MathNet 2019-part-08 — 179 problems (EN–VI)

Prove that if the line joining the circumcenter $O$ and the incenter $I$ is parallel to side $BC$ of an acute triangle, then $\cos B + \cos C = 1$.

To play the lottery game Sum Thing, you choose five distinct numbers from $1$ to $50$, then the lottery master chooses five distinct numbers from $1$ to $50$. If there exist a nonempty subset of your five numbers and a nonempty subset of the lottery master's five numbers such that both subsets have the same sum, then you win.
Can you choose five numbers that guarantee a win? Either demonstrate such a set, with a proof of validity, or prove that no such set exists.

Let $ABC$ be an acute scalene triangle with incenter $I$. Show that the circumcircle of $BIC$ intersects the Euler line of $ABC$ in two distinct points.

(Recall that the Euler line of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.)

Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on segments $AB$ and $AD$ respectively, and $AC$ intersects $XY$ at point $Z$. Prove that
$$
\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}
$$

Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \ldots, f(n))$ is a permutation of $(1,2, \ldots, n)$ ?

Annie has a permutation $\left(a_{1}, a_{2}, \ldots, a_{2019}\right)$ of $S=\{1,2, \ldots, 2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple $(y_{1}, y_{2}, \ldots, y_{2019})$ of integers in $S$, and then Annie gives the number of indices $i \in S$ such that $a_{i}=y_{i}$.

a. Show that Yannick can always guess Annie's permutation with at most 1200000 guesses.

b. Show that Yannick can always guess Annie's permutation with at most 24000 guesses.

Do there exist four points $P_{i} = (x_{i}, y_{i}) \in \mathbb{R}^{2}$ ($1 \leq i \leq 4$) on the plane such that:
- for all $i = 1,2,3,4$, the inequality $x_{i}^{4} + y_{i}^{4} \leq x_{i}^{3} + y_{i}^{3}$ holds, and
- for all $i \neq j$, the distance between $P_{i}$ and $P_{j}$ is greater than $1$?

For any angle $0 \lt \theta \lt \pi / 2$, show that
$$
0 \lt \sin \theta + \cos \theta + \tan \theta + \cot \theta - \sec \theta - \csc \theta \lt 1
$$

A cactus is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

Let $p=2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z} \rightarrow \mathbb{F}_p$ is called good if there is $\alpha \in \mathbb{F}_p$ with $\alpha \not \equiv 0 (\bmod p)$ such that
$$
f(x) f(y) = f(x+y) + \alpha^{y} f(x-y) \quad (\bmod p)
$$
for all $x, y \in \mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?

Scalene triangle $ABC$ satisfies $\angle A = 60^{\circ}$. Let the circumcenter of $ABC$ be $O$, the orthocenter be $H$, and the incenter be $I$. Let $D, T$ be the points where line $BC$ intersects the internal and external angle bisectors of $\angle A$, respectively. Choose point $X$ on the circumcircle of $\triangle IHO$ such that $HX \parallel AI$. Prove that $OD \perp TX$.

A convex polygon on the plane is called wide if the projection of the polygon onto any line in the same plane is a segment with length at least $1$. Prove that a circle of radius $\frac{1}{3}$ can be placed completely inside any wide polygon.

Proposed by: Shengtong Zhang

Can the set of lattice points $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \leq x, y \leq 252, x \neq y\}$ be colored using 10 distinct colors such that for all $a \neq b, b \neq c$, the colors of $(a, b)$ and $(b, c)$ are distinct?

Let $p \gt 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo $5$.
Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called compositional if
$$
f(g(x)) \equiv x^{p^2} - x
$$
in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs (in terms of $p$).

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial
$$
P(x) = (2 n) x^{2 n} + (2 n-1) x^{2 n-1} + \cdots + (n+1) x^{n+1} + n x^{n} + (n+1) x^{n-1} + \cdots + (2 n-1) x + 2 n
$$
lie on the unit circle (i.e. $|r|=1$).

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits are the same when read backwards.)

Let $N=2^{\left(2^{2}\right)}$ and $x$ be a real number such that $N^{\left(N^{N}\right)}=2^{\left(2^{x}\right)}$. Find $x$.

Let $x$ and $y$ be positive real numbers. Define $a=1+\frac{x}{y}$ and $b=1+\frac{y}{x}$. If $a^{2}+b^{2}=15$, compute $a^{3}+b^{3}$.