Find all pairs $(a,b,c)$ of positive integers such that $$\frac{a}{2^a}=\frac{b}{2^b}+\frac{c}{2^c}$$

On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$

Let $ABCD$ be an inscribed quadrilateral of some circle $\omega$ with $AC\ \bot \ BD$. Define $E$ to be the intersection of segments $AC$ and $BD$. Let $F$ be some point on segment $AD$ and define $P$ to be the intersection point of half-line $FE$ and $\omega$. Let $Q$ be a point on segment $PE$ such that $PQ\cdot PF = PE^2$. Let $R$ be a point on $BC$ such that $QR\ \bot \ AD$. Prove that $PR=QR$.

Let $\Delta ABC$ be an acute-angled triangle with $AB < AC$, $H$ be a point on $BC$ such that $AH\ \bot BC$ and $G$ be the centroid of $\Delta ABC$. Let $P,Q$ be the point of tangency of the inscribed circle of $\Delta ABC$ with $AB,AC$, correspondingly. Define $M,N$ to be the midpoint of $PB,QC$, correspondingly. Let $D,E$ be points on the inscribed circle of $\Delta ABC$ such that $\angle BDH + \angle ABC = 180^{\circ}$, $\angle CEH + \angle ACB = 180^{\circ}$. Prove that lines $MD,NE,HG$ share a common point.

Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of $$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$

Let $p$ be an odd prime. Suppose that positive integers $a,b,m,r$ satisfy $p\nmid ab$ and $ab > m^2$. Prove that there exists at most one pair of coprime positive integers $(x,y)$ such that $ax^2+by^2=mp^r$.

Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define $$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$. Prove that $\lambda \geq 44$ and provide an example in which the equality holds.