VTMO 2021 — Grade 7 — cup

Positive integers $k = 1 , k = 2 , k = 3$ sum to $2021$ and $| x = 1 - x = 2 \frac{|}{k} = 1 = | x = 2 - x = 3 \frac{|}{k} = 2 = | x = 3 - x = 1 \frac{|}{k} = 3$. Find $T = \left(x = 1 + x = 2 - 2 x = 3\right)^{2021}$.

Find all positive-integer pairs $\left(a , b\right) , 1 \lt a \le b$, such that $\left(a - 1\right) \left(b - 1\right)$ divides $ab - 1$.

$a$) Can $1 , \ldots , 100$ be arranged in a row so adjacent $a , b$ satisfy $| a - b | 2$ or $\frac{a}{b} = 2$ or $\frac{b}{a} = 2$? $b$) Is a circular arrangement possible?

Given six irrational numbers, prove that three $a , b , c$ can be chosen with all of $a + b , b + c , c + a$ irrational.

$a$) Acute $ABC$ has median $AM , M$ on $BC$, and altitude $BE , E$ on $AC$. $K = AM \cap BE$ and $MAC = 30 ^{\circ}$. Prove $AC = \sqrt{3} BK$. $b$) Acute $ABC$ has $C = 45 ^{\circ}$ and $AB \lt AC$, altitude $AH , H$ on $BC$, and bisector $BD , D$ on $AC$. The perpendicular bisector of $BC$ meets $AC$ at $K$. $I = BD \cap AH$ and $KA = KD$. Prove $AIB$ is isosceles.