CMI Entrance Exam 2019 #4

-
Let $ABCD$ be a parallelogram. Let $O$ be a point in its interior such that $\angle AOB+\angle DOC=180^\circ$. Show that $\angle ODC = \angle OBC$. 



We translate $O$ to $O'$ and $D, C$ to $A, B$, respectively. Thus $AO'OD, O'BCO$ are parallelograms. Now $\angle AO'B = \angle DOC$ because translations preserve angles, and it's given that $\angle DOC + \angle AOB = 180^{\circ}$, so $AO'BO$ is cyclic. This means $\angle O'AB = \angle O'OB = \angle OBC$ where the last equality results from parallelogram $O'OCB.$ Thus we are done. $\square$

Comments

Post a Comment

Popular posts from this blog

1995 IMO #2

-
 Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc = 1$. Prove that \[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}. \]
Read more

2014 IMO SL #C2

-
  Problem : We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
Read more

2015 IMO SL #A1

-
 Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Read more