MathNet 2026 — Part 10

Denote by $f(n)$ the integer obtained by reversing the digits of a positive integer $n$. Find the greatest integer that is certain to divide $n^{4}-f(n)^{4}$ regardless of the choice of $n$.

Given a triangle $ABC$, we draw three circles with respective diameters $AB$, $BC$, and $CA$. Prove that there exists a point that is inside all three circles.

A castle has infinitely many rooms labeled $1,2,3, \ldots$, which are divided into several halls. Suppose room $n$ is on the same hall as rooms $3 n+1$ and $n+10$ for every $n$. Determine the maximum possible number of different halls in the castle.

Two concentric circles have radii $r$ and $R \gt r$. Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find $\frac{R}{r}$.

On a distant planet, there are $2014$ cities, some pairs of which are connected by two-way roads. It turns out that the population of each city is the average of the populations of the cities to which it is connected by a single road, and moreover that it is possible to travel from every city to every other city by a sequence of roads.
Prove that all cities have the same population.

Find all pairs of integers $x, y$ such that $2 x^{2} - 6 x y + 3 y^{2} = -1$.

Let $a \gt 2$ be given, and define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ by
$$
a_{0} = 1, \quad a_{1} = a, \quad a_{n+1} = \left(\frac{a_{n}^{2}}{a_{n-1}^{2}} - 2\right) \cdot a_{n} .
$$
Show that for all integers $k \geq 0$, we have
$$
\sum_{i=0}^{k} \frac{1}{a_{i}} \lt \frac{a+2-\sqrt{a^{2}-4}}{2} .
$$

If $n$ is a natural number such that $2n+1$ and $3n+1$ are perfect squares, prove that $5n+3$ can't be a prime number.

Suppose that $b$ and $c$ are real numbers such that the equation
$$
x^{2}+b x+c=0
$$
has two different solutions $x_{1}, x_{2}$. Suppose that


a. The (positive) difference between $x_{1}$ and $x_{2}$ is $1$;


b. The (positive) difference between $b$ and $c$ is also $1$.

Find all possible values of $b$ and $c$.

We have a row of boxes that is infinite in one direction, as shown.

Determine if it is possible to fill each box with a positive integer such that the number in every box (except the leftmost one) is greater than the average of the numbers in the two neighboring boxes.

Let $N$ be a positive integer. Define a sequence $a_{n}$, $n \geq 0$, by
$$
a_{0}=0, \quad a_{1}=1, \quad a_{n+1}+a_{n-1}=a_{n}\left(2-\frac{1}{N}\right) \quad(n \geq 1)
$$
Prove that $a_{n}\lt \sqrt{N+1}$ for all $n \geq 0$.

You are given an $m \times n$ chocolate bar divided into $1 \times 1$ squares. You can break a piece of chocolate by splitting it into two pieces along a straight line that does not cut through any of the $1 \times 1$ squares. What is the minimum number of times you have to break the bar in order to separate all the $1 \times 1$ squares?

Show that $n$ divides $\varphi\left(a^{n}-1\right)$ for any integers $a$ and $n$, where $\varphi$ is Euler's totient function.

As shown in the diagram above, the vertices of a regular decagon are colored alternately black and white. We would like to draw colored line segments between the points in such a way that


a. Every black point is connected to every white point by a line segment.


b. No two line segments of the same color intersect, except at their endpoints.

Determine, with proof, the minimum number of colors needed.

Pentagon $A B C D E$ is cyclic, i.e., inscribed in a circle. Diagonals $A C$ and $B D$ meet at $P$, and diagonals $A D$ and $C E$ meet at $Q$. Triangles $A B P$, $A E Q$, $C D P$, $C D Q$, and $A P Q$ have equal areas. Prove that the pentagon is regular.

A society has $100$ members and every two members are either friends or enemies. Prove that there are two persons from the society that have an even number of common enemies.

Determine all triples $(x, y, n)$ of integers such that $x^{2} + 2 y^{2} = 2^{n}$.

The sum of the digits of all counting numbers less than $13$ is
$$
1+2+3+4+5+6+7+8+9+1+0+1+1+1+2=51
$$
Find the sum of the digits of all counting numbers less than $1000$.

Does there exist an integer such that its cube is equal to $3 n^{2} + 3 n + 7$, where $n$ is an integer?

In the triangle $A B C$ the angle $B$ is not a right angle, and $A B : B C = k$. Let $M$ be the midpoint of $A C$. The lines symmetric to $B M$ with respect to $A B$ and $B C$ intersect $A C$ at $D$ and $E$. Find $B D : B E$.