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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 \( nn^{\prime} \)?
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 \( a^2+b^2c^2d^2=0 \) and \( a^2b^2c^2+d^2=\frac {56}{53}(bc+ad) \), 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 \( 100m+n \)
20052021
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