# 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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