MathNet 2026 — Part 11

How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles?

Prove that
$$
A=\sqrt{4-2 \sqrt{3}}-\frac{\sqrt{3}+1}{\sqrt{3}-1}
$$
is an integer.

Let $\omega$ be a circle with diameter $A B$. A circle $\gamma$, whose center $C$ lies on $\omega$, is tangent to $A B$ at $D$ and cuts $\omega$ at $E$ and $F$. Prove that triangles $C E F$ and $D E F$ have the same area.

Let $ABC$ be an equilateral triangle, and let $P$ be a point on minor $\operatorname{arc} BC$ of the circumcircle of $ABC$. Prove that $PA = PB + PC$.

Let $D$ be the intersection of the two tangent lines of $(O)$ at $B$ and $C$. The circle passing through $A$ and tangent to $BC$ at $B$ intersects the median passing $A$ of the triangle $ABC$ at $G$. Lines $BG$, $CG$ intersect $CD$, $BD$ at $E, F$ respectively.


a) The line passing through the midpoints of $BE$ and $CF$ cuts $BF$, $CE$ at $M, N$ respectively. Prove that the points $A, D, M$ and $N$ lie on a same circle.


b) Lines $AD, AG$ intersect the circumcircle of $DBC, GBC$ at $H, K$ respectively. The perpendicular bisectors of $HK, HE$, and $HF$ cut $BC, CA$ and $AB$ at $R, P$ and $Q$ respectively. Prove that the points $R, P$ and $Q$ are collinear.

Let $ABC$ be a triangle and $N$ be a point that differs from $A$, $B$ and $C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. We define $B_c$, $C_b$, $A_c$ and $C_a$ similarly. Let $m_a$ be the line passing through $N$ and perpendicular to $B_cC_b$. Define $m_b$, $m_c$ similarly.


a) Assume that $N$ is the orthocenter of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through each bisector of angles $\angle BNC$, $\angle CNA$ and $\angle ANB$ are coincident.


b) Assume that $N$ is the nine-point center of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through the lines $BC$, $CA$ and $AB$ concur.

Some squares of a $n \times n$ table $(n\gt 2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers.

The answer is $3 n^{2}-5 n+2$.

In a scalene triangle $A B C$, $I$ is the incenter and $C N$ is the bisector of angle $C$. The line $C N$ meets the circumcircle of $A B C$ again at $M$. The line $\ell$ is parallel to $A B$ and touches the incircle of $A B C$. The point $R$ on $\ell$ is such that $C I \perp I R$. The circumcircle of $M N R$ meets the line $I R$ again at $S$. Prove that $A S = B S$.

Find all the functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(4x+3y) = f(3x+y) + f(x+2y)$ for all integers $x$ and $y$.