IMOmath

Number Theory

1. (3 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

2. (25 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 \( 100m+n \)

3. (1 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 \( n \).

4. (25 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.

5. (43 p.)
Suppose \( m \) and \( n \) are positive integers with \( m> 1 \) such that the domain of the function \( f(x) = \text{arcsin}(\log_{m}(nx)) \) is a closed interval of length \( \frac{1}{2013} \). Let \( S \) be the smallest possible value of \( m+n \). Find the remainder when \( S \) is divided by \( 1000 \).





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