MathNet 2024-part-05 — 168 problems (EN–VI)

Find all the positive integers $m$ and $n$ so that $n = 3^m + s(n) + 35$, where $s(n)$ is the sum of the digits of $n$.

Solve the equation
$$
((((((((n - \frac{1}{2}) \cdot 2 - \frac{2}{3}) \cdot 3 - \frac{3}{4}) \cdot 4 - \frac{4}{5}) \cdot \dots) \cdot 2023 - \frac{2023}{2024}) \cdot 2024 = 1.
$$

The path from Little Red Riding Hood to Grandma has ups, downs and flat portions. When walking, LRRH goes up two times slower and goes down two times faster than on flat ground. When riding her bike, she goes up three times slower and goes down three times faster than on flat ground.

LRRH noticed that, when walking, the way back home from Grandma takes 18 minutes less than the way from home to Grandma, and when she rides her bike, the way back home from Grandma takes 20 minutes less than the way from home to Grandma. By what percentage is LRRH's speed larger by bike than by foot?

Let $n \ge 3$ be an integer. A 3-element subset $A$ of the set $M = \{1, 2, 3, \dots, n\}$ will be called interesting if there exists a set $B$ of positive integers so that:
(1) if $a \in A, b \in A, a \neq b$, then $(a+b) \in B$;
(2) if $x, y, z \in B, x \lt y \lt z$, then $2 \cdot y = x + z$.
Denote $A_n$ the number of the interesting sets.

a) Compute $A_7$.

b) Prove that $A_{91} \gt 2024$.

Prove that there exist infinitely many positive integers $n$ so that $n + s(n)$ is a perfect square, where $s(x)$ stands for the sum of the decimal digits of $x$.

The sides $BC$, $CA$, $AB$ of the equilateral triangle $ABC$ contain the points $D$, $E$, respectively $F$, different from the sides' midpoints, so that $BD = CE = AF$. Denote $M$ the midpoint of $EF$ and $N$ the intersection of the lines $BC$ and $AM$. Prove that the segments $ND$ and $BC$ have the same midpoint.

Gheorghe Iurea

A rhombus of side length 1 and internal angle 60° is called a diamond. Suppose that a regular hexagon of side length $n$ is dissected into diamonds. Prove that each main diagonal of the hexagon halves exactly $n$ diamonds.

Find all non-constant polynomials $P(x)$ with real coefficients that satisfy
$$
P(x^3 - 7x) = P(x - 7)P(x - 8)P(x - 3)
$$
for all $x \in \mathbb{R}$.

Let $ABCD$ be a quadrilateral inscribed in a circle ($O$) and some point $I$ lies inside $ABCD$. Consider the lines $d_1, d_2, d_3, d_4$ pass through the midpoints of segments $IA, IB, IC, ID$ respectively and perpendicular to lines $OA, OB, OC, OD$. Line $d_1$ cuts $d_2$ at $P$, line $d_2$ cuts $d_3$ at $Q$, line $d_3$ cuts $d_4$ at $R$ and line $d_4$ cuts $d_1$ at $S$. Suppose that the quadrilateral $PQRS$ is convex and points $I, O$ lie inside it. Prove that $PQRS$ circumscribes a circle.

For every $n = 1, 2, 3, \ldots$, define $a_n = 3^{3^n-1} + 2$. Prove that there are infinitely many prime numbers $p$ for which there exists a natural number $n$ such that $p$ is a divisor of $a_n$.

Let the alphabet has only two letters: $A$ and $B$. Is it possible to choose one word of length 5, one word of length 6, ..., one word of length 30 such that any word of length 300 contains one of the chosen words as substring?

A mail carrier delivers mail to the 19 houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

Let $x$, $y$, $z$ be real numbers such that $x \ge y \ge z \ge 0$ and $2x + y + 2z = 5$. Prove that
$$
5 \le x^2 + z^2 + xy + yz + zx \le \frac{25}{4}.
$$
When does the equality case hold?

Let $ABC$ be an acute, non isosceles triangle with circumcircle $(O)$ and circumradius $R$. Denote $M$, $N$, $P$ as midpoints of segments $BC$, $CA$, $AB$ respectively. Let $(\omega)$ be the circle passes through $A$, $O$ and tangent to $OM$. Circle $(\omega)$ meets $AB$, $AC$ at $E$, $F$. Denote $I$ as midpoint of segment $EF$ and $K$ as the intersection of $EF$, $NP$. Prove that $R = 2IK$ and $IMO$ is an isosceles triangle.

Given a non-isosceles triangle $ABC$, inscribed in circle $(O)$ with $X, Y, Z$ are midpoints of the major arcs $BC, CA, AB$ of $(O)$. Let $D, E, F$ be the tangent points of the incircle $(I)$ of $ABC$ on $BC, CA, AB$ respectively. Suppose that line $XE$ intersects $(O)$ again at $M$ and cuts $YD$ at $P$; line $XF$ intersects $(O)$ again at $N$ and intersects $ZD$ at $Q$. Let $T$ be the intersection of $QE, PF$ and $XT$ cuts $(O)$ again at $K$. Prove that the circumcircle of triangle $AOK$ bisects the segment $MN$.

Find the maximum value of $k$ such that: for all $a, b, c$ are sidelengths of some triangle, then
$$
k \cdot \frac{3(a^2 + b^2 + c^2)}{(a + b + c)^2} + \sqrt{\frac{ab + bc + ca}{a^2 + b^2 + c^2}} \leq k + 1.
$$

On the board, there are written numbers $1, 2, 3, \dots, 2023^{2024}$. At each step, it is allowed to erase any two numbers $a, b$ on the board and then replace them with $\gcd(2024ab, a^2 + 254ab + b^2)$, keep doing this until there is only one number $x$ left. Find all possible values of $x$.

Find all odd integer $n$ such that the number of integers $k$ with $0 \lt k \lt \frac{n}{4}$ and $\gcd(n, k) = 1$ is odd.

With positive integer $n \gt 1$, let $4n - 3$ positive numbers are written on the board (not necessarily all different). It is known that any 4 pairwise different numbers from that list form an arithmetic progression. Prove that some number is written on the board at least $n$ times.

Find the smallest positive number $m$ such that there exists unique function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$
f(x + f(y)) = f(x) + \frac{mx^5 f(y)}{f(x)^2} + f(y^2), \forall x, y \gt 0.
$$