\mathcal{P}$ and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of $\mathcal{P}$) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?

Let $X=\{1,2, \ldots, 100\}$. How many functions $f: X \rightarrow X$ satisfy $f(b) \lt f(a)+(b-a)$ for all $1 \leq a \lt b \leq 100$?

Find, with proof, the largest possible value of

$$
\frac{x_{1}^{2}+\cdots+x_{n}^{2}}{n}
$$

where real numbers $x_{1}, \ldots, x_{n} \geq -1$ are satisfying $x_{1}^{3}+\cdots+x_{n}^{3}=0$.

Compute the prime factorization of $25^{3}-27^{2}$.

Suppose $x$ and $y$ are real numbers satisfying $x + y = 5$. What is the largest possible value of $x^{2} + 2 x y$?

Let $ABC$ be an acute triangle with orthocenter $H$, circumcenter $O$, and incenter $I$. Prove that ray $AI$ bisects $\angle HAO$.

For which prime numbers $p$ is $p^{2}+2$ also prime? Prove your answer.

There is a colony consisting of $100$ cells. Every minute, a cell dies with probability $\frac{1}{3}$; otherwise it splits into two identical copies. What is the probability that the colony never goes extinct?

Let $H, I, O, \Omega$ denote the orthocenter, incenter, circumcenter and circumcircle of a scalene acute triangle $A B C$. Prove that if $\angle B A C=60^{\circ}$ then the circumcenter of $\triangle I H O$ lies on $\Omega$.

Let $n$ be a fixed positive integer. Initially, $n$ $1$'s are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4 n^{2}-4}{3}}$.

A thief starts at the point $x=0$ on a number line, and runs either left or right at a constant rate. One hour later, a policeman who moves twice as fast as the thief appears at the point $x=0$. However, the policeman does not know which direction the thief went and cannot see the thief. Show that the policeman can still catch the thief.