Problems And Answers PHIMO MOCK 2020 SENIOR SECONDARY

6. How many triples \((a, b, c)\) are solutions of the system of two equations

\(a + b = 2, ab – c^2 = 1\)?

A. 0
B. 1
C. 2
D. 3
E. infinitely many

7. Suppose \(f(x)\) is a function that satisfies the equation \(5f(\frac{1}{x})+ f(x) – x = 3\) for all non-zero real numbers \(x\). Determine the value of \(f(4)\).

A. \(\frac{37}{96}\)
B. \(\frac{27}{53}\)
C. \(21\)
D. \(12\)
E. \(\frac{15}{4}\)

8. Let \(a, b, c, d\) be positive integers such that \(\log_a b =\frac{3}{2}\) and \(\log_c d =\frac{5}{4}\). If it is known that
\(a – c = 9\), determine the value of \(b – d\).

A. 47
B. 87
C. 93
D. 105
E. 64

9. Find the number of ordered pairs \((x, y)\) where \(x\) is an integer and \(y\) is a perfect square that satisfies the equation \(y + 4907 = (x – 90)^2\).

A. 0
B. 2
C. 4
D. 6
E. 8

10. Find the exact value of \(\sin^4 \frac{π}{8} + \sin^4 \frac{3π}{8} + \sin^4 \frac{5π}{8} + \sin^4 \frac{7π}{8}\).

A. 1
B. 1.5
C. 2
D. 2.5
E. 3.5