Hint : Forget about the trapezoid. Think about how the second circle looks like in $\triangle DBC$.Apply POP .
Search found 110 matches
- Mon Nov 14, 2016 7:40 am
- Forum: Geometry
- Topic: Prove parallel
- Replies: 2
- Views: 3159
- Thu Nov 10, 2016 1:09 pm
- Forum: Algebra
- Topic: Euclidean algorithm with reals ?
- Replies: 0
- Views: 1922
Euclidean algorithm with reals ?
$4.$ Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $ ...
- Thu Nov 10, 2016 1:08 pm
- Forum: Number Theory
- Topic: $N$ terms of an AP are divisible by product of first $N$
- Replies: 0
- Views: 1932
$N$ terms of an AP are divisible by product of first $N$
Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$:
$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
- Thu Nov 10, 2016 1:06 pm
- Forum: Combinatorics
- Topic: USAMO 2005/4 (Stable table)
- Replies: 1
- Views: 2596
- Thu Nov 10, 2016 1:03 pm
- Forum: Geometry
- Topic: Circle passing through the centers of two others'
- Replies: 1
- Views: 2376
Circle passing through the centers of two others'
$1.$ Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX...
- Thu Nov 10, 2016 12:59 pm
- Forum: Number Theory
- Topic: Functional divisibility
- Replies: 2
- Views: 5224
Re: Functional divisibility
Let $P(m,n)$ be the statement $f(m)+f(n) | (m+n)^k$ . We prove that the only solution is $f(n)=n$. Note that this indeed works. Claim 0: For two polynomials $P$ and $Q$ such that $P \nmid Q$, there are only finitely many integers $n$ such that $P(n)|Q(n)$ . Proof: Apply division algorithom on $P$ a...
- Tue Oct 11, 2016 4:04 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies: 10
- Views: 13398
Re: Combi Solution Writing Threadie
The problem is due to Rahul Saha, from an unknown source. 2016 nonnegative integers are written on a board. In each step, you can erase two of the numbers and replace them with their sum and their difference. For any given set of 2016 nonnegative integers on the blackboard, is it possible, in a fin...
- Tue Oct 11, 2016 4:01 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies: 10
- Views: 13398
Re: Combi Solution Writing Threadie
IMOSL 2007 C1 Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 + n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n; \] \[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i +...
- Tue Oct 11, 2016 3:56 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies: 10
- Views: 13398
Re: Combi Solution Writing Threadie
There is a handout by Evan Chen used at MOP this year to teach students about solution writing. Should be relevant to this thread. You can find it in the first section titled "English" in the following link. http://www.mit.edu/~evanchen/olympiad.html I guess the big takeaways from the note that are ...
- Sun Aug 28, 2016 12:56 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115490
Re: IMO Marathon
I will apply induction on the length of the string, and leave some of the finer details to the reader. Let's just do one case. Take $n=6$ (i.e. the string $123456$). Now let's randomly put some arrow signs inside, and try to apply the algorithm. The essence of this solution lies in knowing precisel...