IMOmath

Number Theory

1. (4 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 \( n \).

2. (47 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

3. (23 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

4. (14 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 \( a + b + c \).

5. (9 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?





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