IMOmath

Number Theory

1. (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 \( 6^n \).

2. (21 p.)
Let \( \tau (n) \) denote the number of positive divisors of \( n \), including 1 and \( n \). Define \( S(n) \) by \( S(n)=\tau(1)+ \tau(2) + \dots + \tau(n) \). Let \( a \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) odd, and let \( b \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) even. Find \( |a-b| \).

3. (17 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

4. (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 \( 100m+n \)

5. (26 p.)
It is given that \( 181^2 \) can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.





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