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Number Theory
1.
(26 p.)
Let \( a,b,c \) and \( d \) be positive real numbers such that \( a^2+b^2c^2d^2=0 \) and \( a^2b^2c^2+d^2=\frac {56}{53}(bc+ad) \), 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 \( ba \) 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 \[ f\left(\dfrac{xf(x)}{1f(x)}\right)=f(x)+\dfrac9{52}\] 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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