Problems And Answers PHIMO MOCK 2020 SENIOR SECONDARY

16. In \(ΔABC\) it is known that \(AB = 13, BC = 14\) and \(AC = 15\). Let \(D\) and \(E\) be the feet
if the altitudes from \(A\) and \(B\), respectively. Find the circumference of the circumcircle of \(ΔCDE\).

17. A \(6\; cm × 12\; cm × 22\; cm\) rectangular block of wood is painted blue and then cut into small cubes, each of which ahs a surface area of \(6\; cm²\). Find the number of small cubes that have blue paint on exactly two faces.

18. Given a regular hexagon \(ABCDEF\), compute the probability that a randomly chosen point inside the hexagon is inside triangle \(PQR\), where \(P\) is a midpoint of \(AB, Q\) is the midpoint of \(CD\), and \(R\) is the midpoint of \(EF\).

19. A regular dodecagon is inscribed in a circle of radius \(10\). Find its area.

20. Find the minimum value of \(xy+xz+yz\) given that \(x, y, z\) are real numbers that satisfies the equation \(x^2 + y^2 + z^2 = 1\).