MathDB

2

Part of 2008 May Olympiad

Problems(2)

center of a rectangle lies in the segment connecting the projections

Source: May Olympiad (Olimpiada de Mayo) 2008 L2

9/11/2018
Let ABCDABCD be a rectangle and PP be a point on the sideAD AD such that ∠BPC=90o\angle BPC = 90^o. The perpendicular from AA on BPBP cuts BPBP at MM and the perpendicular from DD on CPCP cuts CPCP in NN. Show that the center of the rectangle lies in the MNMN segment.
geometryrectangleprojections
grades of exams in olympic school

Source: XIV May Olympiad (Olimpiada de Mayo) 2008 L1 P2

9/22/2022
In the Olympic school the exams are graded with whole numbers, the lowest possible grade is 00, and the highest is 1010. In the arithmetic class the teacher takes two exams. This year he has 1515 students. When one of his students gets less than 33 on the first exam and more than 77 on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the 3030 grades and obtained 88. What is the largest number of students who passed this class could have had?
combinatorics