IMOmath

Number Theory

1. (20 p.)
If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

2. (12 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

3. (20 p.)
Let \( 0 < a < b < c < d \) be integers such that \( a \), \( b \), \( c \) is an arithmetic progression, \( b \), \( c \), \( d \) is a geometric progression, and \( d - a = 30 \). Find \( a + b + c + d \).

4. (7 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. (38 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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