# Number Theory

 1. (26 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 $$100m+n$$

 2. (5 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 $$a + b + c$$.

 3. (44 p.) Let $$f$$ be a function defined along the rational numbers such that $$f(\tfrac mn)=\tfrac1n$$ for all relatively prime positive integers $$m$$ and $$n$$. The product of all rational numbers $$0< x< 1$$ such that $f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}$ can be written in the form $$\tfrac pq$$ for positive relatively prime integers $$p$$ and $$q$$. Find $$p+q$$.

 4. (8 p.) Find the sum of all positive integers of the form $$n = 2^a3^b$$ $$(a, b \geq 0)$$ such that $$n^6$$ does not divide $$6^n$$.

 5. (14 p.) If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

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