examday

Community mocks

Euclid Sample Paper 2 · Q1

Paper 2Logic & proof

Let x,y>0x,y>0 and

x+y=1.x+y=1.

The following statements may be used to prove

1x+1y≥4.\frac1x+\frac1y\geq4.
P:(x−y)2≥0,Q:x2+y2≥2xy,R:(x+y)2≥4xy,S:1≥4xy,T:1xy≥4,U:1x+1y=x+yxy=1xy.\begin{aligned} P:&\quad (x-y)^2\geq0,\\ Q:&\quad x^2+y^2\geq2xy,\\ R:&\quad (x+y)^2\geq4xy,\\ S:&\quad 1\geq4xy,\\ T:&\quad \frac1{xy}\geq4,\\ U:&\quad \frac1x+\frac1y=\frac{x+y}{xy}=\frac1{xy}. \end{aligned}

Which ordering gives a valid proof?

AP,Q,R,S,T,UP,Q,R,S,T,U
BP,R,Q,S,U,TP,R,Q,S,U,T
CQ,P,S,R,T,UQ,P,S,R,T,U
DR,P,Q,T,S,UR,P,Q,T,S,U
EU,T,S,R,Q,PU,T,S,R,Q,P

Read the questions freely. Sign in to check your answer, see the worked solution and save your progress.

Sign in to check your answerPractise the whole paper →