IMOmath

Number Theory

1. (30 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. (30 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.

3. (16 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.

4. (2 p.)
Let \( n \) be the largest positive integer for which there exists a positive integer \( k \) such that \[ k\cdot n! = \frac{(((3!)!)!}{3!}.\] Determine \( n \).

5. (20 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).





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