IMOmath

Algebra

1. (5 p.)
The set \( A \) consists of \( m \) consecutive integers with sum \( 2m \). The set \( B \) consists of \( 2m \) consecutive integers with sum \( m \). The difference between the largest elements of \( A \) and \( B \) is 99. Find \( m \).

2. (13 p.)
Real numbers \( x,y,z \) are real numbers greater than 1 and \( w \) is a positive real number. If \( \log_xw=24 \), \( \log_yw=40 \) and \( \log_{xyz}w=12 \), find \( \log_zw \).

3. (54 p.)
Let \( a_1,a_2,... \) be a sequence defined by \( a_1=1 \) and \[ a_{n+1}=\sqrt {a_n^2-2a_n+3}+1\] for \( n \ge 1 \). Find \( a_{513} \).

4. (18 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).\] Determine \( |a| \)

5. (8 p.)
Find the product of the real roots of the equation \( x^2+18x+30=2\sqrt{x^2+18x+45} \) (the answer is an integer).





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