MathNet 2026 — Part 04

Can we draw $2003$ segments in the plane so that each of them cuts exactly to three others?

Suppose we have a board with sixteen boxes arranged in four rows and four columns.
a) proves that seven tiles can be placed, never two in the same box, so that when you delete two rows and any two columns, there is always a piece left without deleting.
b) proves that if six tiles are placed, never two in the same box, you can always delete two rows and two columns so that all the tiles are deleted.

Let $x, y, z$ be positive real such that $x+y+z=3$. Find the maximum value reached by $$
\sqrt{x}+\sqrt{2 y+2}+\sqrt{3 z+6}
$$ For what values of $x, y, z$ is said maximum reached?

We consider the polynomials $P(x)=x^{3}+A x^{2}+B x+C$, $Q(x)=3 x^{2}+2 A x+B$ ($x$ is the variable $A$, $B$, $C$ are parameters). Suppose that, yes $a, b, c$ They are the three roots of $P$, those of $Q$ They are $\frac{a+b}{2}$, $\frac{b+c}{2}$. Determine all possible polynomials $P, Q$.

Calculate all possible values of $f(2004)$, where $f: \mathbb{N} \rightarrow \mathbb{N}$ is a function that fulfills: - $f(n m)=f(n) f(m)$ for all natural numbers $m, n$. - $f(n) \leq n^{2}$ for all natural numbers $n$. - $f(1002) \geq 1003969$

Determine the triangle with the smallest perimeter between all those with the circumference inscribed with the same radius and the same value of an angle.

Encontrar todas las soluciones enteras positivas de
$$
\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b+c-2}=1
$$

Two radius spheres $r$ are exterior tangents. Three other radius spheres $R$ are outer tangents to each other, two to two. Each of these three spheres is also tangent outside the first two. Find the relationship between $R$ and $r$.

Find, reasonably, two positive integers $a$ and $b$, such that $$
\begin{aligned}
& b^{2} \text{ be a multiple of } a, \\
& a^{3} \text{ be a multiple of } b^{2}, \\
& b^{4} \text{ be a multiple of } a^{3}, \\
& a^{5} \text{ be a multiple of } b^{4}, \\
& \text{but } b^{6} \text{ Don't be a multiple of } a^{5}.
\end{aligned}
$$

It is requested to find all positive integers $n$ such that $3^{n}+5^{n}$ is a multiple of $3^{n-1}+5^{n-1}$.

We say that three different natural numbers form an additive three if the sum of the first two of them is equal to the third. Find, reasonably, the maximum number of additive lists that can exist in a given set of 20 natural numbers.

We put 23 matches on a table and 2 players play a game: each, in turn, withdraws between 1 and 4 matches (included). Whoever takes the last wins. Is there a winning strategy for one of the two? A winning strategy is a way of playing that allows one of the two to win no matter how their opponent plays.

Let $\Omega$ and $\Gamma$ two secant circles. we note $A$ one of their intersections. Let $d$ any straight line passing through the point $A$. we note $P$ and $Q$ the respective intersections of the line $d$ with the circles $\Omega$ and $\Gamma$ different from $A$. Show that there is a point independent of the line $d$ chosen and which always belongs to the perpendicular bisector of the segment $[PQ]$.

Let $ABCD$ be a trapezoid such as $(AB)$ and $(CD)$ are parallel. It is assumed that $AD \lt CD$ and $ABCD$ is inscribed in a circle $\Gamma$. Let $P$ on $\Gamma$ such as $(DP)$ is parallel to $(AC)$. The tangent to $\Gamma$ in $D$ cuts $(AB)$ to $E$, and strings $[BP]$ and $[CD]$ intersect in $Q$. Show that $EQ = AC$.

Find all $f$ functions of $\mathbb{R}$ in $\mathbb{R}$ such as for all real $x, y$, $$
f(x f(x)+f(y))=f(x)^{2}+y
$$

Determine all triplets of strictly positive $(a, b, n)$ integers checking: $$
a! + b! = 2^{n}
$$

Characterize integers $n \geq 2$ such as for any integer $a$ on AIT $a^{n+1} \equiv a \pmod{n}$.

Let $ABC$ an isosceles triangle in $A$, $D$ the foot of the height from $A$ and $M$ an inner point at $ADC$ such as $\widehat{AMB}$ is obtuse and $\widehat{DBM} + \widehat{DAM} = \widehat{MCB}$. the lines $(CM)$ and $(AD)$ intersect in $P$, and the lines $(BM)$ and $(AD)$ intersect in $Q$. Let $S$ a point of $[AB]$ and $R$ a point of $[AM)$ unsure $[AM]$ such as $\widehat{SQB} = \widehat{DPC}$ and $\widehat{MRQ} = 2\widehat{QAM}$. To show that $QRS$ is isosceles.

Let $ABC$ a triangle and $\omega$ his circle inscribed. we note $P, Q, R$ the contact points of $\omega$ with $(BC), (CA)$ and $(AB)$. A circle passing through $B$ and $C$ is tangent in $X$ à $\omega$, a circle passing through $C$ and $A$ is tangent in $Y$ à $\omega$ and a circle passing through $A$ and $B$ is tangent in $Z$ à $\omega$. Show that the lines $(PX), (QY)$ and $(RZ)$ are concurrent.

$K$ means the maximum value of $$
\left|x_{1}-x_{2}\right| \cdot\left|x_{1}-x_{3}\right| \cdot\left|x_{1}-x_{4}\right| \cdot\left|x_{2}-x_{3}\right| \cdot\left|x_{2}-x_{4}\right| \cdot\left|x_{3}-x_{4}\right|
$$ or $x_{1}, x_{2}, x_{3}, x_{4} \in [0 ; 1]$.
a. Prove that $\frac{4}{243} \lt K \lt \frac{1}{27}$.
b. Determine $K$.