IMOmath

Number Theory

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

2. (14 p.)
Determine the number of positive integers with exactly three proper divisors each of which is less than 50. (1 is a proper divisor of every integer greater than 1)

3. (2 p.)
\( n \) is an integer between 100 and 999 inclusive, and \( n^{\prime} \) is the integer formed by reversing the digits of \( n \). How many possible values are for \( |n-n^{\prime}| \)?

4. (35 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. (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 \( 100m+n \)





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