MathNet 2026 — Part 08

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

Genji the ninja is to jump along the real axis, starting at the point $0$. In doing so, he alternates between the following two types of jumps, his first jump being type 1.

**Type 1:** Genji jumps from the current position $x$ to a point $y$ in the set
$$
\{x, x+1, x+2, x+3, x+4, x+5, x+6\}.
$$

**Type 2:** Genji jumps from the current position $y$ to $(\sqrt{2}-3)y$.

Let $a$ and $b$ be integers. Prove that there exists a finite sequence of jumps that allows Genji to land on the point $a+b\sqrt{2}$.

Let $ABCD$ be a convex quadrilateral with $AC \perp BD$. Prove that there exist points $P, Q, R, S$ on the sides $AB, BC, CD, DA$, respectively, such that $PR \perp QS$ and the area of quadrilateral $PQRS$ is exactly half that of $ABCD$.

Let $n \ge 2$ be an integer. Killer is a game played by a dealer and $n$ players. The game begins with the dealer designating one of the $n$ players a killer and keeping this information a secret. Every player knows that the killer exists among the $n$ players. The dealer can make as many public announcements as he wishes. Then, he secretly gives each of the $n$ players a (possibly different) name of one of the $n$ players. This game has the property that:
(i) Alone, each player (killer included) does not know who the killer is. Each player also cannot tell with certainty who is not the killer.
(ii) If any two of the $n$ players exchange information, they can determine the killer.
For example, if there are a dealer and 2 players, the dealer can announce that he will give the same name to both players if the first player is the killer, and give different names to the players if the second player is the killer.


a. Prove that Killer can be played with a dealer and 5 players.


b. Determine whether Killer can be played with a dealer and 4 players.

Let $P$ be a point inside $\triangle ABC$. Let $A_1, B_1, C_1$ be points in the interiors of the segments $PA, PB, PC$, respectively. Let $\overline{BC_1} \cap \overline{CB_1} = \{A_2\}$, $\overline{CA_1} \cap \overline{AC_1} = \{B_2\}$, and $\overline{AB_1} \cap \overline{BA_1} = \{C_2\}$. Let $U$ be the intersection of the lines $A_1B_1$ and $A_2B_2$, and $V$ be the intersection of the lines $A_1C_1$ and $A_2C_2$. Show that the lines $UC_2$, $VB_2$, and $AP$ are concurrent.

Find the largest positive integer $n$ which is divisible by all positive integers whose cube is not greater than $n$.

Let $S_r(n) = 1^r + 2^r + \dots + n^r$ where $r$ is a rational number and $n$ is a positive integer. Find all triples $(a, b, c)$ where $a$ and $b$ are positive rational numbers and $c$ is a positive integer for which there exist infinitely many positive integers $n$ satisfying $S_a(n) = (S_b(n))^c$.

Show that
$$
a + b + c + \sqrt{3} \geq 8abc \left( \frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} + \frac{1}{c^2 + 1} \right)
$$
for all positive real numbers $a, b, c$ satisfying $ab + bc + ca \leq 1$.

Alice and Bob play a game on a $1 \times m$ board using $2012$ cards numbered from $1$ through $2012$. At each step, Alice chooses a card and Bob places it on an empty square of the board. Bob wins the game when numbers on the cards on the board are in an increasing order after $k$ steps where $1 \leq k \leq 2012$, otherwise Alice wins. Find all pairs $(k, m)$ for which Bob can guarantee to win the game.

In an acute triangle $ABC$, let $D$ be point on the side $[BC]$ different than the vertices. Let $M_1, M_2, M_3, M_4, M_5$ be the midpoints of the line segments $[AD], [AB], [AC], [BD], [CD]$, respectively; $O_1, O_2, O_3, O_4$ be the circumcenters of the triangles $ABD, ACD, M_1M_2M_4, M_1M_3M_5$, respectively; $S$ and $T$ be the midpoints of the line segments $AO_1$ and $AO_2$, respectively. Prove that $SO_3O_4T$ is an isosceles trapezoid.

Show that
$$
(x^2 + y^2)^3 \geq 32(x^3 + y^3)(xy - x - y)
$$
for all real numbers $x, y$ satisfying $x + y \geq 0$.

Let $A$, $B$, $C$, $A'$, $B'$, $C'$ be distinct points on the plane satisfying $ABC \cong A'B'C'$ and the point $G$ be the centroid of the triangle $ABC$. If the circle of center $A'$ passing through $G$ and the circle of diameter $[AA']$ intersect at point $A_1$, the circle of center $B'$ passing through $G$ and the circle of diameter $[BB']$ intersect at point $B_1$, the circle of center $C'$ passing through $G$ and the circle of diameter $[CC']$ intersect at point $C_1$, show that
$$
AA_1^2 + BB_1^2 + CC_1^2 \le AB^2 + BC^2 + CA^2.
$$

Determine all positive integers $m$, $n$ and prime numbers $p$ such that
$$
\frac{5^m + 2^n p}{5^m - 2^n p}
$$
is a square of an integer.

Find all positive integers $n$ for which an integer that can be written as sum of squares of $n$ integers with each of them is divisible by $n$, can also be expressed as sum of squares of $n$ integers with none of them is divisible by $n$.

Find the number of partitions of $\{1, 2, \dots, 2012\}$ into two sets such that none of the sets contains two distinct elements whose sum is a power of $2$.

Let $Z^+$ denote the set of all positive integers and $P$ denote the set of all prime numbers. For subsets $A$ and $S$ of $Z^+$, $A$ is called $S$-proper if there exists a positive integer $N$ such that for all $a \in A$ and integer $b$ with $0 \le b \lt a$ there exist not necessarily distinct elements $s_1, s_2, \dots, s_n$ of $S$ satisfying the conditions $b \equiv s_1 + s_2 + \dots + s_n \pmod a$ and $1 \le n \le N$. Find a subset $S$ of $Z^+$ for which $P$ is $S$-proper but $Z^+$ is not.

Let $a$, $b$, $c$ be the lengths of the sides of a triangle; $r_a$, $r_b$, $r_c$ be the corresponding exradii, respectively, and $r$ be the inradius. Prove that
$$
\frac{a+b+c}{2\sqrt{a^2+b^2+c^2}} \leq \frac{\sqrt{r_a^2+r_b^2+r_c^2}}{r_a+r_b+r_c-3r}
$$

Let $A = \{1, 2, \dots, 2012\}$, $B = \{1, 2, \dots, 19\}$ and $S$ be the set of all subsets of $A$. Determine the number of functions $f : S \to B$ satisfying the condition $f(A_1 \cap A_2) = \min\{f(A_1), f(A_2)\}$ for all $A_1, A_2 \in S$.

The incircle of a triangle $ABC$ touches the sides $[BC], [CA], [AB]$ at points $D, E, F$, respectively. The circle passing through point $A$ and touches the line $BC$ at $D$ intersects the line segments $[BF]$ and $[CE]$ at the points $K$ and $L$, respectively. The line passing through $E$ and parallel to $DL$ and the line passing through $F$ and parallel to $DK$ intersect at the point $P$. Let $R_1, R_2, R_3, R_4$ denote the circumradius of the triangles $AFD, AED, FPD, EPD$, respectively. Prove that $R_1R_4 = R_2R_3$.

Find the greatest constant $M$ such that
$$
a^2 + b^2 + c^2 + 3abc \geq M(ab + bc + ca)
$$
for all nonnegative real numbers $a, b, c$ satisfying $a + b + c = 4$.