IMOmath

Number Theory

1. (35 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. (11 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

3. (28 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 \( |a-b| \).

4. (4 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

5. (19 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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