# Number Theory

 1. (9 p.) Let $$a$$, $$b$$, $$c$$ be positive integers forming an increasing geometric sequence such that $$b-a$$ is a square. If $$\log_6a + \log_6b + \log_6c = 6$$, find $$a + b + c$$.

 2. (3 p.) The square $$\begin{array}{|c|c|c|} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array}$$ is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find $$x$$.

 3. (25 p.) Let $$0 < a < b < c < d$$ be integers such that $$a$$, $$b$$, $$c$$ is an arithmetic progression, $$b$$, $$c$$, $$d$$ is a geometric progression, and $$d - a = 30$$. Find $$a + b + c + d$$.

 4. (15 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 5. (46 p.) Let $$a,b,c$$ and $$d$$ be positive real numbers such that $$a^2+b^2-c^2-d^2=0$$ and $$a^2-b^2-c^2+d^2=\frac {56}{53}(bc+ad)$$, Let $$M$$ be the maximum possible value of $$\frac {ab+cd}{bc+ad}$$ ,If $$M$$ can be expressed as $$\frac {m}{n}$$,$$(m,n)=1$$ then find $$100m+n$$

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