# Warming up TMO-Set8

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Q1. Several pairs of positive integers (m, n) satisfy the condition 19m + 90 + 8n = 1998 . Of these, ( 100, 1 ) is the pair with the smallest value for n . Find the pair with the smallest value for m .

Q2. Determine the unique pair of real numbers (x, y) that satisfy the equation

(4x^{2}+6x+4)(4y^{2}–12y+25)=28.

Q3. Determine the leftmost three digits of the number $1^{1}+2^{2} +3^{3} +...+999^{999} +1000^{1000}$.

Q4. There are infinitely many ordered pairs (m, n) of positive integers for which the sum

m + (m + 1) + (m + 2) + … + (n – 1) + n

is equal to the product mn . The four pairs with the smallest values of m are (1, 1), (3, 6), (15, 35), and (85, 204). Find three more (m, n) pairs.