MYTS 2017 - Grade 7 - Round 2

Find all ordered quadruples of positive integers (a,b,c,d) satisfying both $ab=c+d$ and $a+b=cd$.

Five real numbers, not necessarily distinct, have sum 18. Every choice of three of them has nonnegative sum. Let ℓ be the smallest number. Find the minimum possible value of ℓ.

ABCD is a square. E lies on CD and F lies on DA. The areas of triangles BCE, EDF and ABF are respectively 21 cm², 44 cm² and 42 cm². Find the area of triangle BEF, in square centimetres.

2017 students stand in a circle numbered 1 to 2017 clockwise. Student 1 stays, student 2 leaves, the next stays, the next leaves, and so on around the shrinking circle until only one remains. What is the number of the surviving student?

The diagram is a pyramid of 21 squares. Mr Thuận fills each square with a natural number, with every entry above the bottom row equal to the sum of the two entries immediately below it. The filled diagram is an example. What is the smallest possible number of even entries in the pyramid? Explain why.

Triangle ABC is isosceles with AB=AC. D lies inside the triangle with AD=BC, ∠DAB=α and ∠DAC=β. Given α+3β=60°, express ∠DCA in terms of α and β.