18. Find the least positive integer \(π\) such that the equation \(β\frac{10^π}{π₯}β = 98\) has integer solution \(π₯. βπβ\) is the largest integer smaller than or equal to \(π\).
(A) 3
(B) 4
(C) 5
(D) 6
(E) None of the above
19. How many positive integers \(π < 100\) such that \(2(5^{6π})+ π(2^{3π+2}) β 1\) is divisible by \(7\) for any positive integer \(π\)?
(A) 12
(B) 14
(C) 18
(D) 19
(E) None of the above
20. π΄, π΅, πΆ and π· are four distinct points lying on the circumference of a circle such that chords π΄π΅ and πΆπ· are perpendicular at point πΈ. Given that πΈπ΄ = 4, πΈπ΅ = 2 and πΈπΆ = 6 , find the radius of the circle.
(A) \(\sqrt{221}\)
(B) \(15\)
(C) \(270\)
(D) \(18\)
(E) None of the above
21. You need to tile a \(10 Γ 1\) hallway with a supply of \(1 Γ 1\) red, \(2 Γ 1\) red tiles and \(2 Γ 1\) blue tiles. Find the number of ways you can tile the \(10 Γ 1\) hallway.
22. π₯, π¦ and π§ are real numbers such that
\(π₯ + π¦ + π§ = 7\)
\(π₯^2 + π¦^2 + π§^2 = 19\)
\(π₯^3 + π¦^3 + π§^3 = 64\)
Evaluate \(π₯^4 + π¦^4 + π§^4\).
23. In \(Ξπ΄π΅πΆ\) shown below, \(π΄π·, π΅πΈ\) and \(πΆπΉ\) intersect at \(π\) . SupposeΒ \(π΄π = π\), \(π΅π = π, πΆπ = π\) and \(π·π = πΈπ = πΉπ = π₯\). Given that \(π₯ = 3\) and \(π + π + π = 20\) , find \(πππ\).
24. Positive integers π, π and π are randomly selected from the set \(\{1,2,3, β¦ ,2020\}\) with replacement. Find the probability that \(πππ + ππ + 2π\) is divisible by \(5\).
25. \(π΄π΅πΆπ·\) is a convex quadrilateral such that \(π΄πΆ\) intersects \(π΅π·\) at \(πΈ\) . \(π»\) is a point lying in the segment \(π·πΈ\) such that \(π΄π»\) is perpendicular to \(π·πΈ\). Suppose \(π΅πΈ = πΈπ·, πΆπΈ = 9, πΈπ» = 12, π΄π» = 32\) and \(β π΅πΆπ΄ = 90Β°\) . Evaluate the length of \(πΆπ·\).
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SEAMO PAPER F 2019 [PROBLEM And SOLUTION]